Polarized mixed Hodge structures: on irrationality of threefolds ...

Polarized Mixed Hodge Structures: on Irrationality of Threefo!ds Via Degeneration (*).

FA]~IO ]3ARDELLr (Firenze) (**)

Sunto. - l~ questo lavoro si studia una degenerazione di varieth lisee e proiettive di dimensione ire e la degeTberazione delle eorrispondenti strutture di Hedge polarizzate. .~a polarizzazione sulla struttura di Hedge ~ista <~ limite ~) viene poi applieata allo studio dell'irrazionalit& della generiea varieth della ]amiglia.

O. - I n t r o d u c t i o n .

The main theme of this paper is the applicat ion of the polarized mixed Hedge s t ruc ture on the asympto t i c cohomology of a one-variable degeneration of threefolds to problems of i r ra t ional i ty of the generic smooth degenerat ing threefold.

Essent ial ly we consider an algebraic fami ly of threefolds ~": ~ -> ~ , with F a smooth curve a n d

i) for a point P o e F z"-l(Po) = Xo is a reduced divisor with normal crossings,

if) Vte F \ { P 0 } , ~"-l(t) = Xt is a smooth project ive threefold with x~(Xt) = 0, H*(Xt , 0 x ) = 0 V i > I , H a ( X t , Z) torsion-free.

In section 1 we s tudy the Hedge theory of the fami ly and we introduce (following [Zu]) i ts jaeobian bundle p: J - ~ F .

We show t h a t J (Xo) -= P-~(Po) = {the generalized jaeobian of Xo} can be defined in te rms of the mixed Hedge s t ruc ture on the cohemolcgy of Xo only and does not depend on the nea rby fibres X, .

We then res t r ic t our a t t en t ion to a par t icular divisor with normal crossings Xo i n section 2, namely we allow for Xo at most three components {X~}~=~,~, a (and so at most a unique curve of tr iple points on Xo) and we assume tha t Vi, j, i # j, R~(x~ n x~, z ) = o.

1%r this special Xo we prove tha t its generalized jacobian J(Xo) is an extension 3

of the (abelian) va r i e ty @ J ( X ~ ) by a torus ~. i=1

We come to the point of polarizing the relevant mixed Hedge structures occurring in the one-variable degenerat ion in section 3.

(*) Entrata in Redazione i l 22 lugl io 1983. (**) Partially supported by a C.N.R. fellowship at the University of Georgia, Athens, USA.

1 9 - .Annal l d i M a l e m a t i c a

288 FABIO :BARD:ELLI: Polarized mixed Hodge strgctures, etc.

Yt e ~\{Po}, H3(Xt) is (( polarized ~> by the usual cup-product pairing, which we will denote by O(t). We see that the family {O(t)} specializes to the cup-product

3

on Hs(Xo, Z) and, as a consequence, to the natural ((polarizations> on ~J(X~) . i = 1

The next step is to polarize the ]:lodge structure corresponding to ~. The geometric meaning of ~ is the following: Hx(~, Q) is the Q-vector space of (~ transverse cycles ~> in Xo. H~(% Q) is polarized by a bilinear symmetric form FZo which is defined by the following rule: for two transverse 3-cycles ~ , i = ~, 2, in Xo consider s 3-eyles 5~, i = 1, 2, in Xt (for t e ~ near Po) with ~ specializing to ~ , then

=

where N----log i ' and T: H3(X~)--->Ha(Xt) is the local Picard-Lefschetz transformation around Poe ~ . Thus ~Xo is defined in terms of O(t) and of the local monodromy of our family. Section 3 is mainly devoted to the proof that ~xo is well defined and tha t there is a subspace V c_ H1(% Q) (roughly V is constituted by those transverse cycles missing the triple point locus of X0) on which ~xo is intrinsic to Xo only and it is in fact determined by the intersection pairings on the various H~(Xg n Xg, Z), Vi, j, i V= j. We remark that YJx. is essentially the polarization on the graded quotient Gr4 of the asymptotic eohomology of our degenerating family of threefolds (see [Gr], [Sol, [G-S] and [C12]), but we have preferred to (~read)> it in homology because it seems to us to be geometrically more intuitive and relevant for the applications. In section 4 we appiy Carlson's theory [Ca 1] and [Ca 2] to compute the one-motif functorially defined by the extension of mixed ttodge strttc- tures dual to the extension naturally associated to the exact sequence of complex

Lie groups 0 -~ ~ -~ J(Xo) ~ ~ J(Xg) --> O.

We then investigate the behaviour of the polarized jacobian bundle p: J - - ~ under the assumption that each X~ for t e /~\{Po) is rational. This is taken up in section 5 where we prove that under this assumption the polarized jucobian bundle p: J - ~ is essentially (over a suitable Zariski open subset Z c t ~, Z~Po) the pdlarized jacobian bundle of ~ degenerating family of curves ~: C - > Z with C smooth, ~-~(Po) semistable and ~-~(t) smooth projective for t e Z\{Po), (more pre- cisely JIz is a fibred product over Z of the polarized jacobian bundles of a finite number of such families of curves). In particular J(Xo) is the polarized generalized jacobian of a semistablc curve Co, and we prove that ~Zo coincides in this case with the natural analogous polarization on the space of transverse l-cycles on Co.

In the applications, given in section 6, the irrationality os a generic threefold of a given family is proved by showing that ~xo is not the natural polarization on the space of transverse l-cycles of any semistable curve. This is done by a direct comparison using the intrinsic characterization of ~Xo on V and the one-motif computed in section 4 (see argument (6.2.1.1.) in section 6). The families treated in section 6 include several classical varieties as cubic and quartie threefolds, c o r n -

FABIO BARD:ELLI: Polarized mixed Hedge struetures~ etc. 289

plete intersections of a cubic and a quadrie in P~, complete intersections of three quadrics in P~, smooth quartic double solids. The irrationality of such varieties is of course well known and treated in ([C-G], [I-M], []3el).

We also prove the irrationality of the generic hypersurface of bidegree (p, 3), p > 2 , in PI• for which I do not know of any conic bundle structure nor of any degeneration to a conic bundle.

This shows that the invsrian.t ~xo can be used to detect generic irrationality for threefolds which are not necessarily conic bundles or which do not admit a degeneration to a conic bundle. The main reason which led me to study the inwr ian t ~Xo has been the wish to understand the irrationality of threefolds from a viewpoint which might possibly be generalized to higher dimension. In order to make this remark more precise I will insert here some considerations concerning threefolds. After the first examples of unirational threcfolds which are not rational (given in [A-M], [C-G] and [I-M]) the first systematic treatment of irrationality of threefolds, which are conic bundles, has been given in [Be] using the link between Mumford's theory of Prym varieties, intermediate jacobians and the geometry of conic bundles, link sketched (for the cubic threefold case) in appendix C of [C-G]. The first proof of the irrationality of some given family using degeneration techniques appears also in ([]3c] pug. 360-61). As far as I know the only progress in this direction (since Beauvillc's paper) has been Collino's <~ Cheap proof of the irrationality of most cubic threefolds ,> (see [Co]). To see the relationships between the various invariants used in the different proofs by degeneration arguments let A be a neighborhood of Po in F biholomorphic to the unit disk in the complex s-plane, with Po corresponding to 0. As in ([Gr], Proposition (13.3), pug. 266) we can write the period matrix of our degenerating family of threcfolds (restricted to A) in the form

( Id~, 0 Zl,~(s) + A-log s/2zi , Z1,2(8)~ |

0 , I d . _ ~ ,Z~,~(s) , Z~,~(s)]

where the entries of the matrices Z~,j(s) are holomorphie functions of s and A is a m x m matrix with rational entries. Then Beauville's argument in [lee. cit.] uses essentially the period matrix (Idn_~, Z2,2(0)) i.e. the abelian part of a generalized jacobian (which he proves not to be the j acobian of a curve by Mumford's theory of Pryln varieties); Oollino's argument in [Col uses the interpretation (for m-~ 1) of the matrix ~ZI,~(0) in (Id, Z~,~(0)) i.e. the extension data. Thus the only part of the period matrix to be investigated yet was A which is just the matrix of --~vz. in a suitable basis. The main difference befween A and the other blocks of the period matrix is that while ZI.,(0) and Z,,~(0) are analytic objects whose geometric meaning (when nnderstandable) is in general closely related to the geometry of curves, A is defined topologically, thus it has not any particular link with the theory of curves. The fact that ~Zo is essentially a local topological invariant of the family ~ moti- vates the hope that the analogue of A (or ~xo) should work also to compare pola-

290 ~A]~IO ]~AI~DELLI: Polarized mixed Hodge structures, etc.

rized mixed Hodge structures in higher dimension. By using the invariant ~x. (or its analogue) we will hopefully treat the irrationality of some specific fourforld in a future paper.

I t is at this point a pleasure to acknowledge the decisive importance that the work and the results of H. CLEmEns have had for this paper, as well as to thank him for his encouragement to pursue this research. Thanks also go to C. McCI~oRy for helpful conversations concerning the topological questions in sections 2 and 3, to my friend 1~. S~I~H who communicated to me his insight in the geometry of stable curves and in period mappings during many stimulating discussions. Another thank goes to R. VARLEY for patiently and repeatedly trying to explain to me the theory of fine moduli spaces, the stable reduction theorem, and many topics in tiodge theory, and for his helpful comments and criticism during the preparation of the manuscript. My introduction to these problems, my interest for them as well as for all questions concerning algebraic cycles ~re the results of many beautiful conversation with my teacher, F. GI-I-ERARDELLI, tO whom I express special thanks.

1 . - The basic one-variable degeneration of threefolds: its Hodge theory and its jaeobian bundle.

Let ~ be a smooth projective fourfold, _F be a smooth projective curve, z: �9 be a surjeetive, proper (and of course projective) morphism. Let {/Do, 1)-1, ...,/)',} c P be the finite set of points on _F for which ~-i(/).,) is singular.

Let ~ = : E\.{/PI, . . . , /) .~} and ~* = : 2~\{/Po} = P\(/Po,/P~, ...,/P,}. Accordingly define

r r

= : ~ \ U ~ - ~ ( / ) . , ) a n d ~ * = : ~ \ ~ - ~ ( / ) ' o ) = X , \ U ~ - I ( / ) ' , ) . i = l i~O

There are obvious surjective, projective and proper maps

~": ~ -~2~, induced by ~.

Ler j: xv* r ~ be the inclusion map. We make the following assumptions:

1.1. Vt ~ P*, ~ - l ( t )= : X , is a reduced irreducible smooth projective three]old X , with

z~(Xt) = 0 , H~(X~, Ox,) = 0 Vi~ l and Hs(Xt , Z) torsion-]ree .

1.2. 7~-~(/)'o) =: Xo is a reduced divisor with normal crossings. We will write 27

Xo = U x~ i = l

where the X~ are the reduced irreducible smooth (projective) comlmnents oJ Xo.

I~A~BIO BAI~D~LLI: Polarized mixed Hedge structures, etc. 291

Let S i'~ -~: X~o ~ X~ be the reduced smooth sur]ace along which X~ and X~ intersect transversely; C~'J'k = Xio N X~ (~ X~ be the smooth reduced curve along which X~, X~ and X~o intersect transversely; p~,j,~,z be the analogous intersection point o/X~, X~, X~ and X~o. We do not make any assumption on the singular ]ibres

7~-l(Pi) l < i < r .

1.3.1. We will denote by H~. the i - th Hedge bundle of the family ~': E* -~ F* on F*, i.e. the holomorphie vector bundle whose sheaf of germs of holomorphic sec-

t ions is

R z . C | O~,.

1.3.2. H i will be the extension of H ~ to F as const ructed by SmE~]3~I~K *

(see [St 1] and [St 2]), i.e. H ' is the holomorphie vector bundle whose locally free sheaf of germs of holomorphie section is

where R denotes hypercohomology and 9~l~(log Xo) is the relat ive << log-complex ~ of ~": 3] --~F.

One has (see [St i] and [St 2])

H~[r. : H ~ and Vt e F* H[~} _~ H~(X,, C). *

1.4.1. Le t A be an open (classical) neighborhood of Pc in P biholomorphic to the uni t disk of the complex plane and with Pc corresponding to 0 in the biholomorphic map. We assume tha t P , 6 A for i ---- 1, ..., r. We will somet ime ident i fy

with the uni t disk and Pc with 0. As usual let A* ~ : A\{Po} , and y be a counterclockwise oriented loop in A* based ~t one point t e A* and such tha t the h o m o t o p y class of y generates z~(A*, t). Let

T(i, t; R): H'(X~, R) ~ Hi(X,, R)

where R = Z, Q, R or C, be the m o n o d ro m y t ransformat ion induced by y on the i - th eohomology group of Xt (with coefficients in R). For any 0 4 i < 6 the assig- nement of T(i, t; C): H~(Xt, C) --->H~(X~, C) for each t e A* determines an auto- morphism T,) of the bundle H~]~., so T ( i ) eA u t (Hi]d.) (see [St2]) .

1.4.2. By the monodromy theorem (in the form proved by LA~D~A~ in [La]) and our assumption that Xo is reduced (i.e. each component X~ appears in Xo with multiplicity I) it follows that Vi, 0<i<6, and Vie A* the monodromy transformation T(i, t; Q) is unipotent. By the bound on the index of unipotency given by

292 ~A~O BA~DELL~: Polarized mixed Hodge struetures~ etc.

S c n ~ ] ) (see [Sc], w 6.6.1) or by Ks (see [Ka]) and our Assumption 1.1, which says ~hat the t todge d iamond of X~ for t e ~* is of the form:

O O O H ~,~ 0

O H ~,~ H ~,~ O 0 H ~,~ 0

O 0 ~ 0 ~ 0

we conclude tha t Vi~ 0<i<6~ ir

T(i , t ; Q ) - - I d = O VtEA* and (T(3, t; Q ) - - I d ) ~ = o VteA*.

1.4.3. The bundle HitA extends the bundle H~lz.. Let D~ be the Gauss-Martin connection on Hi[z.. An explicit computat ion (see [St 2]) shows tha t the connection mat r ix of D~ has at worst simple poles at Po E A. Therefore by Deligne's theorem (see [De 1]~ pug. 53) T(il, viewed as a section of Horn (Hilz., Hi[z.) extends to a section T' , l e t t o m ( H ~ [ z ~ Hi[z) whose value at Po is

T~ = : (i),(p0} exp (--2zV'----lreS(po}D~)

See also ([St 2]). Therefore we have also (by 1.4.2)

~ t l 2 �9 ( ( i ) - - T d ) = 0 Vi, O ~ i ~ 6 , i ~ 3 and (T(~)--Id) = 0

1.4.4. We remark t h a t since JCt. carries the integrable Gauss-Manin connection /)t and has uuipotent monodromy, JC i is the canonical extension of Je ~ in the sense $

of D~,LIG~CE (see [De ]]).

1.4.5. We define N , ) e H o m (H~I~ , Hi[A) by

N(i ) = log T~i ~ = ~ (--1) ~+1. (T~i~--Id)J j~1 J

(a finite sum by (1.4.2)) and 2f i = N(i),(p~ = log T~i),(po}. The definition of 2f(i), Ni and (1.4.2) gives:

Vi~ 0 < i < 6 ~ i =/= 3 ~

N(~) = 0 and in part icular N i = 0

~Y~3~ = 0 and in part icular (N3)~ = 0 ,

! because N(3 ) -- T(8 ) - Id.

FABle BAR])ELL~: Polarized mixed Hedge struetures, etc. 293

1.5.1. Vp, 0 < ? holomorphic subbundle of H ~ and ~ H ~ be the (~filtration ~> holomorphic subbundle of H ~ as defined in [St 1] and [Zu]. Again one has

$

and Vt e/~*

lo i �9 i /~ H{t } -= 2' H (X~, C) --= {the usual I todge fi l tration on H~(X~, C)}.

1.5.2. We define

H ~ -- {the fibre over Pc e F o f the holomorphie vector bundle H ~} ;

and

9 ~H~ : E~H{e~}~ = ( the fibre over -Poe F of the homorphie vec tor bundle _F~H~}.

1.5.3. We denote by

{ ( 0 } c , , , , . . . . . /v H~ F ~ H _F H~C C = -----

the fil trution induced on the C-vector space H~ by the fi l tration subbundles {E~H ~} of H ~.

Let

W . H ~ = O} c WoH~ c W~H= c . . . c W~_~ = c W~ r

the weight f i l t rat ion on H ~ (as defined in [St 2] or equivalent ly in [Se]). * i t : j [ / The triple (H~, F Ho~, W. ~) determines a mixed Q-Hodges t rue tu re (see [De 2],

pag. 31, for a definition and [St 2], pag. 240-245) on H ~ . We remark tha t F ' H ~ is defined over C and tha t W . H ~ is defined over Q.

However H~ has not a canonically defined integral structure, bu t ra ther a (( nil- po ten t orbit ~) of such structures (see the remark in [St2] , pag. 248-49, or [Sc], pag. 255).

1.5.4. In the sequel we will denote by Gr~H ~~ the graded quot ients W~H~/W~_IH ~ of a given mixed t todge s t ructure (H k, F ' H ~, W.Hk).

PROPOSITI0~ 1.6.1. - The mixed Hedge structures ( H ~ , F ' H ~ , W.H~) have the [ollowing properties:

i) let i = 2l ,

WkH~ = 0

W i i

F~ = 0

0 < / < 3

/or k ~i,

/or 1-~ l < m < i ,

294 l~A]~IO :BARDELLI: Polarized mixed Hodge structures, etc.

in particular H~ ~_ C.

ii) H~ -~ 0

iii) W~H~ = 0

W , ~ L = HL 3 3

and H~ = 0 .

]or i ~ l ,

]or i > 4 ,

3 2 , 3 2 3

iv) The ineh~sion map W s H ~ ~---> H~ induces an isomorphism

WaHL ... HL ~2 W3 H~ ---> ~ and Ker 2(3 : W3 H~ .

PR00r. - i) N,: H~ -->H~ induces isomorphisms (see [ S t 2 ] or [Sc])

_~" r i Gr i kH~ Vk>O. i" G i+kHo~ --+ _

But N r (T , - - Id ) = 0 by (1.4.5), so 2~---- 0, V k > 0 , Vi, 0 < i < 6 , i ~ 2 l .

I t follows

H i i i i 0 = W o H ~ = W~H~ . . . . . W~_~ ~ c W~H~ = W~+~H~ = . . . . H ~ .

By our ~ssumption (1.1) it follows ~VmH~(X, C) = 0 for 1 + l < m < i ---- 21. There- fore we have ~lso ~ H ~ : 0 for m ~ l + l , i : 2 1 . H ~ C follows from

H~ C) ~ C , Vt e 2'*.

ii) (1.1) implies H~(X~, C) : O, \tt ~2"*, so we have also H~ = 0. Poincar6 5 dual i ty gives H (X~, C) 0, Vte /7 , and so H~ : 0.

iii) Vi > 0 there are isomorphisms ([S~ 2] or [Sc])

Since _~r~ = 0 for i > 2 , this gives

H~ = W e l l ~ -~ WsH ~ ---- W~H~ and W~H~ : WoH~ -~ (0}.

The assumption H3(X~, Ox,) = 0 = H~'~ implies F3H~ = O. / ~ H ~ = 0 implies t ha t Gr~H~ is a t todge structure of a pure type (2, 2~ and therefore

r 3 2 , 3 G 4 H ~ / F G~4H ~ - O .

I~ABIO BARDELLI: Polarized mixed Hodge struvtures, etc. 295

The isomorphism (of Hodge structures) 2~3: GrtH% --> W2H ~ which is of bidegree

( - -1 , - -1 ) implies tha t ~2W2H~ = O.

iv) The inclusion map WsIt~ ~->H~ induces a map

W~ H~ H~ ~: ~v~W~H~ ~ ~ ~ FHr

which is easily seen to be in jcc t ive and whose cokernel is

3 3 3 2 3

2 3 2 3 = 3 2 3 F H ~ / f W3H~ W 3 H ~ / f W~H~

so ~ is an isomorphism. F ina l ly

N.(W.HL) = WIHL r~ I m N . = 0 , so W~HL c K e r N 3 .

On the other hand Ns induces an isomorphism N~: Gr~H~ ---~ W~H~ and so i t is

clear t ha t Ker N 3 = WsH ~. Q.E.D. 1.7.1. Le t

2~'H~(Xo, C) = {0 c 2~H~(Xo, C) c

c_P~-~H~(Xo, C) c ... c F~II~(Xo, C) c_2'~ C) = H~(Xo, C)}

and

W.H~(Xo, R) = {0 ~ WoJf~(Xo, 1~) c

c_ W~H~(Xo, R) c ... c W~_~/:[~(Xo, R) c W,H~(Xo, i~) = ig*(Xo, R)}

(here/~ = Q, R or C), be respect ively the Hodge and weight fi l tration of H~(Xo, R) as defined in [G-S] and for which (H~(Xo, Z); ~'H~(Xo, C); W.H~(Xo, Q)) is a mixed I todge s t ructure .

In [G-S] is p roved t ha t the cohomology H*(Xo) as well as the mixed i todge s t ruc ture on H*(Xo)are computed in the following way: (here we make a slightly different use of indices): for any mult i index I ---- (io, ~1, ..., i~) II1 = p d- 1 let

x " = n Xo 1 n . . . n

and let

XE~1= I ~ X1 where I i is the disjoint u n io n .

Let E ~'~ d ~ ( X ~ ) , where ~ * ( X E~l) is the usual De-Rham complex of X E~l

296 FA;BIO ]3AI~I)EEZI: Polarized mixed Hodge structure% etc.

Define d: d q ( X [~]) --> dq+l(X ~]) to be d = ( - -1) ~ (the usual exter ior differentia- t ion), and

8: d~(X~J) -> ~q(XE~+IJ) by : Vp e d~(XE~1)

write p as p = ~ p~ wi th p~ e d q ( X I) then , IP=~+1

~-I-1

Then (E o , d, 6) is a double complex and the associated spectral sequence (Eq , dq)

has the following propert ies :

E * 1 -= Hd(Eo) and dl = d ,

E~ = Ro(H~(E.)) and d, = [6].

l~m~thermore one has:

i) Er = H*(E*'*) ~---H*(Xo, C)

where D = d @ 8 and E*'* is the associated single complex

ii) E2"~ E3 ~- "~ E~ ~ ~

The f i l t rat ion W, and /~" on Eo defined by:

w,F,. = @ E ~ ,o, ~,,Eo = | q~i ~,q~O

induce fil trations on E~ _~ E~ which are exact ly the weight and Hedge filtrations

recalled above on H*(Xo, C). So one has clearly

GrqH~+~(Xo, C) = E~ '~

and its tLodge decomposit ion is the homology of the Hodge decompositions in

the complex:

II II II E~-~, ~ > E~ '~ ~ ~+~,~

PROI'OSlTIOSI ].7.2. -- The mixed Hodge str~tctures

(R~(Xo, z ) , ~'H~(Xo, C), W./~'(Xo, Q))

FABIO BAt~DELLI: Polarized mixed Hodge struetures~ etc. 297

have the following properties:

i) Vi, 0 < i < 6 , i r

W~Ht(Xo, Q) = o

In particular all the sequences

.E~-I,q dl dl 1 ) E~ '=

of the E~-term of the spectral sequence of 1.7.1 are qr

ii) tt~ C)

HI(Xo, C)

~ ( X o , C)

iii) W~H3(Xo, Q) =

:g~//~(Xo, C) =

F~ w~HqXo, c ) =

P~ooP. - i) We consider t ia l ly [St 2])

- -~C,

~ 0 ~

= O Vk>4 ,

O V k < l ,

O,

O.

V k ~+l,q - - 1

exact at E~ 'q for all p>l (but

the Clemens-Schmid exact sequence (see [ci 2] or essen-

W~_gHs_~(Xo, C) = Ann Ws_iHs-~(Xo, C) = 0

therefore W~_IH~(Xo, C) = 0 and so WkH~(Xo~ C) = 0 V k H~(Xo, C) (r 1G> > H2 t t~

where all maps are morphisms of mixed t todge s t ructures of bidegrees

p(4, 4) ; ~(o,o); ~ ( - 1 , - - 1 ) .

(Here we consider the canonica l dual mixed t todge s t ruc ture on Hs_,(Xo)). For i # 3, iV~ = O, so a is surjective. Now a(W~s C)) = by the strictness

p roper ty of morphisms = W~_IH~ (~ Im a = Wi_IH~ = 0 b y 1.6.1 i) and ii). I t follows W~_~H~(Xo, C) _CKer a = i m p . Bu t now

p ( w , _ g H s _ , ( X o , c)) = w , _ ~ , ( x ~ c) n I m p .

298 FA~IO :B~DELLI: Polarized mixed Hedge structures, etc.

The second assert ion in i) follows f rom a closer look to t h e spectral sequence, the fu'st pa r t of i), the re la t ion

~'q-- GrqH~+q(Xo, C) and iii)

if) The first assert ion follows f rom No ---- 0; H ~ ~--- C, Hs(Xo, C) ~ 0 and the Clemens-Sehmid sequence. The second follows f rom H~ =- O, H~(Xo, C) -~ 0 and the Clemens-Schmid sequence. The t h i r d asser t ion follows f rom

2r

H~ c) = Q H~(x~, c) 4=1

and the way the Hedge fi l trat ion is induced.

iii) We have W~HS(Xo, C)-~H~(Ho, C), so

W_~HdXo, C) : Ann WsHs(Xo, C)-~ 0 .

Therefore

0 = p(W_eHs(Xo, C)) -~ W~H3(Xo, C) N I m p = W2H~(Xo, C) n Ker a ,

so the map a in the Clemens-Schmid sequence induces injective maps

W~H~(Xo, C) --> W,H~ V i < 2 .

Therefore W~H3(Xo, C ) ~ 0 for i < 1 by 1.6.1 iii) and ~W~H3(Xo, C ) : 0 because ~ W ~ H ~ z 0 and the induced maps are morphisms of Hedge struetttres.

We have to prove ~SH3(Xo, C) ~-- O. The mixed Hedge s t ruc ture on Hs(Xo, Z) has the following fil tration:

H~(Xo, C ) = W ~H~(Xo, C)-= W_3Y~(Xo, C) . . . .

. . . . W_~Hs(Xo, C) z W_oHo(xo, C) = (0}

(just compute the dual weight f i l tration of H (Xo, C) using i)).

Fu r the rmore

~-IH~(Xo, C) = ~v-l( Gr_sH~(Xo, C) ) = Q H ~,~ = 0

because in fact using the last s ta tement of if) it is possible to see tha t :

- -2 --3 QH, (xo, c) . ~(yo, C)= H~, (Xo, C) -~,-~

FABIO BARD]~LLI: _Polarized mixed Hodge structures, etc. 299

Bu t t hen using the m a p p : Hs(Xo, C) -+ H3(Xo, C) one has b y the s t r ic tness prop- e r ty of morph i sms :

0 : p(F-1Hs(Xo, C)) : F3H3(Xo, C) n I m p : FSH3(Xo, C) N Ker ~ .

I t follows t h a t a induces an inject ive m a p

6: F3HS(Xo, C) --> FSH%

and since 8 8 H ~ = 0, i t follows FSH3(Xo, C ) = 0. Q.E.D.

1.8.1. As in Zvc~:E~ (see [Zu]) we consider the na tu ra l m a p s i, q

�9 - 8 , - i T , z q H8 ? ,1~ z , Z -->" ~ ---> ~ H 8 .

DE~'INITI0~ 1.8.2. -- Fol lowing ZUCKE~ (see [Zu]), we define the jacobian bundle o/ the family z": ~ - + F by :

J - - F / ~--H8 q'i(j , Rs z , Z) .

We have the following facts (see [Zu]):

1.8.3. J is a complex ana ly t i c space wi th a holomorphic projec t ion m a p p : J - > t v .

].8.4. Vt e F* p-~(t)=: J(~} = J(X~)= {the in t e rmed ia te jacobian of Xt}.

DEFI~ITIO~ 1.8.5. -- The fibre P-~(Po) of the m a p p : J -~ F will be denoted by J(Xo) and will be called the generalized (intermediate) jacobian o /Xo.

C0~OLLA~Y 1.8.6. -- J is a complex manifold.

I P~ooF. - i t follows f rom (T(3)--:1)2 = 0 and Prop. (2.9) in [Zu]. Q.E.D.

1 . 9 . 1 . We wan t to m~ke the r e m a r k conta ined in (1.5.3) more precise (at least for H%).

Le t I be the Kerne l of the m a p of vec tor bundles

I is i tself a ~ector bundle o~er A and by I , we denote i ts res t r ic t ion to A*. The

300 FABIO ]3ARD:ELLI: Polarized mixed Hodge structures, etc.

Gauss-Manin connect ion Da on H a res t r ic ted to the subbundle I , over A*, extends to a connect ion on I .

Therefore V te F* there is a unique i somorphism, de te rmined b y the connection,

lq'ow

I(t } ~- (~eHa(X , , C)[T(3, t; C ) ( ~ ) = ~?} = (space of i nva r i an t cocycles of H3(Xt, C)}.

I<t } has an in tegra l s t ruc tu re

I~,},z = { v e l m (Ha(X~, Z) --->H~(X,, C))1/'(3 , t; C)(V) = V}

and therefore % and I{t}, z de te rmine an integral s t ruc tu re on I(po}. I t is easy to see t h a t the in tegral s t ruc ture defined in this way does not depend on t. Therefore: H ~ has no t an in tegral s t ruc ture , there is a subspaee 1~~ {invariant cocyles} which carries a canonical integral s t ructure . We will denote the distinguished integral structure on I(po} by I~(Z).

I t is now easy to check t h a t one has an obvious i somorphism

�9 3 : (3,R ~,Z)(~o}_~ I~(Z) .

This is t rue because the e lements of the left hand side space are germs of i nva r i an t

sections�9

PI~Ot, OSlTIO~ 1.9.2. - The map a in the Clemens-Schmid exact sequence

H~ ~ tt~(Xo, C) > H~(Xo, C) a 1Ca , HL HL

induces an isomorphism

a' : tt~(Xo, Z) ---> I~(Z) .

PI~ooF. - a induces a m a p a ' : H~(Xo, Z)-+I(~o} which via the i somorphism

q~71: I(eo}--~ I~t}c Ha(Xt , C) can be seen as the m a p p i n g Ha(Xo, Z) --> I{t}cH3(Xt, C) induced in cohomology by the (~ special izat ion ~) m a p X t - > Xo, therefore it follows

f rom the considerat ions in (1.9.1) t h a t a ' has range in I~ (Z) ; so a ' : Ha(Xo, Z) -+ -+ Ir

To s tudy the m a p a ' we res t r ic t oa r or iginal fami ly to a small closed disk

A~ ~ (z ~ C: [zI<e}, A~ ~ Po and we denote ~X : ~-I(~A~). Then we br ing in the p ic ture the topological descript ion of the Clcmens-Schmid

e x a c t s e q u e n c e as given in ([G-S], pag. 110-121) or in ([CI2]). i n our case it

FABIO BARD~LLI: Polarized mixed Hodge struetures~ etc. 301

becomes

sequence 1)

P H~(Xo)

H~(X~) = T(2, t) -- Id

,._ H~(Xo) H~(X~)

H3(~X)

H,(X,) R4(Xo)

sequence 2) g

sequence z)

T(3, t) - - Id H3(X,)

. H4(Xo)

sequence 1)

Here the homology and cohomology modules are wi th coefficients in any r ing

R = Z, Q, R, C and the maps are expla ined in [lot. cit.]. The two sequences 1) and 2) are exac t over any R as indicated b y the i r defini-

t ion. However in our case we r e m a r k the following facts :

a) Since ~zl(Xt) = 0 one has HI(Xt, Z) ~ 0 and so H~(Xt, Z) is torsion-s

it follows t h a t the m a p H~(Xt, Z) --> H2(X~, C) is inject ive and since we know b y (1.4.2) t h a t T(2, t; C ) - i d = 0, i t follows also T(2, t; Z ) - I d = 0. Therefore fi is

inject ive for R = Z, or Q, R, C.

b) The m a p g is inject ive for R = C the exactness of the Clemens-Schmid sequence and the fac t t h a t T(2, t; C ) - - I d = 0. Bu t since H~(X,, Z) is torsion- free and so injects into H~(X~, C), i t follows t h a t the m a p g is inject ive also for

R ---- Z. So g is inject ive for a n y _~.

e) Since fl and g are injeet ive for a n y R and g = yofl, it follows:

I m fl (~ Ker y - ~ 0 ,

so I m f l f ~ i m a = 0 b y the exactness of 1). Bu t then also K e r 6 ~ i m a = 0 b y the exactness of 2) so t h a t : ~oa = ~ is inject ive for a n y R and a is inject ive for

a n y R. I n pa r t i cu la r H3(X,, Z) torsion-free implies H3(X0, Z) torsion-free.

d) Now using the exactness of the Clemens-Schmid sequence one has

Ker (3(3, t; Q) - - I d ) = Im

and the exactness of 2) gives

K e r (T(3, t; Q ) - - I d ) ---- I m 6 .

302 FABIO ]~s Polarized mixed Hodge structures, etc.

Therefore for R = Q , R , C one has I m S = i m S . The i n j e c t i v i t y of 8 and the

previous re la t ions imp ly t h a t

It3(SX, R) = Im ~ Q I m fi for R = Q, R, C .

Therefore also

r k H s ( ~ X , ~) = r k I m ~ § r k I m f l for R : Z .

I t follows (using ~lso the fac t I m ~ f~ I m fi = 0 over Z)

Hs(~X, Z) ~ Im ~Q im/~ Q Tots

where Tots is a tors ion Z-module .

e) We have a l ready r e m a r k e d t h a t H3(Xo, Z) is torsion-free b y the injec-

t i v i ty of 8. We knew a l ready t h a t H2(Xt, Z) is torsion-free and so we see easi ly

t h a t H3(~X, Z) is torsion-free, i n f a c t if a ~ H3(~X, Z) is a tors ion e lement a ~ i m fi obviously, so ~(a) r 0 and ~(a) would be a tors ion e lement in H3(Xt, Z). i t follows

t h a t Tots = 0 and so

Hs(SX, Z) = I m ~ ( D I m / ~ .

/) Now we finish: t ake a~Ba(Xt, Z) such t h a t (T(3, t; Z ) - - i d ) ( a ) = 0. By

the exactness of 2) over Z 3 an e lement x ~H3(SX, Z): ~(x) ~ a. Wri te x - zl ~ z~ with z l e I m ~ , z 2 ~ I m f l , so z ~ = a ( y ) for some yeH"(Xo , Z). Then ~ ( y ) = x - - z ~ and 6.~(y) = 5 (x ) - -~ (%) = ~(x) = a therefore ~(y) = a, so t h a t

Ker (y(3, t; Z) - - I d ) _tim 5.

The inverse inclusion is clear. The previous considerat ions p rove t h a t d : H"(Xo, Z) -+ --~I~(Z) is an i somorphism. I n fac t we have seen t h a t

/ {~ (xo , Z) -~ Ker ( ~ ( 3 , t; Z) - - Id ) --~ I ~ , z

is an i somorphism, so d : H*(Xo, Z)--->I~(Z) is an isomorphism. Q.E.D.

COROLLARY ].9.3.--

i) H~(Xo, c ) = o.

I~A]3IO BARDELLI: Polarized mixed Hodge structures , etc. 303

ii) Vi 1, ..., iV, 5 i 1 = H (Xo, Z) = O. H (Xo, C) : O and

iii) H3(Xo, Z) is torsion-Jree.

PROOF. -- In the proof of (1.9.2) we have seen t h a t ~he map ~: HS(Xo, C) -> H~ is injective. But then the exact sequence

f f

o = / / ~ >//o(Xo, C) >//~(Xo, C) >//~

implies Hs(Xo, C) = O and so i).

ii) follows from the fac t t ha t HS(Xo, C) = Gr~H~(Xo, C) = E ~ in the spectral 2V

1 5 0,5 sequence of (!.7.1). Bu t E~ '5 : Ker (E ~ -+E~' ) and E~ = @ H s ( x 0i, C), while 1 1,5 _ _ i = 1 E 1 -- O.

I t follows ~ H (Xo~ C) = 0, i = 1, ..., iV, and so ii).

iii) has been proved in (1.9.2 c)). Q.E.D.

PROPOSITION 1.9.4. -

H"(Xo, C) i J<Xo)-r ; c)/H (Xo, Z).

PROOF. -- We consider the Clemens-Schmid exact sequence (see [C1 2])

o Hs(Xo, C) P >H~(Xo, C) a S~ N~>H~ ) co

(~ is injective. W e know by (1.6.1iii)) t ha t WsH ~ = Ker N~----Im a, and so by the strictness p roper ty of morphisms

(~: W~HS(Xo, C) --~ W~H~ A I m ~ = W3H ~

is surjective~ therefore an isomorphism. In par t icu la r a induces an isomorphism

~: W~H~(Xo, C) ~ W~H~ r2w~R~(Xo, C) F~W~L"

B y (1,6.1 iv)) we know tha t the r ight hand side vector space is isomorphic to 3 2 3 H~o/~ H~, and since W~H~(Xo, C ) = Ha(Xo, C) one has an isomorphism

~: H~(Xo, C) --> H~ induced by o . 2 ~H ~ ~ H ~ ( X o , C) ' ~

2 0 - . t n n a U d l M a t e m a t i c a

304 :FABIO ~BAI~DELLI: Polarized mixed 1lodge structures, etc.

~ow: the m a p a in the Clemens-Schmid sequence induces the following commutat ive diagram

O ~l

~(Xo, z) - ~ + i~(z) ,

H~(X3, C) N ~ W~H~ ~ H ~ ,

l ,[ H~(Xo, c) e R~

Since the vert ical maps are injections (Lemma 5 [Ca 2]) or (2.7 in [Zu]) (referring in H~ to the mixed Hodge substructure W~H~) it follows

H~(Xo, c) /R3 x H~ _ ~ ~HS(Xo,C) ( 0 ) , Z ) - - ~ F ~ ~ I=(Z) (~.R ~.Z)}po}=:J(Xo) Q.E.D.

I%E~A~K 1.9.5. -- The :proposition says essentially tha t J(Xo) is intrinsic: it can be defined in terms of the mixed Hodge structure on H3(Xo, Z) only. This is not obvious a-priori and gives an answer (for oar part icular kind of threefolds) to a question parenthet ical ly posed by ZUCKER in [Zu].

2. - A special one-variable degeneration of threefolds: the topology and the generalized jacobian of its singular fibre.

We now restrict our a t ten t ion to a very part icular kind of one variable degeneration. In addi t ion to the assumptions (1.1) and (1.2), we will assume from now on tha t :

2.1.1. Vi, j, 1 < i , j < ~ , i C j , H l ( ~ , , ~ , z ) = 0 .

2.1.2. Xo has at most 3 components, so 1 <.N <3.

Therefore the dual graph F of Xo is one of the following:

/ / h A a) N = I b) Y = 2 c) N = 3 d) N = 3

FA]~IO BAI~I)]~LLI: Polarized mixed Hodge structures, etc. 305

In fac t

H~(F, C) --- GroH~(Xo, C) = { C for i = 0 ,

0 for i > 1 ,

so F has to be connected and simply connected. In par t icular the case

in which HI(F, C) ~ 0 cannot occur. In the following we will s tudy d) snd view b) and c) as easier par t iculsr cases

of d).

2.1.3. Vi~ 1 ~ i ~ 3 ~ 8 i H (Xo, Z) is torsion-/ree.

2.1.4. 3i, ~ e (1, 2, 3}, i r ~, such that o~ the surface 8 *,j 3 a divisor D /or which (D, C~.~,8)~,., is an odd number (here ( . , . )~,,~ is the intersection pairing on H2(8 *,j, R), 1r = Z, Q, c ) .

The essential assumption is (2.i .1); the other assumptions are ma4e to simplify the proofs and the notations. We think that , assuming (2 .1 .1 )on ly , a lmost everything which can be proved with (2.1.2 to 2.1A) should be t rue also in general.

In fact using only (2.1.1) one can prove

PROPOSITION 2.2. --

i) Each intersection curve C ~,j,~ is rational.

ii) Gr~H3(Xo, Q) = @H3(X~, Q).

PROOP.- i) F rom (1.7.2 i ) ) fol lows the exactness of the sequence El ' 1 ~ --1E"1--> -->E~ . Now in our c~se:

E~ ,1 = @ HI(~ ~,~, C) = 0 = ~),1 = @ Hl(p~,~,~,,, C) = 0 . i,~,/~ li1=3

Therefore --1~2'1 = 0, SO HI(C i'j'k, C) = 0, Vi, ~; k. I t fol lows C *,j,~ ra t iona l .

if) By (1.7.2)

1B Gr3HS(Xo, C) = E~ '~ : Ker (dl: E~ '3 -~EI ' ) ,

306 FA~IO ]~AI~DELLI: Polarized mixed Hodge structures, etc.

b u t 2r

E~ ,~ = @ ~ " ( x o, c ) , i = 1

b y our a s s u m p t i o n a n d Po inca r6 dua l i ty .

z~,~ = @ ~ ' ( s ~,~, c ) = o

Q.E.D.

2.3.1. A s s u m i n g now (2.1.2) to (2.1.4) we i n t r o d u c e the fo l lowing no ta t ions . L e t

C = t h e u n i q u e c u r v e of t r i p l e p o i n t s on Xo , i.e. C = C1,~, ~ ,

(., .)z,,j = t he i n t e r s e c t i o n p a i r i n g on H~(S ~,~, R ) , R = Z, Q, C ,

( . , . )~ , = t h e i n t e r sec t ion p a i r i n g on Ha(Xt, t~).

LEPTA 2.3.2. - There exists a pair i, ] and a divisor D on S ~,j such that

(C, D)~,,~ = 1 .

PROOF. -- B y a s s u m p t i o n we k n o w t h a t for s o m e i, ] we can find a d iv isor E

on S *,j such t h a t (C, E)z,,j = l wi th I odd. Of eo~rse we can wr i t e 1 = 2r -~- :[. T h e n cons ider t h e d iv isor Ka,,j + C. O b v i o a s l y (Ks,,~ + C, C)z,,~ = - - 2 b y the a d j u n c t i o n fo rmu la . The re fo re (r(Ka,,~ + C) + E, C)z,,j = 1. Q .E .D.

LEPTA 213.3. - 3

i) ~ (c, c)~,~ = o, i,j=l

ii) I / i, ] are such that (C, C)~,,j>0 then S ~,j is a rational sur/ace.

iii) 3 at least one pair i, ] /or which (C, C)a,,j>0 and so S ~,j is rational.

1J~ooF. - W e cons ider t h e l ine b lmd le s 0~(Xo) fo r 1 < i < 3 . D e n o t e b y Oo(Xo) t he r e s t r i c t i o n of O~(X~) to C. O b v i o u s l y s ince Xo ~ -{- Xo ~ + Xo 8 ~ X , oi1 3~ one h a s

(*) o~(xo ~) | o~G~) | o~(xo ~) ~ oo.

F u r t h e r m o r e deno t i ng b y Noja,,j t he n o r m a l b u n d l e of C in S ~,j, one has

N o w degNa;~,~ = (C2)z,,~; t h e r e f o r e one ge t s eas i ly f r o m ( . )

3

~: (e%, , = o . i , j = l i < ]

I n p a r t i c u l a r i t fo l lows t h a t 3i, ] : (C ~) ,,j>~O.

F A ~ o BA~])E]~L~: Polarized mixed Hodge structures, etc. 307

Then we c la im t h a t the surfaces S ~,~ for which (C2)s,,~>O are ra t ional . I n fac t let S ~,~ be such a surface and consider the exac t sequence

0 ~ Os,,~ ~ O~,~(C) -~ Oc(C) -~ 0 .

Since H~(S~, j, C) : 0, i t follows H~(S ~,j, 0z,,j) : 0; therefore one h~s an exac t sequence

Now C is ra t iona l and O~(C) has degree (C~)~,,j>O by assumpt ion , i t follows t h a t

o(s% = 2 > 2 .

But t hen S ~,~ has a pencil of r~t ional curves (with the generic one, as C, smooth) and therefore b y ~oether~s theorem is ra t ional . Q.E.D.

2.4.1. Le t T(Xo) be a t r i angula t ion of Xo as a s trat if ied object (for definitions

and proof of existence see [Go] or [Ma]). T(Xo) induces then t r iangula t ions T(X~) on each X~, T(S ~'~) on each S ~'j and a t r i angu la t ion T(C) on C. ~o r a n y ring R

we consider

/ / S ~ ( T ( . ) , R ) = Zr~adr~eR , a~ an oriented k-simplex of T( . )

and the oriented chain complex (S . (T( . ) , R), ~ ) w i t h the nsnal bounda ry operator . We cons t ruc t a complex (K, ~) of chain complexes

i~j=l i ~ l

S.(T(Xo), R) >0

like indicated in the following d iag ram

s.(T(s1,~), R)

|

0 ~ S. (T(C),R) .. S.(T(S~,3), R)

s. (T(S~-,~), R)

+ ~ ~. (T(X~), R)

- - - ~ - s . ( r ( x ~ l , R) ~ S.(T(X~), ~)

_, S. (T(X~), R)

~ 0

where each ar row is the m a p induced b y the obvious inclusions, the dot ted arrows are 0 maps , and the signs are the signs to be assigned to each map . Each ~ is the sum of all the maps a t the corr isponding stage.

308 I~ABIO ]3AI~DELLI: Polarized mixed Hodge strueturcs~ etc.

NOTA~I0~. - F r o m now on we will denote by

Hi(Xo(r), It) the I t -module

~ H i ( X ~ , It) for r ~ 1 , k = l

�9 ~ Hi(S k'j, It) for r = 2 k,~'=l k<J

and analogously by H~(Xo(r), It) the corresponding direct sums of cohomology

groups.

L E n A 2.4.2. -- The complex (K, ~) is an exact sequence o] chain complexes.

t)R00F. -- i t is an exercise in elementary algebraic topology.

2.4.3. We consider the complex

(,) H~(r It) (~)* > ~(Xo(2) , It) (~)* ~ H~(Xo(~), It)

induced in homology by the complex of chain complexes (K~ 2~).

We define LR to be the homology of the complex (,), so:

LR = (Ker (~2),)/im (~1), �9

L~ is a It-module. Analogously there is an obvious complex of cohomology It-modules

~ ( x o ( 1 ) , It) (~*)* > H~(Xo(2), It) (~1)* > H~(C, It)

and we define L ~ to be the homology of this last complex.

Final ly we consider the dual of the complex (,) i.e. the complex

H,(Xo(1), It)* + ~2(Xo(2), It)* + ~ ( C , It)*

and we denote by MR its homology.

~ o ~ . - F r o m now on we consider It = Z, Q or C.

There are maps

R~(Xo(1), It) > B~(Xo(2), It) > ~ ( ~ , It)

H~(x.(1), It)* > R~(Xo(2), It)* , , H~(r It)*

FA~IO ]~ARDELLI: Polarized mixed Hedge structures, etc. 309

which induce an i somorph ism

L R - ~ Ma.

F u r t h e r m o r e we see easi ly t h a t there is an obvious m a p

M~ ~ 95*.

This m a p is sur ject ive if

R) R))

is torsion-free, which is cer ta in ly t rue in our case because H2(Xo(1), R) is torsion- free (by the a s sumpt ion (2.1.3) H3(Xo, Z) is torsion-free Vi).

Therefore for R = Z one has a surjee~ive m a p

_s Z -~ L *z

and since the two modules have the same r a n k its Kerne l is exac t ly the tors ion submodule of L z.

PROPOSITION 2.4.4. -- -~Z i8 torsion-]tee.

PROOF. - The eohomology m a p

z) H (C, Z)

is surjective. I n fac t the m a p (X1)* res t r ic ts first Chern classes of line bundles on

S ~,j s C, so i t is enough to see t h a t there is a line bundle ~ on some S t,j, whose res t r ic t ion to C has degree 1. This is equivalent to say t h a t there is a divisor D on some S~, j such t h a t (D, C)s,,j ~ 1 and this is L e m m a (2.3.2).

Therefore one has an exac t sequence

0 - + K e r (2~)* --> H~(X0(2), Z) (X~)* > H2(C, Z) --->0.

Now H~(S~, ~, Z) are torsion-free because HI(S~, J, Z) = 0, H~(C, Z) obviously is torsion-free and Ker X* as a submodule of a torsion-free modil le is torsion-free.

We look a t the d i ag ram

0 z)

H~(G, Z)*< ~

�9 H2(Xo(2), Z) +- Ker (A~)*~- 0

H2(Xo(2),Z)*+- Kercv +- 0

in which a and fl are the obvious maps (induced by Kronecker products) .

310 FABIO :BARDELLI: Polarized mixed Hodge structure 6 etc.

Since ~i1 H~'s are torsion-free ~ and fi are isomorphism~ therefore F is surjective and Ker (2~)*, Ker ~ are isomorphic.

Fur thermore all the modules in Shis diagram are torsion-s Therefore taking duals again one gets

o -~ H~(r Z)* -~ ~ (Xo(2) , Z)* -~ (Ker (~)*)* -~ 0

0 ~ H~(C, Z)** ~ H2(Xo(2), Z)** -~ (Ker ~)* ~+ 0

]3ut now the double dual of a torsion-free module is canonically isomorphic to the module itself: therefore one has

0 --> H~(r Z)* ~ H2(Xo(2), Z)* -+ (Kcr (2J*)* -+ 0

0 ~ H~(C, Z) --~ H~(Xo(2), Z) ~ (Ker ~)* -+ 0

I t follows tha t Q : eoker (H~(C, Z) -o.H2(Xo(2), Z)) _~ (Ker ~)* which is certainly torsion-free.

Therefore L z = {a submodule of Q} is torsion-free. Q.E.D.

PROPOSITION 2.4.5. -- Let R = Z, Q, R~ C. The complex O/ chain complexes de- lined in (2.4.1) induces an exact sequence

o -~ H.(Xo(1), ~ ) -~ H.(xo, R) - ~ Z . -~ 0 .

PROOF. -- The exact sequence of chain complexes (K, 2~) gives two short exact sequences of chain complexes:

S

~) o->s.(~(o),R) ~1 @s.(~(s~,~),R) ~ ,Ker ; .~O,

2) 0 - + K e r 2 ~ - + @ S . ( T ( X ; ) , R S.(T(Xo),R) -+0. i z l

We consider the homology long exact sequence given by 2) gett ing:

H~(Xo, R) -+

-+ H~(Ker 2 , ) -+ H3(Xo(1), R) ~ H3(Xo, R)

-~ H2(Ker 28)-+ H2(Xo(1), R) -~H~(Xo, R)

-> Hl(Ker 2,~)--~HI(Xo(1), R) --~ HI(Xo, R) -+

~A]~IO ]~ARDELLI: Polarized mixed Hvdge structures, etc. 311

while 1) gives:

-> H3(Xo(2), R) -> H~(Ker ~ ) - ~

---> H~(C, R) -~ H2(Xo(2), R) -> H~(Ker )~a) -+

--> H~(C, R) ---> H~(X0(2), R) -~ H~(Ker 2~) -->

Now assuming R = Z, Q, R, C one gets

H~(S ~,~, R) ~ H~(S ~,~, R) = 0 Vi, j ,

Furthermore C rational implies H~(C, R ) ~ O.

so ~(2Co(2), R) = o .

Since C is an effective slgebraic cycle in each S ~,~, we have that

H~(C, R) -~ H~(Xo(2), R) is injective.

Therefore H3(Ker ~3)~ 0 and one gets an exact sequence

0 -+ K2(C, R) -~ H~(Xo(2), R) -~ H2(Ker 43) -+ 0

which allows to identify H~(Ker ~) as

H~(Ker ~) ~ H2(Xo(2), R)/H~(C, ]~).

So the first long exact sequence of homology becomes

0 -~ H~(Xo(1), R) "~ H3(Xo, R) +H~(Xo(2), R)//t~(C, R) -~H~(Xo(1), R ) .

Calling NR the Kernel of the map

H~(Xo(2), R)/H~(C, R) -~ H~(Xo(1), R)

one has an exact sequence

o -~ ~ ( X o ( 1 ) , R) -~ H3(Xo, n ) -~ N ~ - ~ o .

Here iVa can be also viewed as the homology of the complex

By definition of LR~ one has LR ~- N , and so the proposition is proved. Q.e.D.

312 FABIO BARDELLI: Polarized mixed Hodge structures, etc.

COrOLLArY 2.4.6. -- There is an exact sequence

o -~ ~* -+ HalXo, R) -> Ha(Xo(1), .R) -+ o

induced by duality by (2.4.5), ]or R = Z, Q_, C.

PROOF. - Dualize the sequence in Proposi t ion (2.4.5). Since for i----1, 2, 3 3 i H (Xo, R) -+Ha(Xo, R)* are isomorphisms. The H (Xo, R) is torsion-free, the maps 3 ~

same holds for Ha(Xo, 1~) ---~ Ha(Xo, R)*. Since L~ is free, the map Ha(Xo, R) -+ HS(Xo(1), R) is surjective. Q.E.D.

P~oPosITIO~ 2.4.7. - The generalized ]acobian J(Xo) is a generalized complex torus and there is an exact sequence o/ complex Zie groups

a

o -~ ~ ~ J(Xo) -+ 0 J(X~) + o i = l

where ~ is the torus LOlL z and for i : 1, 2, 3 J ( X i) is the compact complex torus underlying the intermediate ]aeobian o] X~.

P~oo~. - The weight f i l tration (over C) on ~the mixed ]~odge s t ruc ture of H~(Xo, C) computed in (1.7.2 iii)) gives an exact sequence

0 -+ W2Ha(Xo, C) -+HS(Xo, C) --> Gr3Ha(Xo, C) -+0.

A closer look to the spectral sequence (1.7.1) in this case and a comparison with the complex (K, 2~) and its maps shows tha t the sequence above is the exact sequence

o -~ z ~ -+ ~a(Xo, c ) -~ ~8(Xo(1), c ) -~ o

of Corollary (2.4.6) af ter the identification

L*c-~ L c = W, ttS(Xo, C) .

Therefore we have a commuta t ive diagram

o > z i ~ Ha(Xo, Z) -+ ~a(Xo(1), Z) + 0

o ~ N~Ha(Xo, C) ~ B:a(Xo, C) -+ Gr.Ha(Xo, C) ~ 0

3 3 i H~(Xo, c ) - ~ (Xo, c ) 0 > Lc > B2Ha(Xo, C) ~" ( ~ - ~ , ~ a l ~ ~ -+0

I~A:BIO ~BAI%DELLI: Polarized mixed Hodge structures, etc. 313

in which the first row is the exact sequence of Corollary (2.4.6) over Z, the second row is the exac t sequence defined by the weight f i l t rat ion on Ha(Xo, C), the th i rd row is go t ten by the second by dividing out E~H3(Xo, C), and is easily seen to be exact using (1.7.2 iii)) and the fact t ha t

P~HS(Xo, C) -~ F~Gr~H3(Xo, C).

The ver t ical maps are defined by the obvious extension of scalars; in par t icular the m a p / ~ z -+ Lc is defined by the identification of L z with LZ/Tors L z and by the map Z z - + L c, so in par t icu lar has the Same image in L c as the map Lz-+ L c and is injective.

Now the composit ion of the ver t ica l maps in the diagram are injective (use [Ca 2], Lemma 5) and so taking eokernels of the compositions of the vert ical maps we get a sequence

H3(Xo, C) / ~ A H~(x~, c) I~...~

which is easily checked to be exact and thus proves the s ta tement of the proposi- t ion using Proposi t ion (].9.4) to ident i fy J(Xo) and the definition of the intermediate jacobian of X o for i = 1, 2~ 3 (see [C-G]). Q.E.D.

3. - The topology of the one-variable degeneration of threefolds: the polarization of the jaeobian bundle and the polarized mixed Hodge structures of the singular fibre.

P~EVIEW 3.0. -- In this section we will define a na tura l polarizat ion O on the jacobian bundle p ' : J* -+ ~* of the fami ly of smooth threefolds zr': 3~* -+ F*. We will consider the exact sequence

8

0 --> ~ --+ J(Xo) ---> ~ J(X~) -> 0 i~1

of Proposi t ion (2.4.7), and we will see tha t :

i) O can be extended in a na tura l way over Pc to give the principal positive 8

polar izat ion of ~J(X~) (so a polar izat ion on Gr~H~(Xo)). i = l

if) O defines a symmetr ic bil inear form ~vx0 on H1(% (2) = Lq (so a polariza- t ion on Gr_~H3(Xo)).

The main p roper ty of Vxo is t ha t we can single out a subspace V _c L e such tha t V and V'xom are intrinsic to X0 only i.e. do not depend on the family ~ ' : 3~ - + F .

314 FA]~IO ~BARDELLI: Polarized mixed Hodge structures, etc.

j~: X~ ~ X o

a: X , ~ Xo

J i , * ~ i f *

5 ~

Then (see 2.4.1) for

3.1.1. The first step is to study the relationship between the various cup- products in the one-variable degeneratien of w 1 and w 2. For this let t e A* cxv* and

be the inclusion maps,

be the specialization map,

the obvious maps induced in homology,

the obvious maps induced in cohomoJogy.

we have

(~3)*: 11~(Xo, R) -+//~(Xo(1), R)

8 8

= Z j 2 a n d = i = 1 i = l

We define the following evaluation maps:

6 i Vx~: t t (Xo, Z) --> Z by Vx~.(d) = d([X~])

where [Xo] is the fundamental class of X~ in Hs(Xo, Z),

Vx : H~(X~, Z) ~ Z

by the analogous formula

and

Sx~: tt6(Xo, Z) -~ Z

Sx : tt~(Xo, Z) ~ Z

with the obvious meaning. 3

i) Sxo= ~Sx ' . , i=1

ii) Vx, oj* = Sx'o, Vi.

8 i V(~ e/ / (Xo, Z)

by Sx~((~) = 6([X~])

3

i = 1

The following facts are obvious:

For any space V = {Xg or Xo or X,} we denote by

Av: H3(V, Z)XH3(V, Z) -+ 11~(V, Z)

the cup-product map.

FABI0 ]~ARDELLI: Polarized mixed Hodge struetures~ etc. 315

We then define the following skew symmetr ic bilinear forms:

Q~ = Vx,ooAx, on tP(Xo, Z) ,

Q = S~.oAx. on Ha(Xo, Z ) ,

Q~ = VxoAx~ on Ha(X~, Z ) , 8

Q~ = ZQ~ on Ha(x0(]), z ) , i = 1

with the obvious meaning.

P~oPosITIo~ 3.].2. -

i) T(t, 3; Z)*Q~=Q~,

ii) Val, a~H3(Xo, Z) Q(a~, a~)= Qt((y*(a~), a*(a2)),

iii) let 1 < i < 3 , Va~, a~Ha(X~, Z) and g~l, ~2~Ha(Xo, Z) such that

i a~ /or ~ = i , i = 1 , 2 , [ 0 for k r

one has Q,(al, a~) = Q(al, ~ ) = Qt((~a(gl), aa(~z2)),

iv) Val, a2, fieH3(Xo, Z) with (tla)*(~--g2) ---- 0 one has Q(a~, fi) -- Q(~, fi),

v) Va~, a~, Ha(X(]) , Z) there are ~ , a~eHa(Xo, Z) such that (23)*(ai) = at, i = 1~ 2 and

Q,da~, a~) = Q(al, ~2) = Q,(a*(al), a*(a2))

/or any such choice o/al~ ~ .

P~oo~: i t is an exercise in e lementary algebraic topology, using the surjecti-city of the map

(~a)*:/~(x. , z) - . Ha(Xo(]), z)

and formulas i)~ ii) of (3.1.1). Q.E.D.

3.2.]. We are now in a loosition ~o polarize the jacobian bundle p: J - ~ / ~ defined in (1.8.2) and (1.8.3). We consider first the bundle p ' : J* -+2~* restr icted to 2~*. Let ff be the locally free sheaf (on/~*)

R~p',Z | 0 ~,

316 FABIO BARDELLI: Polarized mixed Hodge structures, etc.

and 0 the global section defined by:

Vte F* , O(t) = Qt e H~(J(Xt), Z)

where Qt is the skew-symmetric integral nnimodular bilinear form on TIs(Xt, Z) _~ ~-- HI(J(Xt), Z) defined in (3.1.1) Vt ~ ~*.

I t is obvions that:

i) 0, being integral, is a section of the subsheai t~2p.Z,

if) 0 is <~ locally constant ~> and invariant under Inonodromy so <~ constant ~>,

iii) 0 is a holomorphie section of R2p.Z@z Os..

Furthermore Vte/~* O(t) = Qt determines the principal polarization of the complex torus J(Xt), underlying the intermediate jacobian of Xt (as defined in [C-G]).

Since H~,~ H3,~ O, the polarization is positive by [C-G] i.e. makes (J(Xt); O(t)) a principally polarized abelian variety. Therefore (J*, O) is an analytic family (over F*) of principally polarized abelian varieties.

3.2.2. We now extend the polarization O of (3.2.1) to all of J . For this we restate the Remark (1.9.1) in the following form: let I(Z) -= {global integral sections a of the bundle I . with a flat with respect to D3}. Obvionsly I(Z) is also the Z-modnle of global integral flat sections of I , and as such it is canonically isomorphic to Ion(Z). The section 0 defines an integral, skew-symmetric bilinear form on I(Z) (independent on t eA*) and therefore on I=(Z). Using the isomorphism a': H3(Xo, Z) --> I~(Z) of Proposition (1.9.2) we conclude that 0 defines an integral skew-symmetric bilinear form on Hs(Xo, Z), which, by Proposition (3.1.2ii)), is easily seen to be Q (as defined in (3.1.1)). By (3.1.2 iv)) it is possible to see that Q defines a billnear form on H3(Xo(1), Z) which by (3.1.2 v)) is simply QA. There- fore 0 determines the integral skew-symmetric llnimodular bilinear form Qx defining

3

the principal polarization of the complex torus (~ J(X~), where each J(Xo) is the

intermediate jacobian of X o as in [C-G]. Furthermore by Corollary (1.9.3ii)) Vi, HI'~ = 0, and by Proposition (2.2) and Proposition (1.7.2 iii)) Vi, H~'~ ---- 0,

3

therefore the polarization determined by Q~ is positive by [C-G] and makes (~ J(Xo)

a principally polarized abelian variety canonically isomorphic to the snm of the intermediate jacobians of X o (as principally polarized abelian varieties).

3.3.1. For the rest of section 3 all homology and cohomology gronps will be with rational coefficients and therefore denoted simply by H*(space).

3.3.2. We want to define and discuss a bilinear form on JLq. For this let U~

:FA:BIO ]~ARD]ELLI: Polarized mixed Hodge structures, etc. 317

be the closure o] a tubular neighborhood o] Xo in ~-~(A): we will use here the closure of the tubu la r neighborhoods defined by It. C L E ~ s in ([C1 2], Theorem 5.7).

Denote by

U~,~ = U~ (~ U~ ---- (a closed tubula r neighborhood of S ~,j} ,

by W(2) -~ [J Ui,j and by i ,~

W(3) ---- U~ n Uj (~ Uk---- (a closed tubular neighborhood of C}.

As in ([C1 2], Theorem 6.9), there is an act ion 0 of the semigroup S : [0, 1] • on s-~(A). Via this act ion the specialization mgp a: X t - ~ X o for t e A * can be identified with

0(0, O)]**: Xt -> Xo ,

while the Pic~rd-Lefschetz t ransformat ion X, -~ X~ is induced by 0(1, 1). Fltrther- more by ([C12], Theorem 5.7) a as above induces continuous maps

a(i): x , \ w ( i ) ~ X o \ W ( i ) , i = 2, 3.

NOTA~xO~S. - Throughout section 3 we will denote by: T the homology Picard- Lefsehetz t ransformat ion T: H3(Xt)-->Ha(Xt) by N--~ T - - i d and by ( ' , ' )x , the intersection puiring on H3(Xt).

L E n A 3.3.3. -

i) ~(2) is a homemomorphism, so a(2), : H3(X~'~W(2)) -->Hs(Xo~W(2)) is an isomorphism.

3

ii) X o ~ W ( 2 ) ~ - - I _ [ ( X ~ \ W ( 2 ) ) (I_[ is disjoint union). The maps H~(X~\ i ~ l

~W(2)) - ->Hs(X~) induced by the inclusions are surjeetive for any 1 H3(Xo(1)) is surjeetive.

iii) Let ~ e H3(X~) and assume that (~.(y) e (23).H3(Xo(1)). Then N(7) = O.

P~ooF. - i) That a(2) is a homeomorphism follows from the way the action 0 (and in par t icular 0(0, 0)) is defined in [C1 2].

3

i) Obviously one has X o \ W ( 2 ) = i i (Xg \W(2) ) . Then one considers the homology exact sequence for 1~<i~3 ~=o

~ . ( x t \ w ( 2 ) ) -+ ~.(xg) -+ B, (xg , x t \ w ( 2 ) ) .

318 I~ABI0 BA~DELLI: Polarized mixed Hedge structures, etc.

The Alexander dua l i ty theorem gives

n~(Xo, Xo\W(e)) _~ ~(Xo c~ w(e))

(H denotes Alexander cohomology). Then one applies the Mayer-Vietoris sequence

for ~ # i # j , k # ~

Each space occurring in the sequence is a tubula r neighborhood of some sub- va r ie ty of Xo, therefore by considering tubular neighborhoods containing the given ones (and this can always be done af ter shrinking the original neighborhoods, if necessary) one can replace Alexander cohomology by singular cohomology, essen- t ia l ly by definition. Each Xg n U~,~ has a project ion map Xg n U~ij--> S~0 j whose fibres are complex disks and Xg n U~,j c~ Ui,~ has a project ion map Xg ~ U~,~ c~ n U~,~ -+ C whose fibres are 2-policylinders. Therefore the eohomology of the fibres in these fibrations is t r ivial and the sequence above is s imply

R~(sr | H~(s~, ~) ~-% H~(c)--~/z~(si,~ u s~,~)~-~ ~ ( ~ , ~ ) Q H~(S/,~).

The map a is clearly surjeetive, because the class of an ample divisor on S ~,~ restr ic ts to a non zero class in H2(C), therefore y is injective, t~ut H3(S ~,j) = O, Vi, j so

0 = ~ ( s i , J u zi,~) ~_ ~ ( x ~ n (u~:~ u ~,,~)) ~_~(xgn w(2)) ~ ( x g , x~\w(2)).

I t follows if).

iii) I~et us consider an element yr such tha t a , (y)a(2~) ,Hs(Xo(1)) . By if) one can represent the class a*(y ) by a 3-cycle T suppor ted in Xo~W(2). Via the homeomorphism a(2): X t \ W ( 2 ) ~ X o \ W ( 2 ) one can l if t ~ to a 3-cycle y' suppor ted in Xt \W(2) . We consider the class ~' of y' in Hs(Xt). :Now /~(~') = :~' because the Picard-Lefschetz t ransformat ion T is induced by 0(1, :t) which is the iden t i ty on X t \ W ( 2 ) (see [CI2]). The same reasoning applies to y - - ~ ' because a . (y - - ~') ----- 0 so T(y - - ~') ~-- y - - :~' and therefore T(F) : y so -Y(y) = 0. Q.E.D.

RECALL 3.3.4. -- i) V7 ~ H3(Xt) with N(y) ---- 0 and V~ ~ H3(Xt) one has

(r, 2v(a))~, = o .

~ F ~ o B~DELLI: Polarized mixed Hodge strq~ctures, etc. 319

This is a tr ivial computation.

if) Va~, a~eH~(Xt) one has

+ = o .

This is also trivial in our ease using i) and I s ---- 0.

3.3.5. We are now ready to define a bilinear form ~0Zo diagram

o - > (~) ,

> H~(Xo) ~ 0

on LQ. Consider the

obtained by the exact sequence of Proposit ion (2.4.5) and the specialization map ~, . Here we deJine u = ~o~,. We remark tha t ~, is surjective because a s we have already seen in Proposit ion (1.9.2 e)) the dual map in eohomology is injec~ive. Then ~ is snrjective.

Let ~ , ~ 2 c I q and consider two elements 5~, ~SH3(X t ) such tha t ~ ( ~ ) = ~ . Then define

---:

One checks easily tha t ~zo(~, g~) does not depend on the choice of t ~ A* and of the 5~eH3(X~) by Lemma (3.3.3) an4 by the recall (3'3.4 i)). Fur thermore ~o is symmetr ic by (3.3.4 if)). However the form ~Pz, defilled in this way is not easy to handle because it involves in its definition the use of the specialization map and the knowledge of the monodromy. Our goal is to determine a subspace V g LQ where ~xo can be easily determined and where i t is in t r ins ic to X0 only.

3.3.6. As in (2.4.]) consider a t r iangulat ion of X~\W(3) as a stratified object. Using this triangulation~ one construe% a complex of chain complexes (K', ~'~)

0 -~ (~ S . (&~\W(3) ) ~ ~ S.(X~\W(3)) ~ > S. (Xo\W(3)) --> 0 i~j=l i ~ l i<J

entirely analogous to the one in (2.4.1).

2 1 - Anna~i di Matemaf i ca

320 ~A:BI0 ]~ARDELLI: Polarized mixed ttodge slructures, etc.

As in (2.4.1) we will denote by

B,((Xo\W(r))(s)) the Q vector spaoe

3 @ H ~ ( x ~ \ w ( r ) ) for s = 1 a n d r = 2, 3 , k~l

It @ / / ~ ( S ~ , J \ N ( r ) ) for s = 2 a n d r = 3 .

We define

L(w(3)) = h/ (3) ,

I t is obvious to see that the complex (K', 2~) is an exact sequence of chain complexes and that therefore one has a surjective map ~3: H3(Xo\W(3)) -+Z(W(3)) which is the analogous of the map ~: Hs(Xo) -+> L O.

3.3.7. complexes

There is an obvious map R: L(W(3))-+Lq induced by the map of

0

R~(c/---, H~(x0(2/) , H0(xo(~))

where the vertical maps are induced by the obvious inclusions. We define the Q-vector space V ff .LQ as V = i m R. We recall that LQ is the Q-vector space whose elements are equivalence classes

represented by elements

(JO1,~, D1,8, D2,a)[Did e J~2(~ i'j)

and

(~),(D~,~, D1,3, ~)~,~) = 0

Vi, j

We remark here that since LQ = W~H~(Xo, Q) which is entirely of type (1, 1) by Proposition (1.7.2 iii)), the triples (DI,~, D1,3, D2,~) with D~,~ e H~(S ~,J) can be chosen with each /)~,j of type ( - -1 , - -1) in H2(S~z).

i t is also easy to see that if we consider an element y e L z c_ LQ then each D;,j in the triple (DI,2, DI,8, D~,3) representing y can actually be chosen to be a divisor on ~i,~.

FABI0 BAI~DELLI: Polarized mixed Hodge structures, etc. 321

P~O~OS~T~O~ 3.3.8. - V ~_ Lq is the subspaee K o] L~ whose elements are represented by

{ (D,,~, D~,z, D~,a)ID,,~eH~(S',~) Vi, ~,

(~).(D,~, D~,~, D~,~) = 0 and (D~,~, C)~,,, = 0 Vi, i .

P~oo~. - V _c K is clear because elements in V can be represented by triples (C~,~, C,,a, C~,a) with C~,~ e H~(S~,r and obviously (C~,~, C)~,,~ ---- 0.

To see the converse it is enough to use Vi, ~ the exact sequence

---~ H2(~ i'', ~ l " \ W ( 3 ) ) ---N ~ l ( ~ i " ~ W ( 3 ) ) ~ ~ 1 ( ~ i'') -~-- 0 .

We have that

H . ( s , , , , s , , , \ w ( 3 ) ) =~ H , - . ( w ( 3 ) n ~, , , ) =~ ~ , - * ( c ) =

Therefore one has

0 for ,=3,

Q for , = 2 .

0 -+ R ~ ( s , , ~ \ w ( 3 ) ) -~ ~ ( s , , , ) -~ ~ ( r -+ ~ ( s , , , \ w ( 3 ) ) -+ o .

Since the map H2(S ~,j) -+ H~(C) is clearly surjeetive (because any ample divisor in H~(S~,O is mapped to a non zero element of H2(C)), i t follows tha t

i) H,(S' ,~\W(3)) = 07

ii) H2(S~,J\W(3)) = {yeH~(S~'J)I(7, C)~,,, = o}.

Therefore any element in K of the form (C1,2, C,,3, C2,~) with (C~,j, C)s,,j = 0 can actually be lifted to H2((Xo\W(3))(2)). I t remains to show t h a t the lifted element

is in ~L(W(3)). For this i t is enough to prove tha t for example for S *'~, S *'8 c X~,CII~ -- C~,8 -= 0

in H2(X~) implies t ha t the l if ted C,,~--C,,3 = 0 in H2(X~\W(3)). But one has an exact sequence

~,(x t , x~\w(3) ) -~ H~(xg\w(3)) ~ H,(xt) ~ Ho(xt, x~\w(3))

and

H.(xg, x~\w(3)) ~ f~~ n w(3)) = ~6-*(c) = 0 f o r . = 2, 3 .

Therefore by looking a t the diagram in (3.3.7) the claim follows. Q.E.D.

322 ~FABI0 ]~ARDELLI: Polarized mixed Hodge s~ruetures, etc.

COROLLARY 3.3.9. - HI(Si,JN, W(3)) = O, Vi, j.

PROOF. - This has been proved in the proof of (3.3.8). Q.E.D.

3.3.10. On the snbspace Vc_LQ there is a billhead' form B deflated by:

V~, ~ ~ V represent ~ by a tr iple ~ i (ai,2~ ai,3~ a2,a)

wi th al, , ~ I m (H~(S~,'\W(3)) --> H2(Sk,')), then

2 B(~, ~) = ~ (a~,~, a~,~)~,,~. Gj~I

One checks easily, using (2.3.3 i)) and the definition of V, t ha t the bilinear form does not depend on the choices of the representatives.

We remark here t ha t bo th the snbspace V and the form B depend only on Xo and not on the whole family.

PROPOSITI0~ 3.4.1. -- ~ x o l v = - - B i.e. ~/g~V~ i = 1 ~ 2

~ o ( ~ 1 , ~ ) = - B ( ~ , ~ ) .

The proof of this proposi t ion reqvAres several ingredients, among those an explicit computa t ion of N: Ha(X~\W(3))-->HdXd. We s tar t with

PROPOS~T~O~ 3 . 4 . 2 . - ~(3),: ttdX~\W(3) ) -+ttdXo\W(3) ) is suriective.

PRoof . - We apply the Leray spectral sequence to the map ~(3). F ro m the description of the specialization map ~ as 0(0, 9) one has tha t Vp ~ X o \ W ( 3 )

a point

s(3)-l(P) = S 1 • I

where 1 = [0, 1] c R.

q

for p c ( U 2~,J)'% W(3) ,

is an oriented cyl inder bandl% therefore

0 for q~2

QI u ~"\~(~I for q ---- 1, ~,~

Q for q = 0 .

lq~A:BIO ~BA;RDiELLI: Polarized mixed Hodge structures~ etc. 323

In the Leray spectral sequence of a(3) one has

~,o = ~,o _ _ E~o

and

= ~ , / i r a d3(E; F: )

5Tow

, Q) _~,' = ~ ' (Xo \W(3) , R~3,.(Q)) = H ~ " ~ =

i , i

by Corol lary (3.3.9). I t follows El2 '~ ---- 0 and therefore E~ ~ ~ E~ '~ = H3(Xo\W(3)) . The fil tration of

H3(X~\W(3)) induced by the spectral sequence gives then an injection

~ o ~ ~3,o = m ( x 0 \ w ( 3 ) ) ~ m ( x ~ \ w ( 3 ) )

and therefore the dual map a(3). is s~trjective. Q.E D.

3.4.3. Now we reinterpret for our ease the description of the monodromy transformation T (or bet ter of iV), as given by H. CLEMENS in [C1 1]. For each i, j and t e A * (for a small enough d), A * c ~ * , X t n (U~,j\W(3)) is an oriented cylinder bundle over S~,~\W(3) with fibre S~• There is a circle bundle T(i, j) (=X~) over S~,J\W(3) and Xt (~ (U~,j\W(3)) is a bundle over T(i, j) with fibre the 1-sim-

plex I = [0, 1]. The s i tuat ion is described by the diagram:

x , n (v~ , , \w(3) )

I T ( i , 5 )

r ( i , j)

[ ~ ( i , J )

S~.\W(3)

Therefore there is an obvious Gysin map which, composed with the homology map induced by the inclusion T(i, ))c_ X~, gives a map

+i+: /L(s~ ,J \w(3) ) - ~ 3 ( T ( i , j)) +H3(x~) .

324 FA~IO B&EDELLI: Polarized mixed Hodge structures, etc.

Then we consider the following commucative diagwam

~(x,\w(3)) ~a(3), ~ )

H~(Xo\W(3)) ~ .~ ~(w(3))

a(3), is surjective by (3.4.2), ~3 is surjecfive by (3.3.6) and v(3) is defined as v(3) = = ~ ' a (3 ) , .

Since

= r : e r - +

v(3) is given by the data of its components

~(3h,j: ~3(x,\w(3)) -~R.(s,,J\w(3)), 3

so that v (3 )= ~ v(3),,j. i,j=l i<j

3.4.4. Now an easy local computation in our case, or ~he application of II. Cle-

mens description of the Picard-Lefschetz transformation, in [011], shows thgt in

our case the map n, de/ined as the composition of the vertical arrow in the following diagram with N

H3(X,\W(3)) ~ > H~(.X,)

/ir~(Xt)

where the vertical arrow is induced by the inclusion, is given by

3

n = - - ~ r i,j=l i<j

Furthermore using general position it is easy to see that the following intersection formala holds: (see [C1 !], pag. 103)

v~ E~4x,\w(3)), vt~eH~(s',,\w(3)),

where each cycle is considered in the appropriate space via the obvious inclusion maps.

Fha3xo ~AI~DELLX: Polarized mixed Hodge s~ruetures, ere. 325

:PROOY 0]? :PROPOSITION 3.4.1. - Consider, for i = 1, 2,

a l e V = I m (Z(W(3)) ,,,,.R > Lq).

The sur jec t iv i ty of v(3): Ha(X, \W(3)) - ->L(W(3)) impl ies t h a t ~here are cycles ~ ,eHa(X , \W(3 ) ) such t ha t Roy(3)(5,)-----&,. Then

~ . (~1 i ~ ) = ( ~ , ;v(~))~,

where ~ are considered in Ha(X~)via the na tura l inclusion. Bu t since ~e e H~(X~\W(3)), 2V(~) is ac tua l ly equal to

8

n ( ~ ) = - - Z q~,,,ov(3),,;(~) i ,J=i

8O

8

~px,(cq, ~) -~ -- ~ (~:, qb,,~ov(3)~,,(~))x , = by the intersect ion formula in (3.4.4) = i,~=I i<j

8

-- - - ~ (v(3),.j(~), v(3),.,(~2)),,,, = --B(v(3)(~I), v(3)(5~)) = - - B ( ~ , ~ ) . Q.E.D. t , ~=1

3.4.5. I t follows from the inclusion L z --->LQ, the definitions of V and B ~nd prop. 3.4.1 that ~Px, is integral-valued on the subspace V z = : L z ( 3 Vc-->Lq.

3.5.1. We check in this section t h a t the polar izat ion O of the bundle p ' : J* --> ~* defined in (3.2.1) determines the bilinear form Fx~ on LQ.

Vte A* we denote by P( t ) : H3(Xt, Q) -+H~(Xt, Q.) the Poinc~r6 dual i ty isomorphism. Then using the definition of ~Vx~ we get: V~I, ~ e L q choose ~ e eH3(X~, Q): v ( ~ ) = ~ . I t follows

i t is obvious t h a t the valae so got ten does not depend on t e A* nor on the choice of 5~ such t ha t v(~) = ~ , bu t only on O, on the Poincar6 dual i ty isomorphism and on s However the da tum of the Poincar6 dual i ty isomorphism is equivalent to the da tum of O itself, by the very definition of the polarization.

By Proposi t ion (2.4.7) we have an exact sequence

3

o -+ ~: ~ d ( X o ) -+ | J ( X g) -+ o i ~ 1

with ~ = LC/L z

and this allows to ident i fy L e with Hi(T, Q).

326 FABIO ]~AI~DELLI: Polarized mixed Hedge structures, etc.

Since moreover one has a surjective map

t t~ ( j (x , ) , (2) ~ H~(~, (2)

H~(X~) ) Lq

which identifies H~(z, (2) with H~(J(Xt), (2)/Ker N,: it is clear that the polarization 0 in the jaeobian bundle p : J --> E and its local monodromy on HI(J(Xt), (2) completely determine Y'Xo.

4. - The a d u a l . o n e - m o t i f and the Abel -Jacobi map.

4.1.1. in this section we will recall and use Carlson's theory (see [Ca 1] ~nd [Ca 2]) to compute the extension data associated to the extension of mixed ]~odge structures

o -~ Lz -~ ~3(Xo, z ) -~ B~(Xo(1), z ) -~ 0 .

~:his will be done by computing the one-motif (see [De 2i) associated to the dual extension

o -> H~(x0(~), z)-+H~(xo, z) -+Lz-+o

of mixed ]~odg e structures.

4.1.2. The mixed Hedge structure on the lattiee H~(Xo, Z) dual to HZ(Xo, Z) is defined by

and

W_k_~H3(Xo, Q):AnnW~H3(XO, (2)

E-~+IHa(Xo, C) = Ann FkH3(Xo, C) .

(H3(Xo, Z); ~'H3(Xo, C); W.Ha(Xo, Q)) is a mixed Hedge structure and one cheeks that

W_~H,(Xo, Q) : H3(Xo, (2),

w_~H3(Xo, Q) = ~r~(Xo(:t), q)* = ~(Xo(~), q ) ,

w_,H~(Xo, Q) = 0

and Gr_~H3(Xo, Q) = LQ.

FA:BIO ~ARD:ELLI: Polarized mixed Hodge structures, etc. 327

Thus, as expected, the weight filtration on tI~(Xo, Q) is defined by the exact sequence

o -~ H.(Xo(1), q ) -~ H.(xo, q) -~LQ-+ 0 .

The Hodge filtration is given by

Y:~g.(Xo, c ) = H.(xo, c ) ,

2V-~H3(Xo, C) = Ann Y~H3(Xo, C) ,

~~ C) = 0 .

Therefore on W_aHa(Xo, C) there is a pure itodge structure of weight - -3 , whose ltodge decomposition is given by

w_~H~(Xo, c ) = ~-_~;-~ | H:~ ,-~ .

In particular, by the definition, we get:

8

/~-~g .H~(xs C) =/~-~,-~ = ~g~ ,~cx~ * - - - - 8 ,k.Lz ~ 0 / "

i = 1

On Gr_2H3(Xo, C) the ~odge decomposition is simply -~'-~ 2 - - 2 "

4.1.3. In our case we know by ([C~2], Lemma 5) that Ha(Xo(1), Z) injects into Ha(Xo(1), C)/-~-~Ha(Xo(1), C).

By the definition of the Kodge filtration F" on H~(Xo(1), C) in (4.1.2), it follows easily

~(xo(1), C)/F-1H~(Xo(1),C) = n~(Xo(1), C)/F-~W_~H~(Xo, C) = 8

c)/ | , , , = g (Xo~ 3

= Q~ , l~Xo)* . i = l

We will use in the following the conclusion that:

~(Xo(1), c) /F- l~ (Xo(1) , c ) + H~(Xo(1), Z) = 8 3

= @H~'I(X~)*/Ha(Xo(1), Z) = ~ J ( X ~ ) . i=1 i = 1

4.1.4. We follow here the treatment by Carlson ([Ca 2]). The extension

(,) o ~ }&(Xo(1), z ) .+ ~ ( X o , z) ~ z~ -* o

328 FABI0 BARDELLI: Polarized mixed Hodge structures, etc.

of mixed ]~odge structures (on the respective lattices) is represented by a homomorphism ~z: Z c -+Hs(Xo(1), C), which is unique up to addition of elements in

if~ ( Ic ; H.(Xo(1), C)) -{- Horn (Lz; H3(Xo(1), Z))

because

Ext (L; H.(Xo(1)): ---~ J ~ ( I ; H,(Xo(1))) (see [Ca 2], Theorem 2) .

3

The one-motif ~ : ~z--->@J(X~o) is defined by ~he ]ormula (see Proposition 3 in [Ca 2]): Vfi e s z ~-~1

u,.(/~) = ~,.(/5) rood F-I(Hs(Xo(1), C)) -}- Hs(Xo(1), Z) = 3

= ~,,(/~) rood (~H"*(X~) * + H.(Xo(1), Z) .

8

So UH.(fl) is in (~J(X~) by (4.1.3) and it is well defined by the previous considerations. ~=~

The homomorphi~m uH. is functorially defined by the extension of mixed ttodge structures (.) and since L z is of pure type ( - -1 , - -1 ) u~. determines (.) up to con- gruence (see [Ca 2], Proposition 3 and its proof, see also [De 2]).

Using (3.5.1) u~. can also be seen as a homomorphism 3

u~ : H~(~, Z) -+ @ d (xg) . i = 1

4.1.5. We recall here the definition of the Abel Jacobi homomorphism:

_> " i A H2'~(Xio) *

An element y e L z is represented by (3.3.7) by a triple of divisors (D~,2, D~.s, D~,s) such that D~,~ c S t,j, and

D1,2--Dl,,ho~0 in X~,

--D1,2 + D~,sh~0 in X~,

Dl,z--D2,3h~O in X~.

I t follows that there are three 3-chains _Fl~/"2, F~, /'~ e C3(Xg, Z) such that

~F1 = D1,2-/)1,, ,

~/'~ = D1,3 - - D2,, �9

3

Then ~ / '~ is a 3-cycle in C3(Xo, Z) i.e. determines an element in Hs(Xo, Z). i = 1

I~ABIO BAI%DELLI: PoIa;ized mixed Hodge structq~res, etc. 329

3 By definition the Abet-Jacobi homomorphism d : Z z -+ ~ J ( X ~ ) is

i=l

I t is known that ~/ is well defined.

PI~OP0SlTION 4.2. -- Vy eL z

i.e. the one-mort/ uH. is the Abel-Jaeobi homomorphism ~4.

Before giving the proof of Proposition 4.2, we want to make the following

RE~ARK 4.2.1. -- Recall the following facts (see [G-M], 8 and specifically section F): Let

A~(X~) = {C ~ complex valued k-forms on X~}

Consider

r = {(~, 3

~ , ~.)e Q A ~ x ~ ) I (z~)*(~, ~ , ~ ) = 0}

where (2~)*: Hk(Xo(1))~R->Hk(Xo(2))~R has the obvious meaning using De-Rhum cohomology and (2.4.1).

There is an obvious operator d: G~(Xo)--> Gk+*(Xo) defined by

d(al, ~ , ~) = (da~, da~, dad

where d~ is the usual differentiation in A~(X~). Then (G'(Xo), d) is a complex

d d G'(Xo) = {G0(xo) > G1(xo) ~ G~(Xo) -~ ...} and Vi~>0,

]z~i(X~ C) : ~ i ( e a ( X o ) ) (see [G-M], p a g . 108).

This describes a way (alternative to (1.7.1), but essentially the sume) to compute the cohomology of Xo by differential forms.

Using the triangulation of Xo introduced in (2.4.1)it is easy to see that any k-homology class can be represented by a c y c l e / ' e S1~(T(Xo), R) with R = Z, Q, R

3 or C and F can be written as F -~ ~ F i with F i e Sk(T(Xg), R).

330 PA]~IO [BARDELLI: Polarized mixed Hodge structures, etc.

The condit ion ~F = 0 implies t h a t

with y~ ~ Sk_I(T(S'.O).

The usual pair ing Hk(Xo, C) • C) -+ C is given by the following recipe:

3

V[F] e Hk(Xo, C) write /~ = ~ F~, i = l

Vw e H~(Xo, C) represent r by a tr iple (w~, ~o2, r Ker (Gk(Xo) ---> Gk+~(Xo)), then (see [G-M])

One checks easily t ha t the definition does not depend on the choices of representatives.

PROOF OF PROPOSlTIO~ 4.2. -- We follow here the recipe given by Carlson to compute the extension homomorphism and the one-motif. F i rs t we compute a homomorphism ~ which represents the extension of mixed Hodge structures

0 -+ L* ~* ~* , H~(Xo) , ~ ( X o ( J ) ) -+ o .

B y ([Ca, 2], L e m m a 4) any such homomorphism y3n. is of the form ~3u. = rzoS~. where Sp: HS(Xo(1), C)--->H"(Xo, C) is a section of the Hodge fi l trat ion i.e. fi*o o id and Vi Sdr'(H (Xo(1), C)))c C), and Z) G isan

- ,

integral re t rac t ion i.e. rzo~ z = id. We s t a r t by defining S~. i n H3(Xo, C) we choose a complex subspace S such t h a t S h i m s * = O,

fi*t~.: S--->H~(Xo(1), C) is an isomorphism and S DI~2H~(Xo, C), (such an S obviously exists). Then the map (fl.]~)-l: Hs(X0(1), C)--->-H3(Xo, C) defines a sec- t ion of the Hodge f i l t rat ion and we will call it %F. To define r z we look at the sequence

o -~ H~(Xo(1), z ) fl > H d x o , Z) ~ > L~--> 0

and we choose a submodule M in HdXo, Z) with M ~ fiHs(Xo(1), Z) ---- 0 so tha t ~lu: M -+ L z is an isomorphism.

l~A~o ]3iR])]~LL~: Polarized mixed Hodge structures, etc. 331

We call 3: Lz-->H3(Xo, Z) the map (~]~)-1: Lz_+M~_>H3(Xo ' Z). Then we *

define rz: H3(Xo, Z) --~ L z by: V/2 eH3(Xo, Z) rz(/2 ) is the linear functional on L z given by

Vr ~ ~z rz(/2)(~) = </2, 3(~)~

where ( . , .> is the pairing H2(Xo, Z)• Z) -~Z. Then ~a.: H3(Xo(1), C) ->L c is given by

v(/2,, 9~,/23) e H3(Xo(~), C)

~//,(/21, /22, /23) ---- rzoL~_~(/21, /22, /23) : {~ .-.4- (~ , ( /21 , /22! /23), 3(~])}}

~s a complex-valued linear functional on L z. 3 Then by ([Ca2], Remark 3, pag. 115) uH: L z - - ~ ( ~ J ( X ~) is characterized in

i : 1

the following way: V 7 e L z u,.(y) is represented by the linear functional on H3(Xo(1), C)

~..(~)[/2~, /22, /23] = <~..(/2~,/2~,/23), r>

where <.,. > is the obvious pairing LC• L z --. C. Therefore we get

~..(7)[~1, 9~,/23] = <s~(/21,/2~,/23), ~(7)~

where <.,.> is the pairing H3(Xo, C) • Z) -+ C. Now using the Remark (4.2.1) we can represent 3(7) by three 3-chains (F1,/"2,

F3), r~e &(T(X~), Z), s u c h t h a t :

I!l 0 _P~ = 0 , /'~ = 3 ( 7 ) in H3(Xo, Z) and so ~ ~_P~ = 7 , . - - - - g = l "

:By the same remark we can represent the cohomology class SF(/2~, /22, /23) by a triple (o)1, co2, ~o3) ~ G3(Xo) and with the De-l~ham cohomology class of ~o~, [o~] corresponding to /2~ in the De Rham isomorphism H~(Xo)3 ~ --~ H3(Xo,~ C). So

Now since

~,.(7)[/21, /22, /23] = <s~(/21, /22, /23), 3 ( r )} = , . i = l . J

r~

3 3

- i = 1 i = 1

3

we are interested in uH.(7) as a linear functional on @H2'I(Xg) only.

332 :FABIO ]3ARDELLI: Polarized mixed Hodge structures, etc.

I t is therefore clear t ha t the one-motif uH, acts on Z z as the Abel-Jacobi map 3

does i.e. associates to any 7 e Z z the element of @ J(X~) represented by the l inear 3 i:I

funct ional on @J~2'l(X~) i = l

5. - Degenerating families of curves and of rational threefolds.

5.1. Some remarks o~ curves degenerations.

5.1.1. Let ~: C - > A be a proper, project ive, flat, holomorphic map of the complex manifold C onto the uni t disk A c C such tha t

i) Vt e A, t ~ O, Ct- - .~- l ( t ) is a smooth project ive curve of genus g;

ii) Co = ~-1(0) is a semistable curve of genus g i.e. Co is reduced and has only ord inary double points.

I t is well known tha t one can associate to ~: C--> A a jacobian bundle ~: J(C) -> A wi th

~-~(t) = J(C~) the jacobian of C~, V t e A , t ~ : 0 ;

~-~(0) = J(Co) the generalized jacobian of Co.

The const ruct ion is well known (see [C.G. et al.]), and can also be performed along the line of the const ruct ion of the analogous jacobiaa bundle of our degenerating fami ly of threefolds 3C --~ 1 v (although the former mot iva ted historically the latter).

5.1.2. Denot ing by 0o the normal iza t ion of Co (Oo m ay be disconnected) and by:

v: 0o-+ Co the normal iza t ion map,

X = the singular locus of Co,

2 = ~-l(_r),

i : Xr 0o the inclusion map,

~: C~ --> Co the specialization map,

FA~IO BARD~LI: Polarized mixed Hodge structures~ etc. 333

one gets a commutative diagram (see [Ca2], Lemma 8)

0 l ff,

>/ /~ (00 ) ~* > H~(r - , > B o ( ~ )

l 0

where ~. = i . @ (--v.) and homology groups are with coefficients in Z, Q or C. As in [Ca 2], we denote by

3co : Ker O , .

Exactly like in the threefolds case (3.3.5) it is possible to de]ine a bilinear symmetric ]orm ~oc~ on 3Co by:

W, fieS~, choose ~,fleH~(r ~.a,(~)=~, ~.a,(~)=fi

and de]ine ~oc~ fl) = (5, N(fl))c,, where: (.,.)c~ is the intersection pairing on Hl(Ct), N = T - - I d with T: It~(Ct)-->.H~(Ct) the Picard-Lefschetz homology transformation.

In [C-C-K] and [C11] it is shown that

- - 2~'a~ = 8ol5~

where So is the unique bilinear ]orm on Ho(~) ]or which the classes o] points ]orm an orthonormal basis.

5.1.3. The generalized jacobian J(Co)is an extension of the abelian variety J(Co) by the torus (isomorphic to (C*) ~)

= 5~~

giving an exact sequence

0 --> 9 --> J(Co) -->- or(Co) -> 0

and an indentification 3c~ Hi(G, Q). So the bilinear form ~OCo can be regarded as a bilinear form on H1(9, Q) induced

by the skew-symmetric intersection pairing on HI(Ct, Q) @ {the local monodromy of the family}. In particular Fo~ is induced by the usual polarization on J(Ct) @ -~ the local monodromy of the jacobian bundle ~: J(C) --> A.

334 l~A3~IO ]~ARDELLI: Polarized mixed Hedge str~ctures, etc.

5.1.4. The extension

(*) 0 -+ ~ -+ J(Co) --> J(Co) -+ 0

determines an exac t sequence

o -~ H~(g) ~ ~(J(Co)) ~ H~(J(Co)) -~o

o ~ 5 " 0 ~H~(Co) - ~ ( C o ) ~ o

via which one can define an obvious weight f i l t rat ion on H~(Co, Q) coinciding with

the canonical one. J(Co) gives also a way of defining a Hedge f i l t ra t ion on H~(Co, C) in the following way : let U be the universa l covering space of J(Co) (for ins tance

t ake U ~ t angen t space of J(Co) at 0). U is in a na tura l way a complex Lie group. Consider the m a p y: U->J(Co) (so the exponent iM m a p will do). The fibre of y over 0 is the subgroup H~(J(Co), Z) and so one has a h o m o m o r p h i s m tt~(J(Co), Z) -~ U which gives also a C-linear m a p ~: H~(J(Co), C) ~ U.

The one defines ~ H I ( C o , C)----:Ker~ and ~~ C)=:Hi (C0 , C). Therefore ( , ) allows to define a mixed Hedge s t ruc ture on H~(Co) (coinciding

wi th the canonical one) and so also a dual mixed t todge s t ruc tu re on H~(Co)~ thus

de te rmin ing canonical ly a one-moti f

5c. (Z)--->J(~)o

which by [Ca 2] is s imply the Abel -Jacobi homomorph i sm .

5.2.1. I n (1.8.1) and (3.2.1) we have cons t ruc ted the jacobian bundle p : J* -->•* of our f ami ly of threefolds ~ ' : E* -+ F* as an ana ly t i c fami ly of pr incipal ly polar-

ized abel ian variet ies ( f rom now on abbrev ia ted p.p.a.v.) .

I] Vt ~ F* J( Xt) is a reducible p.p.a .v , then it is clear that there are analytic ]amilies o] p.p.a .v . Ps: As--> ~*, i = 1, ..., r~ with

dim p71(t) = gi , Vt e F * , ~ gi = g = din] J ( X t ) , Yt ~ F * , i = l

and with the ]ollowing properties:

i) ~/i, l • i • r exists a Zariski open subset U~ c P* such that, Vt e Us, p~.~(t) -~ = As(t) is an irreducible p.p.a .v .

ii) J * : A 1 x ~ . A 2 x r , A 3 ... x ~ . A , .

The proof is s t r a igh t fo rward and will be omit ted .

l~A~Io BAnDELLI: Polarized mixed Hedge struetures~ etc. 335

PROPOSITIO~ 5.2.2. -- i) Vi, l < i < r , At is an algebraic space and Pt: At -+ ~* is a morphism o/ algebraic spaces.

ii) Assume that for generic t ~ t ~* X t is rational. Then Vi, i - ~ 1~ ...~ r, there are /amilies o/ curves

L: Ct--> Z

where Z is a Zariski open set in F containing Po, with the/ollowing properties:

a) Ct is a smooth 2.dimensional quasi-projective scheme and ~t is a proper~ fiat and projective morphism~

b) Vz e Z~ z ~= Po, )~7~(z) is a reduced irreducible smooth projective curve o/ genus g~ ~7~(Po) -~ C~ is a semistable curve o/ genus g~ (i.e. it is reduced and has only ordinary double points as singularities)~

e) denoting by J(Ct) the jacobia~ o/the curve Ct over Z there is an isomorphism

~: (~J(cJ ~JIz= jnp-~(z), i = l

d) ~)z induces an isomorphism

r

~o: QJ(C~o) -~ J(Xo) i=i

which preserves the bilinear /orms ~ y3c~ and -- Vx. as well as the one-moti/s canoni- t = l

eally determined by the associated mixed Hodge structures. I n particular J(Xo) is the (polarized) generalized jacobian o/ a semistable curve Co and /or i ~ 1 ~ 2~ 3 J(X0) is the jacobian of a curve.

PROOF. -- The proof of Proposit ion (5.2.2) ~il l be divided in several steps (and recalls of general well-known facts).

STEP 1. - Yi = 1, ..., r the analyt ic family Pt: A~-->/~* determines a holomorphic map @t: P * - ~ A , where A - ~ {coarse mod~li space of p.p.a.v, of dimension gt} given by:

@t(t) = {isomorphism class of P:--~(t)} �9

On the other hand the family of threefolds n' : ~* -+ F* gives a holomorphic peciod map ~: A * - + Ag define4 in the obvious way. Since the fami ly u ' : 3~*--~/~* is algebraic and can be completed at each point Pt e F \ F * by assumption (and the local monodromy around each point Pt is quasi-unipotent) one can apply ([Gr]

2 2 - A n n a Z i ell M a t e m a t i e a

336 :FAt~IO BAI~DELLI: Pelarized mixed Hodge structures, etc.

pag. 265 and seq.) to conclude tha t ~ extends to a holomorphic map

~: F --> ~ ~- {Satake compactification of A~}.

Since _~ is a smooth projective curve and ~ is a projective v~riety ~ is a morphism of projective varieties. On the other hand the maps ~ and ~ give a commutat ive

diagram

F* ~ ~g, • .~,. X ... X ~ , ,

where /~ is the obvious m~p describing the locus of reducible p.p.a.v, with types (g~, ...~g~). Since I m # c I m R by construction and ~ is a morphism of algebraic

varieties, it follows easily tha t h ~ is a morphism of algebraic Varieties and so i ~ l

each #~ i ---- 1, ..., r, is a morphism of quasi-projective varieties #~: ~* -+ A~.

R~dALL 2. -- Vi --~ 1, ..., r let g,~> 1 and n ~> 3. We will denote by

by

Ar {fine moduli space of p.p.a.v, with level n-s t ructure}, gt

~4(~) = {universal family of p.p,a.v, with level n-structure over Ac~: ,} g l

and by u~: dr -> A (~) the natura l map. (They will be considered here as algebraic g~ g~

spaces, see [Po 1] and [Po 2]). There is an obvious map 1~: A("~--> Ag, which is gr

simply the forgetful map associating to each isomorphism class [C, ~] the isomorphism class [C], where C is an abelian var ie ty and ~ a level n-structare on it. There is also an action 0 of the group Sp (2g~, Z/nZ) on A (") and A~, is the geometric quo- gt

t ient of A ("~ by this action, so ]~ can be seen as the quotient map. Actual ly the g~

action factors through the na tura l homomorphism

,.(.,..b-,,(.,.. and the act ion 0' of Sp (2g~, Z/nZ)/{~ id} on ~('). gt

Fur thermore one has an act ion 0 of Sp (2g~ Z/nZ) on d (~) compatible with 0 g~

and with the map u~. References for these results are [Mu] and [Po 1] or [1)o 2[.

FA]3IO BARDELLI: Polarized mixed Hodge structures~ etc. 337

S ~ P 3. - Fix an index l<i<~r, and consider the family

P i : A~ -~/~*.

By [Pol l or [Po 2] it is possible to construct a finite unramified Galois cover q~: z$*-+ ~* with the following properties:

i) P* is an irreducible smooth algebraic curve,

ii) the deck-transformation group G~ of q~: F*-~/~* acts simply transitively o n t h e fibres of q~, und is a subgroup of Sp(2gi~ Z/nZ),

iii) the pull-back family ~ : ~ - + - ~ * obtained as in the following diagram

~i~ q~ ~ A~

~V* >, F * q~

is an analytic family of p.p.a.v, with a level n-structure (for a fixed n>3) . Condition iii) gives a map v,: F** -~ A (~, and a commutative diagram

~/ , Yi .4~(n)

in which v~ is a morphism because @~ is a morphi~m. The family /~: ~ - + - F * is pulled-back via v~ by the universal ~amily u~: d (~) -~ A (~) as in the following com- g~ g~ mut~tive diagram

.r

f i

and so ~5~: X,-->-~* is ~ morphism of algebraic sp~ces. One h~s the following formula:

VP ~ , (z)

338 ~FA]]IO ~ARDELLI: Polarized mixed Hodge strgctures~ etc.

The inclusion G~ c Sp (2g~, Z/nZ) defines via 0 an action of G~ on ~/r and this g ~

allows to define an act ion g of Gi on ~ (because ~ is a fibred product !) compatible wi th the act ion of G~ on -~*.

The action Z of G~ on ~ h~s no fixed points because G~ acts s imply transi t ively on the fibres of q~: P * - + ~ * .

The quotients ~/G~ and -F*/G~ exist in the category of algebraic spaces by ([Po 1] or [Po2] or by [Kn]). One can easily see tha t F~/G~ ~- ~* and tha t there is a morphism

Fi/G~ ---- 2'*

compatible with ~ . So one gets a commutat ive diagram:

F*

I t is now easy to define a map r~: A i --> Ai/G i which makes the diagram

ri

commutat ive and to check tha t r~ is biholomorphic. This allows to consider A~ as an algebraic space and p~: A~ --> F* as a morphism of algebraic spaces. Therefore i) is proved.

RECALLS 4. - i) Let g > 2 and ~ > 3 . Let ~tg ---- (coarse mod~li space of smooth curves of genus g}. ~(~) ~- (fine moduli space of smooth curves of genus g with level n-structure}. C(~)~-{universal curve of genus g with level n-structure over g

(n) ~ g }. There is an action of Sp (2g, Z/nZ) on ~(}~)" r is the quotient of ~(}~) by g �9 g g

this action. There is also a compatible action of Sp (2g~ Z/nZ) on r ). We will denote by v: C~ ~) - ~ ( ~ ) and d: ~(~)~ - ~ g the obvious maps. For these results we refer to [Mu]~ [Po 1]~ [Po 2].

if) Let V~ ~- ~ ) / { • id}, where { • i4} c sp (2g, g/ng), as considered by Oo~T and SmEE~I~I~I; (see [O-S]).

FA]3Io ]3AI~])~LLI: Polarized mixed Hodge structures, etc. 339

Let V~ V. \{se t of points parametrizing isomorphism classes of hyperelliptic curves with level n-structures}. Using the compatible actions of Sp (2g, Z/nZ) on C~ ) and on ~(~)~, o n e can consider the following situation: let C '(~-g -- v-1(~(%(~1\ \<~ hyperel]iptic ~ points) and let C (')'~ be the quotient of C '(~)~ by the action of the subgroup { • id} r Sp (2g, Z/nZ). I t is easy to see that C~ )'~ is an Mgebraic space with ~ morphism v~ C~)'~ V~, whose fibres are smooth curves of genus g with level n-structure rood { • id}. There are also obvious compatible actions of S O (2g, Z/nZ)/{• id} on C~ )'~ and on V~ ~ induced by the actions of Sp (2g, Z/nZ) on C (~) and J(~) respectively. We will need this in the sequel of the proof.

iii) In [O-S] it is shown that the natural map

(n) . 0 (n) J r " V n --> Ag

associating to each curve of the family V~ its jacobian with the corresponding level n-structure rood {• id} is an immersion.

Furthermore the family of jacobians

is

(n),O 0 J(e~ ) -> V~

(n),o o A( ,q (n)

i.e. it is pulled-back via J~ ) from the universal family

g : d ( n ) ._> ~(n) g g "

The action 0' of Sp (2g, Z/nZ)/{• id} on A(~ ) gives via the map Jr ) a natural action of the same group on V ~ (coinciding with the action induced on V~ ~ by the action of Sp (2g, Z/nZ) on JL~ )) and so also compatible with the action of

~l~l,o of recMI (4.if) 013. ~r

STEP 5. - Under the assumption that Vte 2"* Xt is rational, we know that J(Xt) is a principally polarized jacobian of a (may be disconnected) curve Vte 2'* (see [C-G]). I t follows that Yi the family p~: A~-+/~* is a family of ]acobians and by (5.2.1 i)) for generic t ~ 2"* p~.l(t) is the jacobian of an irreducible smooth projective curve.

We consider now 3 eases separately:

5a) The generic jacobian in the family p~: At-~2"* is the jacobian of a non- hyperelliptie curve of genus g~>~3.

340 ~ABIO :BARDELLI: Polarized mixed Hodge structures, etc.

5b) All the irreducible jacobians in the fami ly p~: A i --+ F* are jacobians of hyperelliptic curves wi th g~>2.

5c) gi = 1.

5a) Let W~ c F* the Zariski open set 9arametr izing irreducible jacobians of non- hyperclliptie curves and I~i = qTI(W~). Then the map ~.~: I ~ -~ A (") can be lifted g~

to a map #/: l~r,-+ V 0. We consider the fami ly of curves m/: ~ - + l ~ pulled- o. ~(~1,o _+ Vo of recall (4 if)). Thus we back via /h by the (~universal fami ly ~) v~. ~

have a fibred product diagram

~ ( n 7, 0

l ~ > V ~

e~ is a two dimensionM algebraic space wi th a morphism m, on l ~ whose fibres are smooth curves of genus g~. A simple local computa t ion shows tha t e; is smooth, and so e; is a scheme ([Kn]). I t fo!lows from the recall (4 iii)) tha t for the family of jaeobians of m~: e~ -+ l ~

J(e~) ~_ A~I~ = ff~n ~TI(I~,).

As in Step 3 the act ion of G, on I ~ and the representation G/ -~ Sp (2g~, Z/nZ)/ /{~= id} define an act ion of G~ on C~ compatible wi th the act ion of Gi on l ~ . Let e~ = : e~/G, (the quotient being considered in the category of algebraic spaces by [Po 1], [Po 2] or [Kn]). I t is very easy to check tha t : the (algebraic) space C~ is smooth (because locally over W~ is isomorphic to e;), t ha t there is a morphism n,: e~ -~ W~ (which is projective), and tha t e~ is a scheme (by [Kn]).

5b) Call Wi the Zariski open subset of F* parametrizing irreducible jacobians of hyperelliptie curves and l ~ = q~l(Wi). Using the map (2 to 1 for g>3) , ~b(~: ) --~ ~ A ('~ whose ramification is the (( set of hyperelliptic points ~>, it is easy to see tha t our assumption gives a map l~i "(~) -+J%, and therefore a fami ly of curves m~: e~ --~l~r~ pulled-back from the universal family on ~IY ~) ' ~. e~ has the same properties as the analogous scheme gotten in 5a) and via the act ion of Sp (2g~, Z/nZ) on the set of hyperelliptic points on ~( ' ) ~,, one defines an action of G~ on e, compatib]e with the act ion of G, on I~ri. Then one concludes as in 5a).

5e) In this case gi = 1, the fami ly Pi: Ai --> F* is already an algebraic family of elliptic curves in which A~ is a scheme and p~ is a morphism. I f E is the 0-section of p~, the sheaf O~(kE) for k > 3 on A, allows one to eonstrnct an embedding A, ~->Pk-I• and so the morphism Pi is projective.

FABIO BAR])ELH: Polarized mixed Hedge str@etures, etc. 341

i n each case 5a), 5b) (and for 5e) is trivial) denoting by J(e~) and J(5~) the

jacobian of m~: C i - > l ~ i and n~: 5 ' ' ~-* W~ respectively, one gets ~ commutat ive

diagram:

J(C',)

J(e~)

id

W~

r~

which defines in a na tura l way an isomorphism

STEP 6. -- NOW let T~ be a Zariski open set in Wt on which the locally free sheaf n~).Jo)| k , ( ,--, ~i]~] , k > 3 , of k-canonical relative differentials is free (in case 5a or 5b),

for case 5c) use the sheaf (n~),(O~i(kE))). Then one constructs ~ commutat ive diagram

n i

~' P-~* • T ,

Ti

and n~ is a proper, projective, fiat morphism. By the properness of the Hilbert scheme ([Ha]) we can complete the family Cilr, over Po to a family C~Iz,, where Z~----T~ ~3 {Po} which is a Zariski open set in ~ . We then consider a minimal desingularization of C-~lz ~ which will be denoted by C, and ~,: C/--> Z~ the natura l

pro jec t ion morphism. (Here by minimal we mean: no Pl ' s with self-intersection - - 1 contained in the fibres of 2,: 5i -+ Z, live on C,). 2,: C~ --> Z, is a proper and fiat morphism and we denote by

- * = = n T ~ ( T , ) n ~ . C~ -~ Z71(P0) and by C~ = 171(Ti)~ CI/~, C'

S T E P 7 . - - We consider the jscobian J(5*) of the curve 2~: 5" -+ T~. Let (see [S.G.A. 7]):

J~'J(5*) = the Neron minimal model of J(C*) over Z~ (we extend over Po);

342 FA~IO ~BAI~DEL]SI: _Polarized mixsd Hedge str~wtures, ere.

5 ' J ( ~ ) the connected component of 0 in 3/'J(C*);

J(e~){eo} = the connected component of 0 of the special fibre of J~ over Po.

We have the following facts: since Vi, l < i < r ,

J(C*) ~ J ( , I~ , ) ~ by step 5 ~ A~]~, ,

one has

r - $ r r

@J(Ci [ f ) ~ - - - @ ( A i I T ) where T=NT~. i = l i = l ~ i = l

But we also know by (5.2.1 ii)) tha t there is a biholomorphism of complex mani-

folds @ (A~[~)--~ J[~ and so i ~ l

r

@J(C**'I~") ~ J ] T as complex mani fo lds . i = l

This gives to JIT the structure of an abolish scheme over :T, therefore we consider

its Neron minimal model 3~~ over the set Z = N Z~ = T k3 {P.}. So one has i - - 1

and so

{ = 1

I~ow it is obvious tha t as complex manifolds 37'~ J (X o) (see [S.G.A. 7], p~g. 484-490) and therefore J(Xo) has an induced strnctnre of algebraic group, and ~here is an isomorphism of algebraic groups

@dso J( * I J (X ) ~, i T Po} = o �9 i = l

Since g(Xo) i s an extension of an abelian var ie ty by a torus by (2.4.7), and each 2~(J(C/I~)){po } is an extension of some abolish var ie ty by a unique connected

normal linear subgroup L~, it follows by Chevalley's theorem (see [Re]) tha t ( ~ L / i = l

is a torus and so each L~ is a torus. Then ~~ } is an extension of an abelian var ie ty by a torus and therefore by the stable reduction theorem (see [D-M])

:FAP, IO BAgDELLI: Polarized mixed Hodge structures, etc. 343

Ci has stable reduction at Po i.e. Cg is reduced and has only ordinary double points as singularities, so Co is semistable. In this condition we know that

W~ ~_ J(eilz)

where J(C~lz) is the jacobian of the curve A~: C,Iz-+Z over Z, whose complex space is constructed locally as in [C-G et al.].

Therefore we get an isomorphism

Z

Oz preserves the natural polarization of corresponding fibres over T and therefore the induced isomorphism oi the fibres over Po

qSp~ ~J(C~) -~ J(Xo) i = 1

preserves the natural polarizations of the abelian <( quotients >> of each algebraic

group. Furthermore it is now possible to see that the bilinear forms ~ Vcj and

--t0Xo defined in (5.1.2) and (3.3.5) respectively correspond under the isomorphism ~bpo. In fact the itodge-Riemann bilinear relations imply that:

W l , ~2 e O H I ( C ~ , Q) i = 1

writing ~ = ~ ~ with ~eH,(C~, Q), one has

" --1" --1" �9 = ~ (~2))~, (~, ~;)c~ - (r %), i = l

and so an easy computation shows that

~ tVcj = - - t~Xo �9 i = l

By (5.1.4), the analogous consideration for J(Xo) and the isomorphism

~)~o: 0 J(Cb ~ J(Xo) , i = l

one sees easily that Cpo preserves the one-motif defined by the du~ls of the mixed

tiodge structures defined by ~J(C~o) and J(Xo). Q.E.D. i = l

344 FA]~IO BA~DELLI: Polarized mixed Hodge structures, etc.

6 . - A p p l i c a t i o n s .

I n th is sec t ion we w a n t to show how ~11 t he p rev ious cons t ruc t i ons curt be

~ppl ied to specific famil ies of threefolds . We need u p r e l i m i n a r y

LEZ~L~ 6.1. -- Let ~" be a smooth projective/our/old and IVt}tep. a pencil o/hypersur/aees with the ]ollowing properties:

i) for generic t ~ P~ Vt is a smooth projective three/old,

ii) V o = W o u V2o u V / i s a divisor with normal crossing, each V~ being a smooth projective three/old. Let V~ 'j be the smooth sur]ace along which V~ and V~ intersect transversely and V ~,~,3 the analogous smooth curve,

iii) the generic Vt intersects each V~ transversely along a smooth sur]aee B ~ and

B = V, n Vo = :buse locus of the penci l (V,)~p, = B ~ W B ~ W B 3 ,

iv) each B ~ intersects transversely each V~ with j V: i along a smooth projective c u r v e

B i ( ' ~ V~o = V t ('~ V 0 ('~ V~ = V t (-~ V~ '~ ,

v) each B ~ intersects transversely each V~o 'z with i V: k, i ~ 1 along a finite number o /poin ts

Then alter blowing up successively B 1, the proper trans/orm o / B 2 in the ]irst blow up, and finally the proper transiorm o / B 8 in the composition o/the ]irst two blow ups one resolves the singularities o/ the rational map p: ~--+ PI determincd by the pencil {Vt}t~el , thus getting a commutative diagram

y,

P

with the ]oIlowing properties:

i) f f is a smooth projective ]our/old and ~: ~ - + P ~ is a projective proper morphism,

~q~ABIO :BARDELLI: Polarized mixed Hodge structures, etc. 345

if) a is the composition o] the three blow ups,

iii) ]or generic t e P~ X t = z~-~(t) ~ V, ,

iv) X o = 7~-~(0)= X~ (J X~ ~ X~o is a divisor with normal crossings with:

x o ~ ~_ Vo ~ ,

Xo ~ ~ {V~ blown up along the smooth curve V, (~ Vo '2} ,

X~ = {V~ blown up first along the smooth curve V, n Vo ~'3 and then ~long the proper t ransform of the curve V, (~ V~ '3} ,

Xo Xo 1,3 $1,3____ i n 3=__Vo ,

1,2 ,3 S 2 ' 3 = X ~ c ~ X o ~ ( V 2 3 '~ blown up at the points V~N V o },

c = x~ n x~ n x~ ~ m , - = - - 0 ~

The proof is easy and will be omitted. We simply observe tha t Lemma (6.1) enables one to consider a pencil (easy enough to constrnet) {Vt}t~p.C ~- and to replace it by a family (Xt}ta~, c f f to which ~1] the constructions of the previous sections can be ~pplied.

6.2. We now apply all the previous constructions and Lemma (6.1) to specific families of threefolds~ star t ing with the classical ones.

6.2.1. Cubic three]olds.

Let ~- = P~ {V,}t~p1 be a pencil of cubic threefolds with: V~ smooth for generic t ~ P~, V o = Vo ~ u V~ ~3 V~ each V o being ~n hyperplane V o ~ p3 satisfying e~ch of the properties if), iii), iv), v) of Lemma (6.1).

Consider the blown up var ie ty f f and the morphism x: f f - -~P~ with fibres

X, = :z-~(t) ~ V,

for generic t e P~,

8

Xo = ~-1(6) = U Xg i ~ l

the divisor with normal crossings described in Lemma (6.1). In our case we have:

x ~ - p3

X~ ~_ {p3 blown up along the smooth plane cubic curve V~ (~ V~ '~ E 1'~}

346 :FA]3IO BAI~DEL:LI: Polarized mixed Hodge structures, etc.

X~ ~ {p3 blown up first a long the smooth p lane cubic curve Vt and then along the proper t r a n s f o r m of V t

smoo th curve of genus 1},

%1,~ ~ p~

S1,3 ~ p~

1,2,8 $2,~ ~_ (P~ blown up along the three collinear points V~fh V o },

c = n n p 1 .

c~ Vlo ," = E 1,3

n V~ ,3 = E~,",

I t follows t h a t the jacobian bundle of oar f ami ly of cubic threefolds is p : J -+ F

wi th

p-l(t) = J ( X ~ ) = the in te rmedia te jacobian of the smooth cubic

threefold X , for generic t =/: 0 ,

p-l(O) -~ J(Xo) is the generalized jacobian of Xo, a complex Lie group of the fo rm

8

0 --> ~ --> J(Xo) --+ ~ J(X~) -+ O. i = l

Here we see easi ly t h a t :

J ( X ~ ) = 0 , J ( X ~ ) ~-- E 1'2 , J ( X ~ ) ~ E ~'3 | E 2'3 .

So in this case ~11 the abel ian variet ies J(X~) are jacobians of curves. We com-

pu t e L z. L z is the homology of the complex

z ) -+ H2(Xo(2), z ) -+ H2(Ho(1), Z).

i i For S 1,~ ~ S ~,3 --~ P~ we have H~(S, , Z) -= [C]s,,j.g (because C corresponds to a

line in S ~,~ and $1,3).

Since

1,2,3 $2,3 = {p3 blown up a t 3 eoll inear points ly ing on V o }.

calling el, e2, e3 the three except ional divisors of the respect ive blow ups, one has

H2($2, 3, Z) = [C]~, , .Z | [e l ] .Z | [ c A ' g O [e~].Z.

We now need an explici t descr ipt ion of the homology classes [ei] and [C] in X~ and Xo ~. As for X~: since this is the blow up of Vo ~ ~ p3 along E 1'2, we see easi ly

t h a t [e~] in H2(X~, Z) is the homology class of a (( line ~> on the rnling of the excep-

t ional divisor and Jell, [C] are l inear ly independent in H2(X~, Z).

:FA~I0 ;BA]~DEL/SI: _Polarized mixed Hodge structures, etc. 347

X] is: first blow up V~ ~ p3 along the elliptic plane cubic curve E~,3: let D~ be the exceptional divisor of this blow up. Then take the p roper t r ans fo lm of E2, a

and blow it up: thus get t ing X]. Le t /)~ be the proper t r ans form of D~ in the second blow up and D~ the excep-

tional divisor of the second blow up. Calling r~ the homology class of a generic (smooth irreducible) line on the (( ruling ~> of /)~ and r~ the homology class of a line on the ruling of D2, one can see easily tha t

V i = ~ , 2 , 3 , [ e J = r l - - r ~ in H~(X~,Z)

and [eel and [C] are l inearly independent in tt2(X~, Z). i t follows af ter some computa t ions t h a t

i = l

} e~(Xo(2),z) # ~ , = o ~ <(o, o, [e~- e~]); (0, o, Ee~-e~])>,, i

so L z is a free rank 2 module. Therefore ~ = (C*) 2 and dim J(Xo) = 5 (as should be expected).

we see t ha t in this case we have (with the notat ions of (3.3.7))

In par t icular

V = LQ

because (Eel--ej], C)s . . . . O, Vi, j, and so Proposi t ion (3.3.8) applies. Therefore by

Proposi t ion (3.4.1) we can compute

~,~ o, E~,-e;]), (0, O, Eel- e,l)) = - ( [ e , - - e ; ] , Eel-e,l)~,~ = 2 .

3

We evaluate the one-motif % 2 Lz ~ O J ( X g ) on one element (0, 0, [ e ~ - e~]) e L z.

;By (1.2) this is the Abel-Jacobi homomorphism, so

ei e~

o, (If., f., f.), 0 ej ej

where means f . with 0/~ = e ~ - ej and F in the appropr ia te component X~. e$ F

Now: the first os these (( l inear functionals ~> is 0 because J(X~)= 0. The second

is an element in J(X~) ~--- E ~'~ which is just

/ ) j

(=L)f" e E ~,~ Pt

where P~, -Pj are the points in El. ~ pa iamet r iz ing the ((lines ~> e~ and ej in the

348 Fa]~o B~ROELL~: Polarized mixed Hedge structures, etc.

exceptional divisor of the blow up of V~ along E ~'2. Clearly since P~ :/: P ~ this is a non-zero element in E ~,~. Analogously the th i rd integrat ion

e~

f . e J(X~) e~

and it is e~sy to see tha t

where

et ~ l R l

sr s 2tj

~j N$

and T~, Tj are the points on E ~,~ p~r~metrizing the fibres of

/ ) l "--> El '3

containing c~ and e~ respectively; while Ri, R~ are the points on E ~,3 parametr iz ing the fibres of

.D 2 --> E2 ,3

~" a r e n o n - z e r o in E1,3. incident to e~ and e~. respectively. As before bo th , Tj ~$

and E 2,8. I t follows tha t : u~(0, 0, [e~-- ej]) h~s non-zero project ion in each irreducible component E~, J of J(Xo).

The proof of the i r ra t iona l i ty of the generic smooth cubic threefold is then completed by the following:

A ~ G 1 : ~ m 6.2.1.1. - i f Xt were r~tion~l for generic t ~/~*, then by Proposi- t ion (5.2.2) J(Xo) would be isomorphic ~i~ ~bpo to the polarized generalized j~co- b ian of a 8emist~ble curve Co and L z would correspond under the i somorph i sm ~bp0 of (5.2.2) to the Z-module of its <, t ransverse cycles ~> 3r Then (0, O, [e~ - - ej]) e JL Z would correspond under ~ 0 to a t ransverse cycle 7 in 3o0, whose image in El,2 O OE~,8OE2,s under the functor ia l one-motif (i.e. under the Abel-Jacobi map) h~s non-zero project ion in each component E~, j by (5.2.2) and the previous discussion.

i t is ve ry e~sy to cheek tha t for such a cycle one has

- - ~o(Y, 7 ) > 3 �9

l~A:~io ]3AR])]~]~ZI: Polarized mixed Hodge structures, etc. 3~9

In fact the (( t ransverse cycle ~> y would l if t in ~o --= {normalization of Co} to a 1-chain f of the type

= § § § f , and =

where supp ~, j c E ~,j and supp ~Rc {possible ra t ional components of Co}. Thus t he Abel-Jacobi map on ~ gives

Since we know t h a t

Vi, j | " :/: O in E i,r

we see t ha t for each i, j 3~,j is some 0-cycle of degree 0, and 3~,,~. V: 0, Yi, j. Therefore if we compute ~VCo(y, ~) by using the bilinear form So (as in (5.1.2)), we get easily

- - ~ c 0 ( 7 , 7 ) > 3 �9

However by (5.2.2) ~bpo is an isomorphism Preserving the bilinear forms ~fc~ and --~vx~ and since qip~ carries y in e ~ - - e j , one gets

3 < - - Wa~ y) = ~x~ es, e~-- es) = 2

which is absurd.

Therefore the generic J(X~) is not the jacobian of a curve (not even reducible) and now it is easy to see tha t the generic X~ is not rat ional .

6.2.2. Quart@ three]olds.

Here we propose 2 proofs: the first one by induct ion <~ on the cubic threefold ~> and the second one using again the polar izat ion ~VXo. We have chosen to present bo th proofs to show different ways of using the basic construct ion: ba t in par t icular to show tha t the invar iant ~VXo works for most families once one has chosen a severe enough degenerat ion (so to have at least 3 dis t inct irreducible components in the abelian pa r t of J(Xo) and same class like e~ - - ej in L z in order to apply the argument (6.6.1.1)).

FIRST e~OOF. - Choose a cubic threefold V~ocP 4 such t h a t J(V~) is not the jacobian of a curve, and V~ ~ P~ an hyperp lane t ransverse to Vo ~. Then choose a smooth quar t ic threefold V c P~ t ransverse to V~, V~ and to V~ (3 V~. We con-

350 ]~ABIO :BARDt~,LLI: Polarized mixed Hodge structures, etc.

sider 5 ~ P~ and in ~- the pencil spanned by V and Vo~U V~ which w e will denote by {Vt}tep~. Here we consider V"o = O.

Then af ter two blow ups (as in Lemm~ 6.1), one gets a fourfold ~ with a morphism ~: 5 ~ -> P1 and with

X, = ~ - ~ ( t ) ~ V, a smooth quar t ie threefold for generic t e p 1 ,

Xo = z - ~ ( 0 ) = X0 ~ W X~ a divisor with normal crossings with:

Xo ~ ~ Vo ~ ,

Xo ~ ~ {V~ = p3 blown up along the smooth carve V, (~ V~ ~ V~ i.e. along a (4, 3) complete intersect ion in ps, so a curve K of degree 12 and genus 19}

and S ~'2 = Xo ~ n X~ _~ the cubic surface Vo ~ (~ V~. Therefore we get an associated jacobi~n bundle p: J - > / ~ with

p-~(t)--~ J ( X t ) ~ - { t h e jaeobian of the smooth qaar t ie threefold Xt}

for generic t e P~ and p-~(0) = J(Xo) ~ complex Lie group of the form

o ~ ~ -> J ( x o ) -~ J (X~) | j ( x D -+ o .

Here

J(X~) ~ ( the in termedia te jaeobian of the cubic threefold Vo ~}

and

J(X~) __~ {the jacobian of K } .

i n this case

~ Z 0 1 2 ~. -~ H~(S ' , Z) ----~(the pr imit ive homology of the cubic surface},

so a Z-module of rank 6. i n part icl l lar dim J ( X o ) ~ 30. In this case J(X~) is not the jacobian of a cm've and so by Proposi t ion (5.2.2)

J(Xo) is not the generalized jacobian of a carve. Hence the generic quurt ic threefold is not rat ional .

SEC0~r PROOF. -- Here choose ~ = p 4

Vt ~ - s generic smooth qas r t i e threefold c P~,

:FA:BIO BARDELL$: Polarized mixed Hodge structures, etc. 351

V~ = a smooth quadrie hypersurface c p4,

V~ --~ a hyperp lane ~ P 3 c p4,

V~ : a hyperp lane ~ P~ c P~,

sat isfying i), ..., v) of Lemma (6.1).

Then applying Lemma (6.1) one ge~s:

a smooth project ive four fo ld ,

xo ~ ~ v~,

X o ~ {Vo ~ blown up along the curve K~, = V~ n V~o '~ oi genus 9},

Xo s ~ {V~ blown up first along the curve K , ---- Vt n V~o ,s of genus 9

and then along the proper t r ans fo rm K~, 3 of V t ~ V~ '3 oi genus 3}.

Then

S ~'~ --~ X~ n X~ ~_ the quadric surface V~ '~ ,

S ~' = Xo ~ n X~ ~ the qu~dric surface Vo ~'8 ,

S2,s ~ {p2 ~ V~o,S blown up at eight dis t inct points (intersection of u quart ie and

a conic)}, C --~X~v~Xo ~ n X o ~ P l _ ~ ( a c o n i c o n V ~ 's).

Then, as usual, the generalized jacobian J(Xo) is defined by a sequence:

3

0 --> T -+ J(Xo) --> ~ J ( X o) --> 0 i = 1

and

] (x~) = o , j ( x D _~ ](KI 3), J(X~) _~ J(K~,3) Q J(K2,3) .

Again we compute L z. Le t

H~(s1, ~, z ) = Z[r'd | Z[r~] ,

~ ( $ 1 , ~, z ) = g[r~] | Z[r'~], B

/ !

where rl~ r 2 are two lines belonging to the two dis t inct rulings of $1,~ analogously '~.' $1,3, ~ p2 for r~ and r~ m and finally r is the proper t rans form of a line in V~ '3 not

meet ing any of the points to be blown up, and ei are the eight exceptional divisors.

2 3 - Anna~i di Malemat i va

352 :FA:BIO :BAI~D:ELLI: Polarized mixed Hodge structures, etc.

A lenghty bu t easy computa t ion shows t h a t L z is the free rank 9 Z-module with generators

L z : <(r~, r~, r--4e~); (O, r~--r:, 0); (0, 0, e~--e~); (0, 0, e,--e3) ; ...; (O, O, e~-- e8) ~

where

(a, b, c) e H~(S~, ~, Z) | H~(S1, ~, Z) | H2(S ~,~, Z ) .

(Here and in the sequel we will denote ~ cycle and its homology class by the same symbol).

Since dim J(X~) : O, dim J(Xo) : dim J(K~,2) ~ 9, dim J(X~) ~- dim J(Ka,~) -~ -~ dim J(K2,8 ) ~ 12, dim ~ --~ 9, we get dim J(Xo) ~ 30.

~ow we proceed as in (6.2.1): i.e. we remark tha t each element (0, 0, e~--e~) belong~ to V c LQ because Vi, j ((e~ -- ej), C)B~.. ---- 0. We compute ~x.((0, 0, e~ -- e~), (0~ 0, e~--e~.)): by 1)roposition (3.4.1) this is equal to

- - B ( ( 0 , O, e , - - e,), (0, O, e , - - e,)) = ( (e i - - e,), (e , - - e,)),,.. = 2 .

3

We evaluate the one-motif ~H.: Lz-->(~J(X~) on the element (O, 0, e ~ - - e ~ ) e L z.

(f(), f(), f (0 0 ej ej

Then as in the proof of (6.2.1) it is easy to see t ha t ( identifying J(X~) with J(K~,2))

e~ P t

el PI

where Pi , Ps are the points in KI,~ parametr iz ing the lines e~ and ej respectively in the exceptional divisor D in the blow up of V~ ~ p3 along K1,2. Therefore

e~

J ( X o) ~ f(" ) V= 0 ej

because Pi:/:P~.

Analogously in Xo 3

e~ R~ T~

ej ~1 s

I~ABIO ]3AI~D:ELLI: Polarized mixed Hodge structures, etc. 353

where R~, R~ e K~,~ are the points parametrizing the fibres of

Jg~ = {proper transform in the second blow up of the exception~! divisor

cf the first blow up in X~ > X r o}

containing e~ and e~ respectively; while T~ T~ e K~,~ are the points p~rametrizing the fibres of

/)2 -~ K2,~ (/)~ = exceptional divisor of the second blow up ) ,

incident to e~ and e~ respectively. In follows tha t :

f " va 0 in J(KI,3) and f . va O in J(K2,3) RJ s

and so J , . (0, 0, e ~ - er has non-zero projection in each irreducible abeli~n component J(K~,~), J(K~,a), J(K~3,) of the abelian par t of J(Xo).

:Now by applying the argument (6.2.1.1), one proves tha t there are t e F* for which J(X,) is not the jacobian of a curve, and so the generic quartic threefold is not rational.

I~EMAnK 6.2.2.1. - In this example we also see tha t (in general)

V LQ

In fact the element (r'~, r~, r - - 4 e ~ ) e L z c L Q cannot belong to V by Proposition (3.3.8) because, for exampl% (r~ C)~,~ = ~.

6.2.3. Complete intersection o] a quadric and a cubic hypersur]aees in ps.

In this case we l imit ourselves to give an indication of 2 proofs like in the quartic threefolds case.

FIRST e~ooF. - One can choose and fix once and for all a cubic hypersnrface 5 ~ c P~ and two hyperplanes H~ and H~ c p5 transverse to 5 and one transverse to the other, such tha t H~ n 5 is a smooth cubic threefold whose intermediate jacobian is not the jacobian of a curve. Then one chooses a smooth quadric hypersurface Q c P~ t~ansverse to 5 , Ho ~, Ho 2 and to e~ch ~- (~ H~, H~ N Ho ~, ~- c~ Ho 1 (~ Ho ~ and considers the pencil spanned by Q and H i u H~.

One consideIs the pencil {V~}~eel in ~- spanned by 5 n Q and (H~ ~ ) H ~ ) n 5 , so get t ing a rat ional map p: ~---~ p1.

354 ~A:BIO ]3AI~a)EL:LI: Polarized mixed Hodge structures, etc.

B y app ly ing L e m m a (6.1) again to resolve the singularit ies of th is map , one gets a f ami ly ~: ~ -+ P~, whose generic fibre is i somorphic to a smooth (2, 3) com-

plete in tersect ion in p5 Qt 53 57, and whose fibre over 0 is a divisor wi th normal

crossings:

Xo = ~-~(0) = x ~ u Xo ~

wi th

X~ ~_ ~ 53 H 1 = a smooth cubic threefold,

X~ ~_ {57 53 H~ blown up along the canonical curve K (of genus 4);

K = 57 n//~) n.H~ n Q~},

~,2 = Xo ~ 53 Xo 2 __~ 57 n / /~ n H~) = a smooth cubic su r face .

o ,~ 253 H i ) ~ (pr imi t ive homology of S~,2), so I t is easy to see t h a t L z ~ . I t 2 ( 3 ' n H o

t h a t L z is a r a n k 6 free Z-module and so the jacobian of Xo will be of the fo rm

0 ---> 7: --+ J(Xo) -+ J(X~) �9 J(X~) --> O.

/ to re

~hus

d i r e r = 6 , d i m J ( X o ) = 5 ,

d im J(X~) = dim J(57 (3 Ho ~ + d im J ( K ) = 9 ,

dim J(Xo) = 2 0 .

Since b y cons t ruc t ion J(X]) is not the jacobian of a curve J(Xo) is not the

generalized jacobian of a curve and so the generic smooth complete intersect ion of a quadr ic and a cubic hypersur face in p5 is not ra t ional .

SECOND P~OOF. -- We fix once and for all a smooth quadric hypersur face 57 c p5 and choose three hype rp lanes Ha, H i , H i c p5 m u t u a l l y t ransverse and such t h a t 57

intersects t r ansverse ly each intersect ion H~, H o 53 Ha, H g 53 H~ 53 H~. Then we choose a smoo th cubic hypersnr face K c p5 t r ansverse to 57 and to

each 57 n Ro, 57 n n ~ n Hi, 57 n ~o ~ n !to2 n n~. We consider in 57 the pencil {V~}mp, spanned b y 57 53 K and b y 57 53 (WlHo).:

We a p p l y L e m m a (6.1) to 5 7 and to the pencil {Vt}r thus ge t t ing a fourfold f f

and a m o r p h i s m g: f f - > P1 whose fibres are

X , ~- ~-~(t) ~ a smooth (2.3) comple te intersect ion for generic t e P~ ,

X o ~-- z- l (0) : Xo ~ u X o U X~ a divisor wi th no rma l crossings wi th

:FABIO :BARDELLI: Polarized mixed Hodge struetq~res, etc. 355

2 r - ~ X 0 =

3 X o - -

~- (3 H~ ~- a qnadr ie hypersm' face in P~,

{~- (3 H~ blown up a long the curve K~,~ = ~- n Ho 2 ~ H~ (3 K = a canonical

curve of genus ~},

{~- n Ho ~ blown up first along the curve K~. 3 = ~- r3 Ho ~ c3 H ~ c~ K = a canonical curve of genus 4, and then along K~.3 = the proper t r ans fo rm of ~- (3

(3 H~ (3 Ho ~ (3 K , a canonical curve of genus 4}.

Once again:

S ~'~ = X~ (3 X~ ~ ~- c~ Ho ~ c~ Ho 2 = a quadr ie surface in p s

S ~,3 = Xo ~ n Xo 3 ~ ~- (3 Ho ~ r3 H~ = a quadr ie surface in p3

S 2'3 = Xg n Xg ~ {~- (3 Ho ~ (3 H~ blown up a t ~he points ~- (3 H x (3 Ho ~ n H~ (3 K lying on a p lane and intersect ion of a quadrie and cubic},

C = X o ~ c 3 X g n X g = ~ - ( 3 H ~ c ~ H o 2 n H ~ _ a conic ~ - P ~ .

Then if we denote by :

/ ! l rl , r 2 a basis of H~(S ~,~, Z) with r~ i = 1, 2, rulings of the quadrie S~,~j

II l/ ff

r l , r 2 a basis of Hs(S ~,3, Z) wi th r~, i = 1, 2, rulings of the quadr ie S 1,3,

r~, %, e~ ..., eo a basis of H2(S ~,3, Z),

If/ Ill

where r 1 , r 2 are the proper t r a n s f o r m of the two rulings of the quadr ie ~- (~ H~ (3 H~

and e, are the six except ional divisors in the blow up, we get (after some computa-

t ions): J5 z is the free Z-module of r ank 8 genera ted b y the cla~ses:

f I! /It I! f! llr /If "~

((r~., r2, r 2 - -3e i ) ; (0, r~--r~, 0); (0, O, r ~ - - r 2 ) ; (0, 0, e~--e2) ; (0, 0, e~--e~); ...

... ; (0, O, e~-- e~)) ..

Now in a way analogous to the previous eases (0.2.1) and (0.2.2) one considers an

e lement (0, 0, e ~ - - e j ) ~ L z. This e lement lies in V o L e and easily, b y apply ing Propos i t ion (3.4.1) one gets:

~o((O, o, e , - e~ ) , (0, o , e , - e ~ ) ) = 2 .

As in the previous cases one can show t h a t the one-mot i f uso(0, 0, e~ - - e~) has non- zero project ion in each irreducible component of the abel ian pa r t of J(Xo). The i r ra t iona l i ty of the generic (2.3) smooth complete intersect ion is therefore p roved

b y the a rgumen t (0.2.1.1) a l ready used in the previous eases.

356 :FA:BIO :BAI~DELLI: Polarized mixed Hodge struetures, etc.

6.2.4. Smooth complete intersections o] three quadrie hypersur]aees in p6.

Let Q1 and Q~ be two smooth quadric hypersarfaces in P6 intersect ing transversely. Le t $r = Q1 (~ Q2. In ~ we consider a pencil of hypersurfaces {V~}~el where V, = ~ - ~ Q~, {Q,}teel being a pencil of quadric hypers~rfaces in p6 with:

i) for generic t e P~ Qt intersects 5 t ransversely and so Vt is smooth,

ii) Qo = Ho ~ k)H~ the union of two t ransverse hyperplanes in p6, so tha t Vo : V~o (J Vo, where V~ : H~ (~ ~- : a smooth complete intersect ion of two quadries in p5 for i - ~ 1, 2.

l~ere we choose the pencil {Q~}t~e~ in such a way tha t the corresponding pencil {V~}~p~ in satisfies propert ies i), ..., v) of L e m m a (1.6). Then, by applying Lem- ma (1.6) to ~- and to the pencil ~V~}~p1 with V~ ---- 0, we get a fami ly ~: f f --> P~ with:

X~ = ~-~(t) ~ V, = ~ n Q~ =

" {a smooth complete intersect ion of three quadrics in P~ for generic t ~ P~}

x o = = X o w X o ,

with

Xo ~ ~ Vo ~ z {~ smooth complete intersect ion of two quadrics in ps} ,

and

Xo ~ _~ {Vo 2 blown up along the curve K ---- Vt n Vo ~ (~ V~ = intersect ion of three

quadries in P t so a canonical curve of genus 5}.

Then

S ~'2 ~ X~ (] X~ ~ Vo ~ (~ V~ ~ (a smooth complete intersection of two qnadrics in p4

so a quar t ie Del Pezzo surface containing K } .

We know tha t S~, ~ ~ {P~ blown up at 5 generic points}, so t h a t

H~(S~'~' Z) -~ [r]'Z | (f~ jell'Z).=

where r is the proper t rans form of a line in P~ (not passing through the 5 points) and each e~ ~ {exceptional line of the blow up}. :Now i t is ve ry easy to check tha t

SO

Lz~--H~(S 1'2, Z)o : (primit ive homology of S 1'~)

and L z is a free Z-module of rank 5.

~~A]3IO ]~ARD]~LLI: Polarized mixed Hodge structures, etc. 357

Therefore the generalized jacobian J(Xo) fits into the exact sequence

0 ~ ~ ->J(Xo) ~ J ( V ~ ) @ J(V~) | -~0,

where J(V~o) is isomorphic tO the jacobian of a genus 2 curve for i = 1, 2; J(K) is the 5-dimensional jacobian of a non-trigon~l genus 5 curve and ~ _~_ (C*) ~. We see therefore t ha t dim J(Xo) = 14. l~ow it is clear t h a t also in this case by (3.3.7) V = LQ and therefore by (3.4.1)

W o ( e , - e~, e , - e~) = - B ( ( e , - - e~), ( e , - - e~)) = - - ( ( e i - - e~), ( e , - - e~))~,. = ~ .

Fur the rmore the canonically defined one-motif uH: Lz --> J(X~) @ J(X~) (which by Proposi t ion (4.2) is simply the Abel-Jacobi homomorphism) when evaluated on the element e~ - - e~ e L z gives

ej ej

Mow J(Xo ~) is the jacobian of a complete intersect ion of two quadries in p5 (which is V~), and e~, ej are two dis t inct lines on Vo ~ and so i t is classically well known th a t

i ' . ~ 0 in J (X~) . eJ

I f ~: X~--> V~ denotes the blow up m a p . a l o n g K c Vo ~ which gives X~, then als,,,: S ~'~ --> V~ n H~ is an isomorphism.

Fo r e, r S ~,~ an exceptional line of the bira t ional morphism S ~,* -+ P~, we denote f ~f !

by e~ = als~.~(e,). I t is easy to check tha t , denot ing by e~ the to ta l t r ans form of e~ in the blow up a and by rp the line in the exceptional divisor of the blow up para-

~! metr ized by P e K, one has e, = e, - - rp~ - - re~, where P~, and P** are the two intersection points of the line e' ~ c V o n H o wi th the curve K = V~c~Ho ~(~Q~. F r o m this description it is easy to check tha t the element

e i

f. e J(x~) ej

is ac tua l ly equal to ' 1 2 e i P i + P i

ej 1 2 P~ + ~Pj

and since f . e

0 in J(V~o) by the same reason as before, and

f " e J(K) , J )14- p2 j - - - - j

358 ~ABIO ]~ARD]~LLI: Polarized mixed Hodge str~ctures, etc.

is not zero in J(K) (because otherwise P ~ q - P ~ ~ P~ + P~ and so being PT:V= P~ i i ~

Vk~ Vs~ K would be hyperelliptic)~ this says tha t uG(e~--es)~ J(V~)Q J(V~)G J(K) has non-zero larojection in each irreducible component of the abelian par t of J(Xo).

Then one applies the argument (6.2.1.1) to conclude tha t the generic intersec- t ion of three-quadrics in p6 is not rat ional .

6.2.5. The quartic double solid.

We present here two proofs of the i r ra t iona l i ty of the generic smooth quar t ic double solid: the first one uses as main ingredient the precise form of the extension da ta and i t is essentially along the line of A. Collino's << cheap proof of the i r ra t ional i ty of the generic cubic threefold >) (see [Co]). The second proof once again uses as a basic invar iant the bilinear form ~z.. We choose to present bo th proofs to show tha t the const ruct ion developed in sections 1, ..., 5 allows one to use only the extension da ta informat ion to s tudy the i r ra t ional i ty of a given family essen- t ia l ly when rk L z = 1, and t ha t in this case the informat ion contained in the extension da ta is ana ly t ic dnd closely ~ied to the geometry of curves. I f rank JL z > ] , to use the extension da ta informat ion one needs to choose a special k ind of basis of L z to compare it wi th the corresponding basis of the (< supposedly isomorphic >> Z-module of t ransverse cycles of a semistable curve. The special kind of basis is cons t i tu ted generally by vectors in L z of minimal length with respect to FXo. On the other hand when rank L z > :l Y;x~ itself provides in general an invar iant which is of an essentially topological nature, i towever for ~Xo to be of some use a rn ther severe k ind of degenerat ion is required.

F I R S T P R O O F . - - We choose in p3 a pencil {Be}e~p1 of quart ic surfaces with the following propert ies :

ii) for generic t e P 1 Be is smooth,

ii) Bo-~ 2Qo is a smooth quadric counted twice,

iii) the generic Be intersects Qo t ransversely along a smooth octic of genus 9 which we will denote by K,

iv) B~ is smooth.

We form in the line bundle L on P~ such tha t Opt(L) --~ Opt(2) the fami ly of quart ic double solids {Ve},~l associated to the pencil {Be}, where each Ve is the quart ic double solid with branch locus Be. Clearly we see t h a t {Ve}e~p1 c P ( L O Opt). Fur- the rmore Vo ~ V~ (5 V~ where each Vg is isomorphic to Pa and Vo ~ (5 Vo 2 ----Qo. It is possible to see that

B * = V , C ~ V ~ B ~ for i ~ I , 2

and t ha t B ~ (~ B 2 = K.

~A]3IO BARDELLI: Polarized mixed Hedge structures, etc. 359

By applying Lemma (6.1), we get a family a: ~-->P~ with:

Xt -- z-:(t) ~ Vt for generic t d Pz,

Xo = z-~(0)= X~ U X~ a divisor with normal crossings with

x~ = vo ~ ~_ p~,

x~ = { q blown . p along the curve K ) ,

X~ n X~ ~ Qo a smooth quadric surface.

Therefore L z = H~(Qo, Z) ~ = (primitive homology of Q0} i.e. L z ---- < r ~ - r2} where r~, r2 are two lines in Qo giving the two distinct rulings of Qo.

J(X D=o and J(X D~J(K).

I t follows that the generalized jacobian fits into an exact sequence

0 --> ~ --> J (Xo) --> J ( K ) --> 0

and so dim J(Xo)- - - -10 .

We now compute the one-motif u ~ ( r ~ - - r~) ~ J (X~) ~ J ( K ) . This is simply

r l

f( . ) e J (X D . r~

Now, as in the previous proof, denoting by a: X~-+ V~ the blow up map, by r'=r a(r,) and by r~"= {total transform of r'r in the blow up a}, one gets:

~! r ~ = r i - - r e ~ - - r e ~ - - r e ~ - - r e ~

where rp~ is the line in the exceptional divisor parametrized by Pj e K and P~, P~, P~, P~ are the four intersection points of r' with K. /

Therefore one sees easily that:

4 y~ e~ f l i = 1

f~ ~ P i=l

Now assuming that the generic Xt were rational, one would get by (5.2.2) that J(Xo) is isomorphic to the generalized jacobian of a semistable curve Co. Since rk L z = 1, one would get that the corresponding Z-module of transverse cycles 3~. is of rank 1. So it is very easy tO see that a generator of 3e. maps under the Abel

360 ~ABIO BAI~DELLI: Polarized mixed Hodge structures, etc.

gaeobi map into

f (. ) c J(~o) ~-~ J(K) S

where ~o is the normalizat ion of Co, and S, T are two points in K (K turns out to be the unique irreducible non-rational component of Co). Since the Abel- Jacobi map in the fami ly of threefolds corresponds to the Abel-gacobi map in the (~ family of curves ~) by Proposition (5.2.2) (because they both are the functorially defined one-motif of the corresponding mixed I todge structures) and r ~ - r, is a generator of Lz, one would get:

4 4

Z Z ~ • (T -- S) in J(K) i=I i=i

where ~ means linear equivalence of divisors in K. Choosing for the sign ~= ( T - S) the ~- sign (otherwise one can interchange the

roles of S and T), one can choose r'4 passing through T and r~ passing through S, so tha t among P~, P~, P~, P~ at least one, say P~, is T and analogously P~ = S.

Therefore we would get

8 8

i "

i=i i=I

Then K has a g~ with s < 3 . Clearly since K is a complete intersection in p3, it ca~mot have a gl 2. Since its

genus is 9, i t cannot have a g~. So the only remaining case is : K has a g~. This is also impossible because the map g: K - > P 1 x P 1 given by g----g~xg~ (K, being a (2, 4) complete intersection in p3, has g14's) would map K birationally onto an irreducible curve 2 / c p1 x P1 of bidegree (3, 4) whose ar i thmetic genus is 6 (while the genus of K is 9). Therefore for generic t e PI J(X~) is not the jacobian of a curve and so the generic quartic double solid is not rational.

SECOND egOOF. -- We have to investigate a laborious degeneration. Any smooth quart ic double solid /~ is an hypersurface in the line bundle L on P~, such tha t Op.(L) = 0e,(2 ). In part icular we can th ink of L (hence of V) as embedded in the projective bundle

L c P ( 0 e . O 0e~(2)) ---- ~-.

i n ~- there are the following smooth hypersurfaces:

S~ = l m (1, o) where (1, o) ~ F(P~; Os. | Oe.(2)).

~A~IO BAI~DELLI: Polarized mixed Hodge structures, etc. 361

(In part icular So ----= I m (1, 0)),

S~o = I m (0, 1) where (O, 1) eT'(Pa, Op,(--2) OOp,)

(here we use the fact t ha t P(0p~| 0p~(2))~_ P(0p~(--2)O 0p,)). Using the natural projection p : ~- -~ p3, we define (for ~ certain set of indices i)

H~ = p-~(h,) for h, an hyperplane in p3. I t is obvious tha t :

that

Pic (Y) ~_ Z[H~] �9 Z[S~] ,

Va e / ' ( P 5 0p,(2)) [S~] = 2[H~] ~- [S=].

Fur thermore for a generic qu~rtic double solid V c L, one has V n S~ = 0 and so IV] = 4[g~] § 2 [ ~ ] = 2[S0].

Therefore we s tar t with the following construction: we choose in p3 a smooth quartie surface ~ , two hyperplanes h~ and h~ with the following properties:

i) h~ and h~ intersect transversely along a line r,

ii) each h, intersects transversely the surface 2~ along a smooth plane quartie C~ (of genus 3),

iii) r intersects transversely E in four dist inct points /1 , T2, T3, T, and 4

i ~ 1

We consider in ~- the following hypersurfaces:

Hi : p-Z(hi) for i = 1, 2 ;

So; S~ and finally a quartic double solid V whose branch locus is 2'. We have the following obvious facts:

i) each H~ ~ P(Op,|

ii) S ~,2 = : H1 (~ H~ = P(0r �9 0~(2)} = {the classical ruled surface F2), the intersection is transverse along S 1,2.

iii) V ~ = : Son H~ ~ P~ and the intersection is transverse along V ~ V ~ can aJso be regarded as the zero-section of the ruled threefold H,.

iv) V | = : S~ n H, ~_ P~ and the intersection is transverse along V ~,~ (which can be regarded as the c~-section oi the ruled threefold H~).

362 FA:BI0 BAI~DELLI: Polarized mixed Hodge structures, etc.

v) S o n S ~ = O

vi) So n Hx (3 H2 = ro ~ P~ and can be regarded as the zero-section of the ruled surface S ~,~.

vii) S~ n H~ (3 H~ = r~ ~_ P~ and can be regarded as the co-section of S ~,2.

Now the divisor in ~- ]7o = So @ Hx @ H2 @ S~ is a normal crossings divisor and i ts dual g raph is:

~co

g ~ 2

So

H2

F u r t h e r m o r e since [So] = [H~] -1- [H~] @ [S j , Vo is l inear ly equivalent to V. We consider the pencil {Vt}t~p~ spanned b y g and Vo. This gives a ra t ional m a p ~---> P~ whose base locus is

v n Vo = ( v n So) u ( v n u ( r n H,) u ( v n s . ) .

By cons t ruc t ion

Bo = V N So ~ 2~ the smooth quar t ie surface c pa we s t a r t ed with~ V (3 S~ = 0 ,

B~ = g n H~ = {is jus t the , res t r ic t ion ~> of g to Hi i.e. a surface which is the

double cover of h~ c pa b ranched along the smoo th quar t ic curve C~ c h~, in pa r t i cu l a r B~ is smoo th} .

Now to resolve the singulari t ies of the m a p ~ - - + p1, we do as in L e m m a (6.1)

i.e. we blow up first B1, secondly the p roper t r a n s f o r m of Bs, and finally the

(~ double proper t r a n s f o r m ~> of Bo. I t is ve ry easy to check t h a t we get a projec- t ive smooth four fo ld ~ wi th a m a p s : ~ --> P~ which resolves the singularit ies of ~- - + P~. Also one has

X , = z~- l ( t )= a smoo th quar t ic double solid for generic t ~ P1, 4

X, = ~-~(0) = a no rma l crossings divisor Xo = m x o with i=l

X 0 = H 1

:FA:B$0 BAI~DELLI: Polarized mixed Hodge structures, etc. 363

2 ~ {H~ blown up a long H 2 (~ H1 (~ V i.e. a long a curve K~ which is a double X0 =

cover of r b ranched a t T ~ T2, T3, T~, so in pa r t i cu la r K~ c F2 and genus

K1 = 1},

X~ ~ {So blown up first a long So ~ H~ N ~ i.e. a long the smoo th quar t ic curve C1

(g(CO = 3) and then along the proper t r ans fo rm of So (~ H25~ T i.e. along the proper t r ans fo rm of C2 wi th g(C2)= 3},

X ~ S ~ .

The dual g raph of Xo is

x~

x~ $2,4

~2,3 xo ~

where

S~,2~ F2~ P(O | (2))

$1, 4 =~ V~ ,~ = p ~ ,

$2,* ~ V~ ,2 ~ p2

$1.3 ~ fo,1 ~ p2

S 2'3~- {V ~ blown up a t the four points T~, T~, T3, I'~} ,

= C 2 ~ ro = p1. and C ~ ~-~ rr --> P~, =

i t is possible to check t h a t all the m a i n results of sections 2, 3, 4 can be generalized wi th l i t t le effort to the no rma l crossings divisor Xo in which we have 4 com-

ponents ( instead of 3) and 2 dis joint curves of t r iple points . The result is t ha t : choosing for H2($2, 3, Z) a basis /, e,, e2, e2, e~ with ] p roper t r ans fo rm of a line

c V ~ not mee t ing T1, I'3, T2, T4 and e~, e2, e2, e~ except ional divisors of the bira- ~ional m o r p h i s m $2,3-> VO, 2, then clear ly each ei--e~, e-5 z (but more e i - - e j e V wi th the not ions of (3.3.7)), therefore

~ . o ( e i - - ej, e , - - ej) = - - ( e , - - ej, e , - - ej)~ . . . . 2 .

On the other h a n d one sees as in the previous cases t h a t u ~ . ( e i - e~) has non-zero project ion in each irreducible componen t J(K1), J(CO, J{C2) of the abel ian p a r t

of J(Xo). Then one applies a rgumen t (6.2.:L1) like in the previous cases to conclude t h a t the generic quar t ic double solid is not ra t ional .

364 I~ABIO ]~A~DELLI: Polarized mixed Hodge structures, etc.

6.2.6. Hypersur]aces o] bidegree (p, 3) in P~•

Here we s t u d y a non-classicM case: threefolds which are fibred over P~ wi th Del Pezzo surfaces as f i b r e s . These k ind of var ie t ies are one oi the m a i n 4 types in ~V[ori's classification of threeiolds whose canonical divisor is not numer ica l ly effec-

t ive [Ko]. There are of course m a n y famil ies of threefolds of this type , bu t here we will confine ourself to s t udy a simple case: name ly threelolds which admi t a

m a p into P~ whose fibres are cubic surfaces. We will also assume for semplie i ty

t h a t these var ie t ies are embedded in a t r iv ia l pro jec t ive bundle. So the threefolds we will s t u d y are in a na tu ra l w a y hypersurfaees of bidegree (p, 3) in P~ • We do not know whether these threefolds are bi ra t ionM to conic bundles of some kind

nor ii t hey a d m i t a degenerat ion to some conic bundle. Le t V be a smooth hypersur faee of bidegree (p, 3) in P ~ • V is an ample

divisor in p~• and so one has the following easily computed homology groups:

H ~ ( v , z ) = o , H~(V, Z) ~ Z | Z ,

so H~,~ = H~ = 0 and HI.I(V) = C | An easy c o m p u t a t i o n (using the ad j tmct ion formula , an obvious exact sequence

and the Koda i r a van i sh ing theorem) proves t h a t H~,~ =-O. Obvious ly H3(V, Z) is torsion-free. Now for p ---- i i t is clear t h a t a smooth va r i e t y of bidegree (1, 3) is

ra t iona l and can be seen as the blow up of a smooth comple te intersect ion of two t ransverse cubic surfaces in P~. Therefore in this case h 1,2 ~-- ~0 ~ {the genus of

the blown up curve in p3}. An usual c o m p u t a t i o n using Chern classes shows t h a t for a n y p>~l ha,2(V) -~ 16p - - 6.

Since for p ---- 1 we have a l ready seen the r a t iona l i ty of the smooth variet ies of bidegree (:[, 3), we will s t udy now hypersurfaces of bidegree (2, 3) and then will app ly an induct ion a rgument .

i) Hypersur]aces o] bidegree (2, 3) in P~•

Let ~ - : P~ • and in ~- consider the following pencil of hypersur faces of bidegree (2, 3):

Vt for generic t 6 p1 is a smooth h y p e r s u r f a e e ,

V 0 = V~ ~3 V~ w V~ a divisor wi th normal crossings in which,

V 0 = {a smooth hypersur face of bidegree (1, 1) (so the blow up of p3 along a line

) c g3)}, V~ : (a smoo th hypersur faee of bidegree (1, 0)} ~ p3

V~ ---- {a smooth hypersur iaee of bidegree (0, 2)} ~ P1 •

where Q is a smooth qnadr ic surface Q c ps , Q t ransverse to ].

~ABIO BAI~DELLI: Polarized mixed Hodge strcwtures, etc. 365

I t follows:

V I , 2 1 2 o = F o r t Vo- -~Q,

Vlo 'a = V~ n V~ ~ {blow up of Q in the two points Q r t } ,

V ~'3 = V ~ (3 V 3 ~ P~ 0 0 0 =

Vol (3 V 2o (3 V 3o -----~'~ a conic _~-~ P1 .

We can choose V, is such a w a y t h a t each intersect ion

v nVo, v, v, nv nv nv

is t ransverse , the base locus of the pencil {f~}tspl then will be:

3 8

V t r ~ V o - - - - U v , ( 3 V g - - - - U B i wi th B t = V , R V ~ / = 1 1=1

and each B ~ is a smoo th surface.

B y app ly ing L e m m a (6.1), we get a smooth project ive fourfold ~ wi th a mor-

ph i sm ~: ~ -+ P~ wi th X , = vr-x(t) ~_ V, for generic * ~ P~ and Xo = ~r-~(0) = X~ u t3 Xo 2 u X~ a divisor wi th normal crossings wi th

X 1 ~ V x 0 = 0 ~

X~ ~- {V~ blown up along Kx,2 ---- V~ n B ~ = V, n V~ n V~, a canonical curve of

genus 4},

Xo 8 ~ {Vo 3 blown up first a long K~,~ = Vt n V~ (~ V~, which is a genus :[4 curve and then along the proper t r a n s f o r m of K2,3 = V~ n Vo 3 n V~ a p lane curve

of genus ] } .

As usual :

= - - 0 =

1 , 3 S 1'3 ~ Vo

S~'3_~ ~V2o '~ blown up a t the 6 d is t inc t points Vo ~ n Vo ~ n V~ n V~ = 2~3 ----intersection of a plane conic and cubic on V o } ,

~ X z t 3 X a ~ = . C ~ X o o o = a c o n i c ~ P ~

Now we need a more precise descr ipt ion of S~'a ~ V~ '~. Let P~, P~ the two dis t inc t r" the lines of the two dis t inct rulings of Q pass ing th rough P~. points ] (3 Q, r~,

~t ~u I n the blow up V~o 'a --> Q let el, e~ be the except ional divisors and ri, r~ the p roper

366 FABIO BA]~DELLI: Polarized mixed Hodge structures~ etc.

! g

t r ans form of ri~ r~ respectiveJy. Then the pic ture of Vo ~'3 is the following (we denote by z the I~roper t r ans form of a plane section passing through ~Pl, P~)

f

~ ; ~Y ~Y ~ f

It is clear that r~ ~- r 2 : z : r~ @ r 2 and that

~IV ~VY ~ f ~ r r~ = r2 @ e 2 - - e l ~ r 2 : r 1 @ e l - e 2 .

Therefore H~(S 1,3, Z ) : Z[el] G Z[e2]O Z[~] o Z[~] and V~ '~ can also be thought of as F~ = P ( O p 1 0 0 p l ( 1 ) ) blown up at two dist inct points lying on the zero- section So with (So)2 = 1.

Analogously denoting by kl and k~ two lines in the two (( rulings >> of $1,2~ we have

H2($1, ~, z ) : [ k l ] . z | [k2].z

and finaHy

where g is the proper t rans form of a line in V~ '3 not containing any one of the points which are blown up, and t~ are the exceptional divisors. A lenghty bu t easy com- pu ta t ion shows tha t

L g = <(k2~ ~ , g - - 3 t l ) ; (0, e l - - e2, 0); (0, 0, t~--t~); (0, 0, t2--t~); ...; (0, 0, t s - - t6)}

so t h a t L z is a free rank 7 Z-module. We remark tha t each element (0, 0, t ~ - - t j ) e V r Lq and therefore

~fxo((0, 0, t~-- t j ) , (0, 0, t~ - - t j ) ) : - - - B ( ( 0 , 0, t - - - t j ) , (0, 0, t~--t~)) : -

: - - ( t i - - t j , t~--tj)~,~ : 2 .

FABIO BAI~I)ELLI: Polarized mixed Hodge structures ~ etc. 367

Now the generalized jacobian of Xo, J(Xo) fits in an exac t sequence

O ---> ~ --> J(Xo) --> J(K~,~) @ J(K~,~) @ J(K~,a) --> O

where

dim ~ = 7 , d im J(KI,~) ~- 4 dim J(K1,8) = 14 d i m J (K . ~) = 1

so t h a t d im J ( X o ) = 26, as expected. Now one considers the value of the one-mot i f uH~ on the e lement (O, O, t ~ - tj).

E x a c t l y as in the previous eases it is easy to check t h a t u~(O, O, t~--tj) has non-zero

project ion in each irreducible s u m m a n d of the abe]inn p a r t of J(Xo). The irra- t iona l i ty of the generic smooth hypersur face of bidegree (2, 3) in p l • is then

conc luded b y the a rgumen t (6.2.1.1).

ii)] Hypersurlaees of bidegree (p, 3) with p > 2 in p~•

Now we app ly an induct ion to conclude the i r ra t ional i ty of the generic smooth hypersur face of bidegree (p, 3) in p~• The rcsnlt is p roved for p = 2.

Assuming it is t rue for a given p, let us prove t h a t i t is t rue for p + 1. We consider a pencil {Vt}t~p. of hypcrsar faees of bidegrce (p + 1, 3) in p~• with Vt

smooth for generic t e P~ and Vo - - Vo ~ u V~ a no rma l crossings divisor in which

Vo ~ = a smooth hypersur face of bidegree (p, 3) such t h a t the jacobian J(V~) is not t h e jacobian of a curve ~

V~ = a smooth hypersur face of bidegree (1, 0) so V~ ~ p a .

Obviously V~ (~ V2o = V~o '~ is a smooth cubic surface. We also wan t : the generic V, t r ansverse to each V o and to V~o '2. Then one sees

ve ry eas i ly t ha t a f ter resolving the singularit ies of the ra t ional m a p P1 • pa _+ p1 given b y the pencil {Vt}~p~, one finds ~ smoo th projec t ive fourfold ~ wi th a m a p

z : ~ - - > P~ such t h a t

X t

. X o

X t N 0

X ~ ~ , 0

~-l(t) "~ Vt for generic t e P~

~ - ~ ( 0 ) - Xo ~ u Xo ~ is a divisor wi th normal crossings and

Vo ~ ,

tVo ~ blown up along the curve K Vo ~ n V o n V~ of genus 10}.

H2(S , , g ) . F u r t h e r m o r e $ 1 ' ~ : X~ (~ X o ~_ Vlo '~ a smooth cubic surfacer and L z - ~ o ~

24 - . A n n a l i d l M a t e m a t i e a

368 ~ABIO ]~ARD:ELI,I: Polarized mixed Hodge structures, etc.

I t follows that the generalized jaeobian J(Xo) fits into the exact sequence

0 ~ ~ -~ J(Xo) -> J(X]) | J(K) ~ 0

and since J(X~) ~--J(V~) is not the jacobian of a curve, J(Xo) is not the generalized jaeobian of a curve, therefore by Proposition (5.2.2) the generic Vt is not rational. So the generic smooth hypersurface of bidegree (p, 2) in P~ • is not rational for any p>2. Q.E.D.

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