USIYERSITY OF ALBERTA

Normal Rinctions and their Application to the Hodge Conjecture

Jason Colwell fJ

,A t hesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science

in

Mat Lematics

DEPARTSIEXT OF bI.4THE'VI.ATICAL SCIESC'ES

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The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantid extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otheMrise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. ABSTRACT

The Hodge Conjecture states that for a projective algebraic manifold .Y. any rationai cohornology class of type p: p is the class associated to a rational algebraic cycle of codimension p.

This thesis explains part of the Griffiths program for proving the Hodge

Conjecture. The task will amount to proving that the image of the class map

contains Prim"*m(S.Q). We retrict the domain of [ j to O(S/P. Q). the preimage of Primm-"(.Y. Q). The Griffith program then factors t hc funct ion int O the following cornposi t ion:

The approach is to prove the Hodge Conjecture by establishing the surjec- tivity of Q and that in& > PrirnmVm(X,Q).

Our object here will be to show that Prirn"*m(X:2) = i& in PrimK(,Y:Z j. ACKNOWLEDGEMENTS First 1 would like to thank rny supervisor. Dr. James D. Lewis. who ha5 taught me most of the algebraic geometry I know, through courses. Iiis book. and persona1 instruction. In this last area he proved ta be very generous wi th his tirne. and willing to discuss my study at unscheduled meetings. As well. 1 am grateful for the good teaching received in courses 1 have taken in the Department of Mathematical Sciences. Background in differential geonietry and algebra has served me well in learning the materid required to wite this thesis. I am thankful to my parents. for their encouragement and support through- out my nork on my b1.S~. Finally. 1 aould like to thank Elizabeth. whose example entoura,.neci me to keep working hard. NOT TO BE USED AS EXAMPCE OF FORMAT CONTENTS

Introduction

Definitions and Prelirninary Results The setting. and two isomorphic sheaves. 18] The complements. The splitting of the Gysin map. rd -4, isomorphisrn of sheaves.

u nInterrnediate Jacobians.

Eu -4 connection. BThe Cech cup product. A diagram of pairings.

An esact sequence.

SI Two sheaves to be compared larer. EN -4 larger algebraic variet- An upper-sernicont inuous funct ion.

The Argument is true

Example Applications

Index of Sotation INTRODUCTION Let -Y be a projective algebraic manifold of dimension 11. Of de Rhaiii co- homology Hi(-Y ). t here is the Hodge decomposit ion HL(S ) = $,+,=, H'.T( -Y). where HP*q(.Y) is the space of closed harrnonic forms of type (p. q). The dr- coinposit ion sat ifies Hpiq(-Y) = HPP(X). where - represent s coiiiplex coiiju- gation. -4 rational algebraic cycle of codimension p is a suiii z = rf=,r,;,. where ri E Q and the zifs are irreducibie codimension-p subvarieties of .Y. Integration of a form 4 E E~^-~~(X)over z is defined

Lié know this integral exists because. using a desirigularizatioii 1) : i - s. of zi. we have $,l,(,,,,ng ;= J, p-d. tvhich esists. This iritegratioii de?;rrrirls to cohomology because J,&,) dd = Jzd(pœ;) = J3Epm~= O. Tc) ail!- such cycle 2. we ma? associate an element of H2p(X). If :is the cycle. rlir associated rlass < is chosen so that the linear functionals

and represent the same element of Hz"-'P(S)=. Poincaré dualit!. HL"-'P( -Y)= 2 H2p(S)guarantees that such a cari be found.

The Hodge Conjecture (the version considered hew i. knowii a:. H~dge~*~(.Y.Q) ) states that for a projective algebraic manifold S. aiiy ratio- na1 coliomolog' class of type p. p is the class associated ro a ratioilal algebraic cycle of codimension p.

iié will show that in provii~gthe Hodge Conjecture ail assuriipt ion nia? be niade about p and n. and that only part of HP-P(SQ) := HP.*(Xj TI H2p(S.Q) need be considered. To the end of justifying these assuiiiptioiit;. which tvill be stated later, we will state the Lefschetz theorenis. Let .Y. being a projective algebraic manifold. have Kahler forn, s E E1.'(X).the space of (1. 1)-type forms on X. Then descends to cohomology. H'(X) + H'(X). J may be chosen so t liat L is dual to the map on homology obtained by intersect ing wit h a srnoot h h!-perplarw section.

The prirnitire eoh oniology is defined to be the kernel of L'+' : H li-' (-Y1 i H"+"~(X)?written Prim(X), or primk(x)in the case of the above restric- tion to HqX).

Lefschetz Decompostion

Strong Lefschetz Theorem

is an isomorphism. 0 The situation is depicted in the following diagram:

~nrnO(x) The left side of the above diagram couid have been writ teii: The Lefschetz decomposit ion is compatible wit h the Hodgr cl~roriiposi- tion. Yamely, set

Then prirnk(~)= $p+,=e PrirnP-q(S).For p + q < n.we also have t tir St rong Lefschetz Theorem. which states that

is an isomorphism.

Weak Lefschetz Theorem Let 1. be a smooth hyperplane section of &Y.with inclusiori riiap J : 1- s S. Then j. : Hi(Y.Z) -t H[(X.Z) is surjective for 1 = r2 - 1. aiid bij~îti\-e for 1 < n - 1. (The correspponding statement for rational horiio1og~-follow:. from this.)O The strong and weak Lefschetz theorems allow us to rediir~the Hoclgr Conjecture to a special case.

We rnay assume 2p 5 rz. Suppose 2p > n. This is equivalent to 2(r2 - p) < r2. By the strong Lefschet z t heoreni ive have as an isornorphistii

Dual to L'P-" is the map on homology which consists of taking 2p - 11 h~-- perplane sections: H2p(-V-Q) + H2(n-p)i-v-Q) Sow suppose we know H~dge~-~*"-~(.Y.Q) and wish to find an algebraic cycle giving a certain class 7 E HPsP(X.Q)c H2p(X.Q). Find { = (L'P-")-'(~) E Hn-pmn-P(X. Q) c ~~("-p)(rj.Q). By our supposition find z E C'"-P(S) :I. Q with < = [z].Obtain y E CP(-y) Q from ,- by taking a siiiooth hyperplaii~ section 2p - n times. This process is dual to L2p-". so that

and y is a codimension-p algebraic cycle dual to q. Thus HodgePeP(X.Q) follows from H~dge"-~."-~( X. Q). Rerall '2(n- p) < n. So we may assume 2p 5 n. O

We May Assume n = 2p. Suppose Zp < n. Let F be an (n - 2p)-dimensional manifold paranietriz- ing an j n - 2p)-dimensional family of complete intersection subvariet ies eacli obtained by n - Zp successive smooth hyperplane sections. rovering S. For any f E F. we have the corresponding (2p)-diniensional intersection subvari- ety -Yj with inclusion jj : -Yj ~f X. SOWsuppose we know H~dge~.~(S~.Q) for smooth Xf and wish to find an algebraic cycle giving us a certain class 9 E NP-P(S.Q). We know that for any general f E F. j;(q) E HP.P(.Yj.Q) is algebraic. By HodgPP(XI),Q) find an algebraic cycle rj of codimensioii p. Hilbert scheme arguments show easily that as f varies in F. s! traces out an algebraic cycle :of codimension n such that jj(>l)= iir] = ji([.-j).Bj- the weak Lefshetz theorem j; : HP+(X.Q ~iH2p(.Yj. Q) (any f E F u-itli -Yr smooth will do) is an injection and therefore [z]= rl. \te liave foiind rlir desired algebraic cycle r. Our assumpt ion of H~dge~.~(S,.j iras sufficiciit to obtain H~dge~'~(S.Q). So ive ma>-assume -Y itself has diiiiension "p. C -4 third assumption can be made.

We Need Consider Only Primmsm(S. Q) C Hmsm(S. Q). Let 1' be a smooth hyperplane section of -Y with inclusioii j : 1- - .Y. By the projection formula. Ive have the cornmutati\*ediagrain

Hn-2 (X.Q) 2- Hn(S. Q) (2 by weak Lefshetz) 1jm /

where j. is obtained via duality hmthe pushforward on 1ioiiiolog-. To ser this! let (! ) be a pairing: then

j. jm({)= (j.jœ(c). .YIx

projection formula= j=(jW7JY-V)X

projection formula= j-R. J-(Y))x Thus j, O j' = L. From the Lefshetz primitive decomposi t ion we know

Hn(X. Q) = L Hn-Z(X, Q) S Primn(X, Q)

By the above diagram we obtain

Since the Lefshetz and Hodge decompositions are compatible. we have (as- suming n = 'Zm)

Son- suppose ive know ~od~e~-'.~-'(1: Qj and wish to find an algebraic cycle giving a certain class rl E H"."(X. Q). Let q~ r j.( 0 :in the abm-e direct sum. By supposition. we can find a :E Cn-'(Yj 2 Q with

Xenow know that the success of findjng an algebraic cycle giving '7 depends on Our finding an algebraic cycle giving 5. For if we can find I E CmiS)2 Q with [il = ( E Primmm(-Y.Q). then. considefing z as E Cm(X)8 Q via j : Y + X. we have our desired algebraic cycle z t x:

Bp induction, then. it suffices to consider only Primmvm(X.Q) c Hmmm(X.Q). (The Hodge Conjecture for Primmm(X.Q). in conjunction wirh Hodgem-la"-' (-Y.Q). implies Hodgemvm(X. Q). ) l2 Ke oow outline the Griffiths program for proving the Hodge Conjecture. We have seen so far that our task arnounts to proving that the image of contains Primmvm(X.Q). So we may restrict the domain of [ ] to the pre- image of Primmqm(X? Q). Note t hat Primm." (X. Q) = ker L : Hm."( X. Q) i ~m+~.m+i(X. Q). It follows t hat the pre-image of Primm."'(S.Q) coiisists of cycles whose generic hyperplane sections are homologically ~quivaientto O (the group of which is denoted 8(Xt)). We will denote the pre-image of Primm~"(S.Q) by O(S/P. Q). So we are interested iri pro\-ing that tlitb image of contains Primmtm(X, Q). Most basicdly? the Griffit hs program split s [ 1. on Q(X/P. Q). into a composition:

Each elenient of O(X/B. Q) is fibered by hyperplane sections over P. a. tlir -4bel-Jacobi hornomorphism. is defined fiberwisc. on smoot 11 fi bers. So we are actually only interested in t hose cycles in O(X/P. Q)for whicli

- establishing the surjectivit!. of and that i& > Prim".". Its surjectivitj- is the staternent of **Poincaré'y existence theorem'. which is conjectured to be t rue in general. ii-hat is not true in general is the fiberwise statement ".lacobi inversion". the surject ivi t~r of 4 Jm(Xt).which would imply PoincaréSs existanre theoreni. We will. in this thesis, demonstrate that Primm-"(-Y$) iniE. The maps iP and Z dlbe defined later. The object here will be to show that Primm'"(X: Z) = imE in PrimL(S:Z). ivhen X is a surface! then since Jacobi inversion holds for curves. we liave the Hodge Conjecture for p = 1, known as the Lefschetz (1.1) Theorem. This case is the subject of the Appendix. Jacobi inversion holds for 0thcases as well. a few of which will be discussed in the section Example Applications. If Q is replaced by Z in the statement of the conjecture. this iiitegral versioii. known as H~dge~*~(X,Z),in general is false for p > 1. A rouiitt.resaiiiple was found by -4tiyah and Hirzbruch. This thesis is cornposed of two main sections. The first rontains defini- tions and prelirninary results: indexed 8,Fl... . The second is the niaiii argument. Its strucure is found in the diagram ivhich precedes it. and its steps are listed in the table of contents. The diagrarn is interpreted as fol- lows. The statement in a box is implied by the conjunctioii of the state- rnents in boxes below and attached to it with lines. For exaniple a follow from and B: and from statements a through m. The statenieiits ~.~.fl.~.@1~1~1:!1(17~C%-Iare the most basic staternents. ini- piied by no others in the diagram. The truth of is the object of the whole argument. Yotation found in the diagram will be explaineci in the firsr section. DEFINITIONS AND PRELIMINARY RESULTS

The sett ing, and two isornorphic sheaves. *Y c Pi'' is a non-singular projective vaxiety of diniension ri = .>ni. Tliere is a linear pencil of hyperplanes given by ~Q:Q + = O where [ao.nl] E P. (zo... .--.y)E k". and in local coordinates f = ol/oo. ij-e set S,as the intersection of -Y with the hyperplane determined 1)- t. LVe ma>- choose this L~fshet:pencil(the collection {& : t E P)) (see [.Andreotti k Frankel]) so that Xiis non-singular for all but finitely ma- points. for al1 t E 1- := P\E, where C is the finite singular set c P. For 1 E E.the sin- lar section -Yt has one ordinar3 doubk point.given in local coordinat es by 71 1 . . . . . x,) E P" : x:=, r; = O> . as its only singdarity. 11% niay also ar- range the pencil so that -Yo. -Y,. and the base locus D = n,,? .\; = .Y ,? {z : -0 = 11 = 0) are sniooth. Let be the blowup Bo(S)of -y nlorig the boar OCU US D := nt,?S, = A' n . . .z.v) E P.' 1 20 = Z, = O). Let 7 Lw t lie -- 1 obvious projection k; -t P. Let Y = f (/-). and f = 7 iy. Lié have

Proposition The two sheaves Or+:I: Rif.@ and Rl f-':,.,(. o\-er [- are isoniorphic. O Before giving a proof. we give the definitions of Or-Z Rif.@ aiid R'j. R;-,(. . The latter is derived from the presheaf Cd- ct H'(f -'( L).ntil- ) aiid R' f.C is derived from the presheaf b' r, Hz(f -'(V).C). (H denotes h>prcohoniol- og!= 1 Rriefly we review the definition of hypercohomology [Griffith h. Harris. p.4461. Let ;CI be a cornplex manifold, and U be a cover for JI. Let (A'.d) be a complex of sheaves on &1 with differential d. CP(U.A?) is the group of cech pcochains with values in dg. The two operators

satisfy h2 = d2 = O'dh + bd = O. Consequentl- we have a double cornples and an associated single cornplex. denoted (Cm(U).D). (Here c" = @,,,=, p.'; and D = d + 6. so that D2 = d2 + d6 + 6d + J2 = d6 + dd = O.) If iw refine the cover to Ut < Li. we get mappings

H'(CW(U))I H'(Cœ(U')). So. rrvemay make the definit ion of hypercohomology: H'(M. A') := Iim- H'(C'(U)o D). If

Proof First. nt,[. is the sheaf of relative holoniorphic difkrriitial 1)-foriii-. defined to be the cokernel sheaf in

Consider the following two complexes of sheaves on 1'-.

wi t h the relative different ials.

is a trivial complex with f'Oc being obtained from the presliraf

V ci {h O f 1 h holomorphic on I-}.

Sow fix. for a tirne: the open subset C.' of I*. Let CI' = f-'( l*j. The natural inclusion f '0;c RF,[- is a quasi-isomorphism by the holoiiiorphic Poincaré lemma. Thus it induces an isomorphism on hypercolioniology:

Examining H9(CKf-0;) more closely [Griffiths 6: Harris. p.41(i]. IVP obsen-r the existence of the spectral sequence abut ting to H-(I4 .. f 'Of-).ivir li the pt h cech cohornology of IV with respect to the qth cohoniology stwaf of the complex f'Otr (see Grifiths k Harris. p.116). LVe have

so that

This shows the spectral sequence 'El.c;- to be trivial. degeneratiiig at the second terrn. \Ve conclude

Combining t his wit h the previous isomorphism H'(Ct: 2 HS(I-I:f 'O;-) gi ves H=(KQF/[*) % H=(w.fROC)

ne non- cease to fix CI c I'. The preceding isomorphisiii indicates ail isomorp hism between the pres heaves given bj.

and I*-i Hz(f-l(I.-). flOL.) and t hus between their respective sheaves R'fœRFlc- and R'/.( fRO(.). Son- let us examine the stalks of the sheaves Rf.(f 'Or-)and Or.'1 R'f.@ over I-. The latter has stalk :3 H1(Xi.@) over t, and the foriii~rlias stalk HZ(St.fœOc lx,) But lx, is (from the definition of fm(3<*)a constant sheaf. 2 Oc,,. Consequentl. Ive can "factor out" f 'O1- ta obtaiii tlir isomorphism of stalks

This shotvs R' f.( TOU) 2 O[-C R1f"@ We already knotv R'~.Q;/, Z R' f.( f'oC-) Combining the last two isomorphisms yields the desireci result. O

The cornplernents. Recall that Y was defined to be BD(S). This is the subset of -Y x P giveii by {(x. S) E X x P 1 x E -Y,). and is also the graph of the liiiear iiieroiiiorpliic projection x * P [z] H [+ zl]. - We define the eomplements to be the manifold := X x P -Y = {(r. s) E X x k 1 x 4 X,). We will be working with the following coniniutatiw diagraiil and its cohomology.

Here.1. i. r. are inclusion rnaps. 7. as- pce~*iously - defincd. is -2 jy;. and ive define g = z2 IF. p = 71 IF. The map i : -Y,i 1' is the fibcr iiicliisioii. -. . where k o ! 1s given erplcitly by r ci (x. t ).

The splitting of the- Gysin map. : -Y the .As before. let p 1- be the projection.. - being restricrioii of rl : -Y x P + -Y.and k :y+ X x B as above. CVe mish to examine the splitting of the Gysin niap P. : ~"(r)i HC+'(X x P) under two decompositions. Mé start by present ing tlieni. The Künneth decomposition of Xn+'(X x P) is. usina H1(P)= O and HO(B) a NZ(P)z @. H"+~(x' x $1

The projection prl = D x B -t D was omitted from the diagram in @ to preserve commutivity. (To see that conimut ivity would iiot Iiold ware pr, inciuded. observe that for s # t and an? x E D. 50 1 o pr,((r.s))= is. t)$ x.5) = ((x.S. From [l].[Z].there is a splitting

\vit h the isomorphism given by p* + r.pr;. .Uso. Hn(S)aiicl H D) are orthogonal under the cup product 1-11. Lemma The Gysin map k. splits accordingly:

O Proof Sotice that P. op- = k. O k- O 7;.using p* = (;;, O k)' = k' O 7;. Bx the projection formula. k.kœ is the cup product witti t lie futiclanient al class of T in H2(.Y x P). This class decomposes into ; ::: [PI+ [-Y] :t: tindrr the Iiunnet h decomposi t ion

where J is the hyperplane class and 7 generates H2(P). So k. is "- id ? L on Hn(-Y)(see the preceding diagrani j. ~rliereHG ( -Y ,I is identified with its isomorphic image in H~Y). Sow to examine the effect of ka on the part of ~"(r)isoriiorpliic to HL-'(D). We have the nori-commutatiile diagram

Son-cornrnutivity was dernonstrated in B. Though it is not roniinut at ive. the cohomologv diagram is. because T. 01. = (701). does not depend on choice of t. Sow the component map Hn-2 _t Hn(X)3 Hn-2(D) E H~Y)is given by r. O pr; and th^ projection Hn+*(X x P) Hn(X)E Hn+2(X)-t Hn+'(S) by -1.. So tlir map we are looking for is

which is equal to i. c 1. from the above diagram. né thus obtain the desired split ting. O .An isornorphism of sheaves.

Proposition Mé also have an isomorphism. for g = g JF-~(lr)obetween Or-:: R1g.@ and R'~~.R',~,.,,.(log Y).O Proof Mè basically repeat the proof of the previous proposition.- rscept tliat ive begin wit h the following two complexes of sheaves on C' := C n -Y x ( *.

is the cornplex of relative differential forms. wit h the relative diRerelit ials.

is the trivial complex wit h gDOc being obtained from the preslieaf

V -? {h c g 1 h holomorphir on I-).

The exact argument we used before will not work because IVP do not ha\.e a quasi-isomorphism between the cornpleses. However. t his can easil~.Iw remedied by restriîting the sheaves in bath cornplexes to -Y r i *'+. TIleii ive caii apply the holomorphic Poincaré lernma and proceed as bcfore. C

Canonical extensions. Now we define some bundles (which of course give rise to sheaves) o\.er I*. Let 'H = Rn-'f.R;,, 2 Or 3 Rn-' f.@. Let E; = Rna2.R>,,r,,(log Y) 2 0,- z Rng.C. The Hodge filtrat ions on Qt-/l.and 91, L-lL.(logY) are { FPO~,~.} and { FPR; xcIr(log Y)). where FPRFIL- is the complex

and FPR>, Z.,U(log Y)is defined similarly. Let P = Rn-'f. FPR>,Lc R. and 3 = Fm. Let EP = Rna2.FPR&,L.(log Y) c K. and E = Çm+'. ive wish to extend these bundles over al1 P. For this we need t lie Canonical Extension Theorem [Deligne. p.911 Given a bundle V on 11' wirh integrable connection and unipotsnt rtioii- odromy. there exists a unique extension to al1 of L such tliat (i) If holomorphic sections of and Ü are expressed in ternis of iiiult i- valued horizontal bases. the coefficients grotv like powers of log 1 f l. (ii ) The connection rnatrix of V. expressed in terms of a basis for C. Ilal; logarit hmic pales: and the residue of the connect ion is nilpotent . Furthermore. the construction V i V is functorial for horizontal maps. is exact. and is compatible with Hom. tensor products. and exterior poxvers. O

(The connection matrix is the matrix 9 of 1-forms E Ei-(V)that in ternis of the aforement ioned lodal frame {e1. . . . E,) for V.sat isfies DF;= O,, C, . j The integrable Gauss-Manin conneetion for R and K. and the -utiipoteiit monodrorn y transformation. will be disrussed in the next section. -/ F i. By making the canonical extension at each point of S. Ive obtaiii exteri- sionc and of 31 and K. Correspondingly. we set

[- Intermediate Jacobians. It follo\r7s from Hodge Theory that the image of the coiiiposite niappiiig

for smooth -Y,' is a lattice. The cokernel J is t heti a comples torus. îalled t lie interniediate Jacobian of .Ys.hing duality the definition rail l>e rewit teii

rvliere ' denotes the dual space. Let @(-Ys)denote the group of codimension m algebraic- cyîl~son -Y, which are homologically equivalent to zero. There is an -4bel-Jacobi homomorphism defined by a process of integration. Specifically. for < E O(X,).by the definition of B(X,) can find a chah q( of reaI dimension Pz - 1 in X, with ( = aq. Define

The inrermediate Jacobians fit together to form an anal~icfiber epace over L'. for which the sheaf of germs of holomorphic cross-sections ic siyen b 1- 3- J := Rh-, f.z- Il;e want to extend J to al1 B. -An extension 7 is defined by the short exact sequence

To interpret the sheaf 7. we examine the local situation around one singular point. Suppose {el.. . .cg) is a basis of r(~.Fjover f(l.OA).Let {ui.. . . ui,} be the rnulti-valued sections of R2"-' f.Z determined by generarors of H2"-'(S:. S '! for some t E A. with the first. say k: elements spanning the invariant sub- space. Using the map j.R2"-' f.Z i 7 from the previous exact sequence. {u,. . . . u~~}can be considered multi-valued functions 5 -. C. The matris [uj(ei(t))]is. for t E A', the pcn'od mate of X,subject to the @en choice of bases. The columns of the period matrix determine a lat t ice in P. and J(X,)is the torus obtained by taking the quotient. For t = O, only the first k columns need be considered? so Ive divide out only these vectors. Weobtain a quotient space J C Cg x 3 with projection ;: : J -t A. The sections of 7; which are locally liftable to holomorphic mappings 7 -t Q-. The fiber ~~'(0)is called the generolkrd intermrdiafe Jacobian of Xo. There is a monodrorny weight filtration on H = Hz"-'(X1) (for fiaed t E 4'). a sequence of Qvector spaces (H = WZn-? > W2n-3 > -.- > O}. The monodromy weight filtration provides a rnized Hodgc s tnicture on Ho (the fiber of 'Fi over O). A mixed Hodge structure consists of (i) -4 finite increasing filtration

called a weight filtration (in this case the monodromy weiglit filtratioii~ (ii) A finit e decreasing filtrat ion

called a Hodge filtration (in this case { fP Furt herrnore. the induced fiitration on the gradai pieces kt: /lt;-, :

musi define a Hodge structure of pure weight 1 on Wi/Hi-i. Thar t lie nioii- odromr weight filtration provides a mixed Hodge structure oti Hu is prmd in [Schmid].

-4 few words first about the monodromy transforn~atioii T 0x1 H. Its effect on any (2m - 1)-cocycle in Hz"-' (-Yt) is that of traiislat iiig it t hrougli different sections .Xs as s E B circles once counterclockwis~around O and returns to its original value t.

By the Monodromy Theorem ([Katz].[Landman]) t here are intrgcrs p > O and q 2 O such that (TP- l)~+l= 0. \\é niake the further assumption that T is actually unipoteiit. tliat p = 1. \\è could always arrange this by passing to the finite covering of 1' given by Set .\' = log T = - C:=,(I- T)"k. We nnow present the inonodromy u-right filtration. defined recursively by W = ker Sn-'+ .VU,- Ul - ;j;n-i-l I.€>n-i-2

starting at O and -1. (-According to [Zucker] ive have .\-" = O. so H = HZnd2= WZ,-, = . . . Proceed downnard from there.) In the Lefschetz pencil case we can be more particular. On -Y,. the subspace of HZm-l(Xt,Q) invariant under T is one-dimensional. generated by the cocycle 6,. It is called the vanishing cocycle because it approarhes O as t + O. We have the

Picard-Lefschetz Formula On H2"-l (Si.Q). T is giwii by

wherr the sign * depends on m. 0 Therefore I - T = &(m. 6,)6,. Since (6,. 6,) = O (beçausr t iie diiiiensioii of -Y,is odd: see [Lewis.l4.20]). we infer t hat .\- = I - T and t hat .\-' = 0. \lé will nom compute the weight filtrations on HZ"-' (-Y:.Q). Hi'"-' (-Yo. Q). and Ho. Applying. we obtain. letting r = 2nz - 1.

w,,= ~2m-1(X,. Q) (for the same reason)

W',-+l = Pm-'(-Yt. Q) Mi, = ker .Y + S lt;,, = ker.V = ~2m-1(-yt. Q)* = ker(e. 6,) (sinceiV2 = O,+ XW>,-l c ker :Y) u.;,-~= ii;-, = SC^.;,^ = SH'~-~(-yt. Q) = im(&(a.6,)d,) (where 6, is the (local) vanishing cocycle (for t ) ) = Q& In summary. the monodromy weight filtration is {O c 115,,-2 c IV2,,.- 1 c IG,) = {O c Qbt c H2m-1(Xt.Q)Tc HZm-lj&.Q)} . SOIVive can sini- plify According to a result of H. Clemens. .Yo is a strong tleformatioii retract of f -'(A). so that via the pullback of the inclusion niap jo : -Yo~i f -l (A).we have the isomorphism HZm-' ( f -'(il). Q)= H2n1-1(-Yo. Q) But we may also note that because of the

Local Invariant Cycle Property (See [Zucker.l. 101 or [Lewis. lec. 141. ) j.R?f.Q 2 ~q7.Q Vq 2 0 (for q = 2rn - 1) the pullback of the inclusion map (j,: -Y,ii /-' (1)is an isomorphisrn

(By the way R"-'T.@ is defined just as RC-l f-@ \vas. wit h f replaced b~-7 and I - by P. It is the sheaf associated to the presheaf I- ci H"-l if -l( l *). C) over P.) Combining the lwt two isomorphisms yields

So ive ma? wite the iveight filtration

This filtration is defined over -1' (where the fiber of R oïer t is Hini-' (-Y,.Q) ). Taking the limit as t -t O. ae obtain a filtration on Ho. the fiber of 'H ovrr 0: W2rn= Ho Z Q2"

>-Q and in particular is for sorne 70 E H'"-' (-yo.Q). We now wish to compute the period rnatrix M(t)for arbitrary t E A'. a matrix whose columns effectively describe R2m-'f-Z as a latticc in F.First. let us fix multivalued sections of R2"-'1.Z. determined (for somp tu E 1' ) by a (multivalued) basis {tq1. . . . ,v2,) of H2m-1(XtO. Z) (the multiple values of L

log t + 2;i- = (C + (*)(dt)61) - (f) T& 27-

since the choices of sign (f)are the same Sow we want a frame of 7 over 1.Sote that since 7 ic: a holornorphiî subhundle of Ü. there is a natural quotient map Ü -7 R/F.-- with the cokernel having rank g. Csing the fact that ly2m-l~2m-1 Fmb"~2m-1(X,) = (,Y,) = invariant classes Fmw2m-1 H2m-1 i (see [Lewis. lec. 141). we may assume without loss of --generality t hat (ivliere [.] denotes image under q) {[q].. . . . [vJ} is a bais of RIF. Sow t-1.. . . . t-~,-~. being invariant under T. naturally extend over O. i.e. over al1 1.and are single-valued. We ma!. write. for 1 5 i 5 g - 2,

- Eramining pre-images under p. u*e obtain, in q-'(O) = 3.the set

ahich. being linearly independant. mus be a frame of 7. Sote that sincc (c. 6,) = 1 and (ri,. . . : L'z~-L} is a basis of HZm-'(X,.Z)T = kerjr: cft). theu {cl. . . . . ~2~-~.C} is a (multidued) basis of HZ"-'(X:. Z). The period nia- trix. then. up to a Gl(W) transformation. is

The Iast row has entries (for I < i < 2g - 1): log t 3 = (tnitl*) - (&)( )(i.i- 6,) - 1hlnil j=l

The last colurnn has entries (for 1 5 i g - 1):

The last. (g.-2g)th. entry. is

log t 3 k- - (*)( .,-- 1st - C h, ib, ) cil ,=1

log t 3 = (2.. t?)- (*ILrg Niv.6,) - 1=12 h,( i*. i*, ) - log 1 3 - O - (Ml---11 1 + 1hjnJ). 1=1 So the whole matrix can be written as The last colurnn of the above sum matrix has unbounded norm as t - 0. The other columns are bounded as t -t O. Consequently. the minimum non-zero norm of the lattice M(t)Z2g does not approach O as t i O. This means the toms J(XJ does not degenerate as t -i O. The quotient spacp J inherits coordinates from C3 x 11 with the projection n : J ;A. J is t herefore a cornplex manifold wit h. recall. the generalizrd ititernirdiat~ Jacobian J(Xo)= n-'(O). lb_l A connection. There is a natural integrable connection on 31 = Or: :: Rn-'f.@. the Gauss-Manin connection D = d 3 1. It gives an exact sequenre

which satisfies the Griffiths infitessimal period relution TF c 91 1 Fp-'. For the estended cohomology bundle 75 ive gei

The different iat ion 7 is no longer surjective. Let S denote t lie iiiiag~of P. so that we may write a short exact sequence

\Ve have. too. the Gauss-Manin connect ion F = 6 1 on h' = Qr :t: Rfig.@. It gives the exact sequence

Let 7 denote the image of v. so that we may write a short exact squenre

LVe wish to compute V explicitly. For a frame of *fl ive will use, as before where cl.. . . s,-l are invariant under T and L* is the dual cocyle to &. Let a = a;oi + a2,C be an arbitrary element of 3L (wherr n, E O1\*). \\ will calculate the image of a in Ri. 8 'H = Ri. ,RZm-l fi@.

Lg- 1 log t = C aidt 2 vi + aigdt 3 u - ' + dt :: Jt i= 1 Lg- 1 = C a:dt~t~~+(a~,dt) i= 1

6,) + a,(& -I i.)

Soting t hat 6, E ker(e. 4) = span{vi. . . . . I*,,-~}. and writing & = rllce, - . - + d+l t*,-l .with di E UA=.we have

Sou- note t hat {

- a2gdi (dt :; ri)+ a;,(& ': i.). '* )zn~-~t \f,'e ssee that T is indeed surjective. Sext. we will compute the extension of V to al1 Ü. Ué have an esart

but V is not surjective. and we will see this explicitly. Our (single-valued) frame is again {vIoc~, . . . . E). So al1 the di's extend holomorphically over 1. This time for an arbitrary elernent of %. ive pick a = C;:;' aivi + argF with ai's holomorphic on al1 A. Using the same computation as before. NT obtain

\lé have eexpressed in terms of the holomorphic frame

for 0: (log 0) z R. -4s proof that the holomorphic coefficient of $ :: i. riiust vanish to order at least 1 at O. Recall S c Ri(log O) :: H is the image of F. 1I.e wrote the short esart sequence: - O i j.RZm-' j,,@ + Ù -S S i O.

\ik clairn Ri 3 R c S. To check this. let 6 = ~:z>~(dt x l*i)+ b2,(dt :I: F:I. where bi E UA.be an arbit rary element of Ri 'fl. Then ive clioosr n = 12;' aivi- + azgCexactlj- as before. so that Vn = b. Ri Z K c 7 is verified in just the same way. The C'ech cup product. \j:e recall the general formula for Cech cohomology cup product . Let A and B be sheaves on a topological space T. wit h cover U. The îup produci is. first. a binary operation on Cech cochains:

u : ~~(2.4.A) :: c"(u.B) i rmcn(U.A :f B j

The cup product descends to cohomology. and to verify this. we must show t hat u takes two cocycles to a cocycle. and takeç a cocycle arid a cotmutidar!- to a coboundary: in other words:

These facts will foliow quiclily from the following obsenration: If n E i"" (Li.A j and .j E Cn(u.B) Then

@

= a(((-1)"~)u 3) is a coboundary. Thirdl- if 30 is any coboundary and .3 is a coq-cle. theti. siniilarly. i)a u 3 is a coboundary. IF-e have verified the three necessary properties of U. Hrnce u descends to cohomology. giving a map:

Taking the direct limit of U.or alternatively insisting that U be Leray (i.e. each member of the family U has trivial higher rohomo!ogy groiips). iva have: a -4 diagram of pairings. The following diagram of pairings cornmutes up to a sign:

where 6 is the connecting homornorphism from 6 is the connecting hornomorpism from

and where 6 is given by inclusion. The isornorphy of H2(P.C) and XL(P'j.lTm-'f.C) arises because F + P has compact fibers Xr!and dimX = 2nz - 1.. nCornmutivity up to a sign' means that, for ct E H1(P.j. RZm-1f.Z 1 and O E Ho(P,Rk @ F),we get a u da = -s'(a u O). We now give a proof of this: Choose a covering U = (C;} that is Leray for dl sheaves involved. Let

Vie wiil first determine da. -4s an elernent- of Tc, (R: 3 7)C Tri (S).we may assume. lifting O] to Ï,. that u: = VÏ, on L;. As a O-cbain. r := (r,)must be passed through the coboundary operator. from C0(Lf.S) to C'(U.S):

- Sorv ne rnusr take the pre-image of this under j. RZrn-'f.C - .H. But this lasr map is essentially an inclusion. so da = dr E H1(U.j-R2"l-' f.Z). whirh means

= rz; - -'1 Csing the Cech cup product formula.

The image of this under the isomorphism H2(P.j.Fm-2f.ê) 5 H2(P. @) is lx,(a U ba)i2k = J', QG A (ri - rj) But now consider a ü a: where cr is identified with its image under O.

Thus we are computing

d (ly,aij A rj) inside C' C P (almost everphere). Sow to calculate d'(a u O). We just saw that the preimage of (a u O),, under d : O-,-t Rb is Jx, A rj. Passing this through the coboundar~- operator. from ~~(24.op)to C2(U.O? j. we get

il-e niust take the preimage of this under C i O?.ivliicli is rssriitial1~-ai inclusion. So ive may write

and commutivity up to a sign is verified. O - a The rnap i. \\é now construct the map whose image will be our tiiaiii ohjrct of scrutin?. The Gauss-hlanin connection V. frorn a.is defined oii R.l.arid desrendc; to a function on F = X/3. Furthermore. P annihilates the iinape of Rn-llmCin 'H. in particular the image of Ra-' f.Z. Consequeiirlj-. T desrend?; toJ=- R,,T,w,.O.The range of the resulting function is deterniined by the infitessimal period relation (see IbJ):

.\ormal Functions are elements of Ho(P. rj. i.e. holoiiiorphic cross- sections of U?J(.Yt) ivith *'moderate" growt h near singular point S. and sat- isfying the horizontality condition VJv = O. In the Lefsclietz pencil case. al1 normal funtions are horizontal. (This will be presented later. after the calculations in the main argument used to show is t rue.) bë \\-il1 be const ruct ing a map taking normal functions to the priniit ive cohoniology of X. This ivill give the topological invariant associated to a norriial functioti. First. from the short exact sequence

ive get the connect ing homomorphism

Second. there is (again) the Local Invariant Cycle Propert!:

Third. there is a splitting orthogonal with respect to the cup product (see- [Lewis]). itè ma!. now construct the composite rnap z. .An exact sequence. Here Ive construct an exact sequence. which will pro[-ide an iitiportaiit step in the main argument.

(1) First we restate the theorem of Lefshetz on the cohomolog~~of hyperplariç. sections.

Theorem Let i : *Yt+ X be the inclusion. Then i' : Hq(-Y) + H7(-Yt)is injective for q = n - 1 and bijective for q < n - 1. 0

(II) Second. we define the aanishing cohornology. We have the map i. : H,-l (*Y,) -t Hn-I (.Y).which. via Poincaré dualit. provides t lie Ciysin niap. also called i,. i. i. : H"-'(S,)- Hfi"( X) The vanishing cohornology is Hn-' (-Y,),= lier i.. It is ralled --\aiiisliing.. because in homology ( H,-' (Xt)). ker i. is the group of (n - 1 )-classes oii S,which .*vanishM(are homologically equivalent to 0) mheri considereci as classes on S. It is generated by the vanishing cocycles {6,,. . . . .6,, } ( where {tl.. . . .tx} is the singular set S c P).

(III) Third. we haïe the follorving. due to the projection foririiila. Fact i.i= : Hn-'(-Y) I Hn+'(-Y)is the cup product ivith the h>-pcrplaiiarlass. by the strong Lefshetz theorem an isomorphism. 0

(IV) \\e put the two facts together to get Lemma Hn-'(-Y,) = CH"-'(X) Hn-'(-Yt), for smooth -Y,.G Proof \iè know that the composite

is an isomorphism (just above). and for t his we must have cok~ri' = ker i, = H "-'( X, ), . T herefore (V) LVe notice the Gysin sequence for the pair (X1*Yt ). aliicli in part is

resid e Hn-2(.Yt)3 Hn(S)i Hn(X --Yt) 4 Hn-'(-Yt)lS HKC1(S,).

The map i. is constructed just as before. The map i' : Hn-'(.Y ) iH "-'(-Y, ) is the pullback. As before we have Fact i.i' i.i' : Hn-2(X) -t Hn(X)is the cup product with the hyperplane class (but this time (recall the weak Lefschetz theorem) need not be ail isonior- phism). Its image is. from the definition and the Lefshetz decoiiipostioii. Prim" (S)I.0 It is said by the previously mentioned theorem of Lefsclietz t liat i' : Hfid'(S) + Hn(Xt)is an isomorphism. So Primn(.i;)L = iiii Li- = iiii i.. and applying the definition of Hn-'(-Ytlu. ive obtain the short exact sequencp

which sheafifies to give

where Primn(X) is the constant sheaf on I-. Tensoring witli Or -.Ive ohtain the exact sequence of sheaves over I -.

where to identify 3t, we note that the direct sum

passes to 72. Thus. there il: a vanishing part 31, c 'H and corespoiiding % c 3.77,. c 32 and corresponding 7, c 7. Putting Hodge filtration levels into the preceding short exact sequence.

O -t FmilPrim"(X) : Or,--t ''9' +FtF, O.

(See for a definition of 9.)

(VI) Lemma There exists an exact sequence

extending the last one. Proof From the exact sequence

we infer Rng.C 2 Primn(X)f (Rn-' f.@), over a punctured disc 1-around a singular point. In this punctured disc

is horizontally split. By the Canonical Extension Theormi. t lie split t ing passes to the canonical extensions over 1.so we have

On 1.. the last exact sequence yields

- -As C modulo (Fm+'Primn(S)3 Oa)f 3, is a free OL-iiiodiil~.ive iiiust have - C 1 (Fm+'Prim"(X) OA)S.&. O

(VII) Take the tensor product of the sequence from the lririrria wit h O: to obtain

Its cohomology sequence (using HO(P.Qf) = O) is But Fm+'Prirnn(X))is a finite dimensional C-vector space. So H1(P.9: -3 F"+'Primn(X)) is equal to H1(P.R&):x Fmc'Primn(S). The exact sequence becomes

TWOresidues. We repeat the computations of the last section, first tensoring the sheaves aith O(1) (pole at m). Since H1(P:R$(l)) = HO(P.O?(- 1)) = O by Serre duaiity. and Rb = O?(--).the last exact sequence beconies

-i HO(P.Q!(I) g Ft)-t HL(~.nb(l)) :; ~"+~~rinf'(X) = 0. implying an isomorphism

( recall 7 := irnr (see IG()) considered (the pole is at infinit),) as aii eleiiimt of Ho((;. 7)gives rise. via the connecting homomorphisrn frorii

and 3 = R~S~-~,({)E Hn(S - .Y,).

Ke will see later (in the main argument) that 0 and 3 r~presentinverse classes in Primn(X) (staternent a). ml Two sheaves to be cornpared later. According to [Deligneop.80]there is an isomorphism on L-: where {Q}is the order-of-pole filtration on RE .Y). with Gm~lR~xI~,L-(~l') being

There is an obvious extension of Ç (using this expression of it) to dl of B:

.A larger algebraic variety. Sow we express k; itself as the fiber of a morphism of a larger algebraic varieti to F'. To trhis end construct a suitable degenerate diviaor rationally equivalent to F Define

We have the diagram of maps

- where :: = x = pq. Define 2, = X-'(s) Sote that 20 = Y and r 2, = Xs x $ u X x (x). YOW?usinp the order-of-pole filtration from &. me set @ An upper-semicont inuous funct ion. Define the funct ion

ljqwr-sernicontinuity (staternent (in in the argument ) wi1I be prowd later. THE ARGUMENT Schema

- The ksqe of the nomd &dior. Ac(?, 31 1 h II'(?,Aff.-:f.Z! *=de: E is t3e sübser anniMaied by the 3a5e d h? [o.9; 8 +j

in Ki(? , R2m-!f. Ç) ZI- EL(?,j,~"-' f. C: [m.

Rqlrsing FD~F,,in m&es no diiference.

/'d(r) = O on ui open neighbourhood cd m. \ [a(X x P,Zo) a (X x P,Z,)YJ 'J 001 'L; U] n The decomposition H1(P.j.RZm-1 f-@) Primn(S)+ Hlb-'( -D), in ia orthogonal. which rneans that under the cup product pairing in [11.

and the decomposition. we have. for r E H1(k.j.R2"-'/.@). pr1a~1n: = ann prlr E Prim(X). (By pr, we mean projection ont0 the first term in the decomposi t ion

Anci **ann'*refers to the annihilator under the cup product piring.) \\ conclude (-Y: Z) = imZ in Primn(S:Z ) fi ann Primmqm(.Y: Z) = ann pr,(imf) in PrimC(S:S )

L I Trivial. C1

By and dL6/:we need -only show that the projection (in u)of t lie image (in O)of HO(P.Rk ç 3ù)in H1(B.j.R2"-' f.ê) under h is cont aiiied in FmCIPrim(X).In other words. we need only show

To t his end. let o E Ho(P. R! 3 7,).T hen form a. .3 as in m. Sorv B.g. and tell us that prl(b(o) Iq) = pr,(a) = -prl(3) = -Res,(a) 11, in PrimR(X).But 1reveals that & 2 es,(^) E HL(P.Ri) ::; FmCIPrini"(-Y). especially that Res,(e) E ~"+~Prixri~(~)This irnplies U e ~~~~~~1016.~1Again. by a and a. wr need oaly show that Fm+'Prirnn(>;)- is rontained in the projection (in m) of the image (in O)of HO(P'ni 2 3,)in HL(P. j.R2m-1fs@) under 6. In ot lier words. ive need only show

To this end. let r E Fm+'Primn(X). Then by 1111: $ r E H1(P.Rf) x FmclPrirnn(X) has a pre-image (call it o)in HO(P.R?, ô ). Forrn a. 3 as in m. Now 110 says that $ @ Res, (5)= 60 = !$ :: r. especially thar Res,(à) = r. ~ext.BBrewalthatpl(*) = -pt1(3) = -Rrs,(ir)

ive use the diagram frorn B. B is the frorn the long exact -irqueiice for

Thus iG = ker0. * ann imZ = im6. since the pairings are dual. This is what we wanted to prove. 13

I - The decornposition Hn-'(.Yt) "- Hn(X)5 Hn-'(-Y,), passes to Ù. Tlius H splits into a constant part und a vanishing part (call it ÜL):

There is a corresponding 7, c 7 (giwn by = Fnnt). So 7 splits thus: -replace 7 by 7,. equivalently Ho(P.Rb :: 7)by HO(P.0; :I 7,). in aiid

1 - There /s a commutative diagram

where the first two vertical arrows are residue maps. S, is definrd as the image of uunder v: and the map 7 i S, is defined by rornniutii-it~.via the exact ness of the rotvs. Yse the cover {l (-1} = {P - {x}.P- {O)). I-sing the prrviouc diagraiil o. - - and the inclusions Oj i R C S and Rb :: K C 7 froni a.ive obtaiii tlie other commutative diagram:

HU((-,-,.R& :: 8)- H1(I;. Rns.@) C H "Il (C- C',

Sote that the map 6 1 is the same 6. restricted to c P-as die b iti m.- bot11- being the connecting homomorphism from the short exact sequences in /. - Sow. refer to and the diagram above. The image of 5 E Ho([ ;. 9; i. Ç! when traced through the diagram. over then down. is R~S~--~_((j = o. When traced down then over. it is 6 Ir.-,, (a) = 6(5) IL-, . This shoi~sthe

What we aim to prove specifically is that for p as in aiid pr, as in m. pr1(pœ(8))= -pr,(a). This is equivalent to p'(3) + n Iying in the subspace Hn-q (D), of H1(P.Rn-' f.C) under the decomposition H1(P.R"-' f-@) 2 Primn(.Y) 3 Hn-2(D),from a. First construct the diagram: H~Y)- 1i Hn-2(X, )

lsFm] 1W.1.1 / XE(-Y)

kijp. ~"(y)7 Hn(X) 5 Hnf2(X x IP) = H~-?(x) Hn(S) CVe specify the maps. f. is the dual map to the pushforward on 1ioiiiolog~- from inclusion T : -Y, + Y-;. i. is the same for i : X', i -Y. p. is the pushforward from any of the inclusions p : li' i X x P as a fiber of z2. k. i.; the Gysin map. the dual to the pushforward on hornology froni the inclusion k : i -Y x P. The rnaps i' are pullbacks of the obvious iiirlusions. Tlir reniaining map is the sum of residues defined from the unioiis

and (x-x,)u(C-CJ=X xP-X, x(oc}. ( AS subsets of X x P. Y - X, and .Y - X, are equal. ) The lower rectangle. intuitive15 cornmutes. Conimut kit? in t lie iipper rightrnost triangle is trivial. The upper iniddle triangle {;.. k.. p.i.) îorii- mutes. as it is the pushforwards of the niaps 3.1;. p o i. whicli coiiiiiiiite:

C pmi. = (p O i). = (kO 1). = k * i. A11 t hat is left is the upper left triangle (1 + p'). k.. k. + p.). Herr. liowm.er. ive have. from a.

So this triangle is not commutative. But if ire replace ~"(r)+ HG(S) L>!- H~(T)+ Primn(S). then since Primn(-!') = ker L : Hn(S)+ Hn+?(X). pgxqand so the whole diagram now cornmutes. O Since ive are looking at a residue at r.we may consider just a siiiall disc 1 c P centred at r. on which f is smoot h. The commutat i\.e diagraiii shows 3 = Re~~-~,(c)= Res,(e). O 1110 iis truel The staternent of 1101 precisely.- is Lemma Let a E Ho(P.Rb {Z 3,)and 5 E HO(P.RpP1(i ) 1:: C)br its lifting under the isornorphism frorn a.Then

where $ E H1(U'0;). and 24 is the covering {Go = P - {cc}.(> = P - {O}) of P. Proof Let us calculate 6~7.Precisely. rr is the O-cochain

- FVe must determine the pre-image of o Il-, -and a Icr, under QI:! C i Qk 1:: 3,. LVé ma? lift o Ir, to 5 Ir:o~ HO(lb.95 2 Ç). By an alrnost identical construction to that which gave us ir. ma'- find r E Ho(P.Rk( 1) 2 g) where the pole is at O instead of at x. Then Î - HO(.(l)) Sow apply the coboundary operator to the cochain

C\è obtain the 1-cochain

FinalIy. the pre-image of the cochain must be identified. that is. under the map - Ri 5 ~"+'Prim"(X)-t Rk â Ç. But this map is essentially an inclusion. so that we finally have has no pole at a (and only a first order pole at O). hence is icientically 0. @

This follows irnrnediately from the exactness of the sequence in B. 2 -1 -1 Ré rvill prove this by showing that Ç1 E C. for we will tlirri Iiaw tlir

-1 ait ti the quotient sheaf supported on 5 (since E Ir-= G = C il-. The lotig exact cohomology sequence will in part be

Then H1(P.Rk g)= O will imply H1(P.Rf '3C) = O. So again. it suffices to prove g1c G- \Ve dldetermine bounds for the growth of the periods of the growtli of the periods of local sections of Ç' near singular points. It is this prop~rty rvhich characterizes the canonical ext.ension in terms of growt li. Let 1 be a local section of Ç' is represented by a relatively closed C'" n-form o on .Y x A - I' that has a specified order of polr along T on eadi local coordinate. In particular. O is a relatively closed foriii wit ti a pole of order at most ri - (rra + 1) + 1 = rn. Let LI- be a small polydisc in -Y x P centreci at the double point of .Yu. vit p()= 1. Sow the image of HJC, - W)in H,(C! j is (for f 0 j a codimension 1 invariant subspace. Chains represent ing t liese classes caii be kept uniformly awa. from -Y,.so the periods of O are bouiicled on A. -\ subspace of Hn(Ct)orthogonal to HJC, - II-) is generated II~the tube on the duai cycle (see [lZ.pp.470.51S]). Again. the periods arising froni tli~ romplement of W is bounded. so we have only a local coiiiputatioti on Su. Pick local "double point" coordinates (xi. . . .x,) on .Y and use t for 1 so that x P is given locally by

Ive may assume 1x1 1 < 1 and x7=,Ir, l2 < 1. On X,. the relm-aiit portion of the dual cycle is giwn by [l?.p.518]

for t = 1tlci< Without 10s~of generality assume t real and positive (i.e. e = O). Since o has a pole of order at most m. we may arite. lorally.

with the hrJ's CX functions. Let = o IKx(+ so that (ResF(o)) /.y,= Resxr(ot). We are interested in the section o integrated over elemeiits of Hn(Ct).But the subspace (image of) H,(C,- W)is invariant and the coni- plementary subspace is generatd by tubey. Thus the integral of interest is

which b>- duality and since E Y.equals

Lié need a result to determine the residue expIicitl>-. It will be proved by reducing the order of pole [Griffiths. p.?YS]. (Simple-pole forms have well-defined residues (see [Lewis. lec. 91).) Let o = ~-~rl.on the unit polydisc An.with coordinates (y. -2.. . . . z,.,. and where >I is a Cr form. First ive need a lemma. which follows from the similar fart almiit Ta>-lor series for real variables. Lemma Let h(y. 2) be a C" function on An. Then there is a partial Taylor espansion

\vit 11 bl.b2 being CE. and al = $(O. -).O .\pplying the lemma to the coefficients of 11. ive write

where ql. . . . .0, are free of fi and Y, with coefficients indepericlant of y: rlj is CX.and every term of O involves Y or dy. Write 01 = pl A dg + ps where pz is free of dy. Proposition Res(o) = pi 0 Proof First. the form ota closed form with a pole of order ni. musr have a logarithmic plus a pole of order rn - 1. Applying t his expression

O= enables us to wite

(Every term of Cl and Subtracting an exact form from o will not change its residiie. So for m > i subtract

The difference contains a pole of orcler only m - 1. 1-sing tlir leinnia. ive nia!- wri t e the difference as

- (thus redefining 01. . . . . qm-- 1. O). Sote carefully. however. tliat rll. . . . .q,- are the same as before? O has been changed. and to qm-~lias I~ertiaddsd

which is independant of dg. 11-e proceed inductivelj-. reducing the order of the pole by one earh t ime. with a process that discards the qi with the highest i. adds sortiething in- dependent of dg to the qi mith the second-highest i. and changes 6. Ué continue the process unt il the pole is of order 1. It is wort h iiot icirig t hat t lie process leaves qi alone until the final step. in which somethiiig is added to rli = pi A dy + pz independant of dy. In other words. only p2 is affected. not Pl - Now we have a form (redefining p, 6)which has the same residue as o. Expressing q1 = p A dg +pl we notice that only pz has been changed from the original p,. and not pl. And the residue of O' is dearly pi. O \iTe may apply the previous proposition to O,. replacing o witli a,. To make furt her use of the result, we change variables. Replace {xi. . . . .r,} by {s = & 1:. 22, . . . r,). Here 21 = whicii is n-ell-defined since rl-is and real there. giveslus a wd-defined branch of J. ij-e have

and may simplify the calculation of Resx,(ot)as follows. To lise the lmitiia and proposition. rve must compute the (m - 1)st derivatives of t lie coefficients of ot with respect to s and e~luatethem at s = t. i.e. at JFi= t. i.e. on -Y,. In doing this, only those terms involving ds contribute to the residue. On 71. t E WC.so that XI E W+.xl.. . . .x, E &TIR+- wlienre dxl = fi,.dxj = -4for j > 1. So an. term containing dr, A clT, will integrate to O on I~.Thus only some of the terms of

those in which dxl.and for j :-1. onc of dr,. fi,. appears. iie~tlIw coiisid- ered. In the substitution

may ignore the dx terms and pretend that

Kewill estimate the growth of term by term. so we will assume that

-- 1 = --(s - t,-m s - 2 z; - t) h(~.t)dzt A - A ds. 3- ( j=l To use the propostion. we must compüte

rvhich bj- Leibniz's rule becomes

where the ck's are constants. The iterated chain rule gives 11s

where the sum is t aken over al1 non-negat ive k-t uples a = (ol. . . . . ne ) sa t is- fying c!=, iai = Ir. 101 = ~f=;=,ûi. and the di's are constants. The last three expressions are to be evaluated at s = t. Gngthe last two espressioiis. the third last. Vse u, = Imxj (j= 2.. . . . n). for coordinates on yt. Xote

Since the loweçt power of (t - xy=ZI:) in (i)is f - m. orcurring onl~-when A- = O. a = O. ive may estimate

for some constant -4. This upper bound can be simplified b>- a change of coordinat es. Replace (u?.. . . . un) by 1.2. . . . . rr.-1 - r ) where

subject to 2 v, + . - + i!;-, < 1 0~r~t/l-t The different ials satisfy du2 A - A du, So our bound is

Since the v's do not appear in the integrand. the bound becomes

( + r2 ) 4-m rn-2 dr.

= . by substituting un= t-Br.

= . substituting uq= tan i.

arc t an J-' ( sec2 i1-i sec' cd<

1 + sin I; arctan t/t-I- 1 = 4 - log [ i 1 -sin< 1, = 4 (- : log = At (- log i 1I+-) - ,/R-

1 B - 4' log t. 3-

for sorne constants -4'. B. What ive have proven is t hat the periods of local sections of F near the singular points grow at rnost like pourers of log t. Therefore c. C

If d (frorn n)is identically O on P. then in particular. ivr ha\-e

dimH1(P. -

L 1 If the pairs (X x P. Zo)and (X x P. 2,) are isornorphic for al1 # x. then & = ~m+lnk,,,(aZo) 2 Gm+'R>xpp(~Zs)= Z, Hence

for al1 s # m. But if also d(s) = O on an open neighboorhood of x. theri d must be identicah Oon P.O This is a direct application of the definition of upper-semicontinuity to a function with integer values. O

Consider the linear automorphisrn. for s E P - {cc}:

(QO. al) H (a0.al - sa0) Sow let us examine the effect of

satisfies (z0.z1) . L,(a) = ao-0 + (al- sao)zl = O which can be rmrittm as

This. by definition. means (z.L,(a)) E Za. Lié daim that 1 x L, in fact transforms 2, isomorphically onto Zu. To see t his consider the inverse transformation

in which ( LJL : P i P is given by (ao.al) ci (ao.al + sao). -4 similar calculation to the above shows that (2.a) E Zo 3 (2.L;'(n)) E Z, so that LW do indeed have the isomorphism between the pairs (Xx P..&) aiid (Sx B. 2, j. 0 is truel Our task is to show that d(m)= O. which means that

= H' (P.Ri z Rnr2.Gm+1R;x?2(a~,)) Sow C' := Gm+lR>,,,,(aZ,) is the complex But as 2, = X, x PUX x (x};we may observe

1vit.h the poles of order 1 in O?(()being at oc. So Gm"R; x- o( 02, ) contains the subcomplel ,Mo := Gmi'CI>(m.Xm) n;0.(1). which is

the poles of order 1 in OI.(l) being again at s.The quotient Cg,/M' is sup- ported on ~;l(z)= X x (2).It is clear t hat R"7ï2.M0 a H"(X.Gm-' n>i .-Y, 1) .:. 0?(1).which indicates, via Serre dualir): thar H1(P.R&Zç)?(l)) * Ho($.O-,! -1) j = 0. =+ H1(P.f?! 2 Rn;r2..M') = O. From the cokernel sequence

ive have. using left esactness,

This is what u*e wanted to prove. I

Theorem for Hypercohomology to the mor- phism 7r3 : X x P x P 4 3 and the complex of sheaves g 3' on X x IP x B. Then we have

being upper-semicontinuou~on P. But for the scheme P' k(s) = ê for ail E P. Applying m, we discover the upper-semicontinuity on P of po1 This is the proof of the Semicontinuity Theorem for Hypercohomology Let V i S be a projective morphism of Xoetherian schemes' and let A'' be a cornplex of sheaves on V, with the Ai"'s being coherent O-modules 0at over S, with differentials linear over f -'(Os)- For ail p 2 0. the function s e dimc(s)H1(&,A:) is upper-semicontinuous on S.O.

with a Lemma Rn+lx,2' = 0 0 Proof The presheaf associated to R"+'xZ0 = O is

U-e know that there e'cists a spectral sequence IqE,) abutting to

=OVp~m+l.piq=n+l would imply *EZpq= O and thus ~ouldimply Hn+l(n-'(L'). GmtLR~xFxi:rxs(~Z)). ivhich would give the desired wishing of RnC1r.Z'. So it suffices CO establish the mkhing of the shed associated to the presheaf 6' * Hq(a-'(C): QXxpXppxr((~- m)Z). name- Rqa.R~,pxF,,,p((p- m)Z),for al1 p + q = n + 1. For this. we ivill establish the vanishing of the stalks - aobl Hq(X 0% ((p m)X,).t = - aobo over all (a:b) E P 8 P. By Nakano's generalization of the Iiodaira Vanishing Theorem [23,p.132] these groups, except for the stalk over ((0, l)?(0.1)) are O. The entire fiber T_~(((O~1). (0' 1)))is contained in the polar locus 2. so a special argument is needed. We can replace Z by a linearly equivalent divisor 2' on S x P x P without changing the cohomology. Take 2' to be giwn by

When a* = bo we get the equation of -Y,. We can now apply the vanishing t heorem again. O This argument is taken, some of it uerbatim. from [Zucker. 4.461. To establish 120 wwe now use the Leray spectral scqucrtci for hypercoho- mology. We have the sequence {Er)with

E = E, = Hn+'(-Yx P.;r;R! :: 3: ) (degeneration at Erj E.kJ = Hi(P?RJT~.(R: 3 2:)) = Hi(P$Rb RJr2.3;) so that Hn+l(xx P.rini z 2:)

But

So the direct sum has a single term: that is. cl From this we may obtain the Corollary -411 normal funct ions are horizont al ( [Zucker]). C) Proof CVe know (statement 1121) that H1(IP.Ri 8 C) = O From a we have the short exact sequence Since H2(P.Rb) = O. we have the surjectivity of

- So from HL(P.Rb. Ç) = O ive conclude

By Serre duality. H'(P,~)= O. From the canonical bundle forniula. ive have the inclusion R$ = @(-2) v OF.then 7,2 R$ LI r.. hi ch implies

Sow Ho is right exact on coherent sheaves because the diiiiensioii of P is -m+l.= 1. Consequent1~-.the surject ioii 5- i 3 gkes a surject iori

which is the 3;anishing of the middle Iink in the foilowing factoriza~ioiiof Ti:

(The reason ive have the factorization is ([Lewis.lec.ll]) that

14&-l consisting of invariant classes under monodromy. so t liat F (on Fn'.-) --m+l.= gives no poles of 7 Rb at C.) Therefore. & = 0. O EXAMPLE APPLICATIONS In t his section we examine an equivalent condit ion for Jacobi Iiiversiorl. and another special case of the General Hodge Conjecture. Recall the Abel-Jacobi map

.Jarobi Inversion is its surjectivity. Define .J,(I(',) to be the image. in .J(S,). of the subgroup < O(Xt) under cP. (O(.Yt)al,is the group of codimension-m algebraic cycles on Xt which are algebraically equivalent to O.) It is known. by an application of Poincaré's complete reducibility theo- rem. that Ja(Xt)is a subtorus of J(X). I-sing Hilbert Scheme arguments. it can be shown that tlir quotient

at most countable. It follows that @(O(-G))

also count able. + .lacobi Inversion is equivalent to .Ja(.Yt) = .J(X,). For if @(.Yt ) + .J( .Y:1 surjective. t hen

both countable. and the quotient of two tori. .J(X,) and .J.(X', i. HCIICPth tori must be identical. the quotient O. What exactly does it mean for a cycle f E @(-Yt)to 1w algebraically equivalent to O? It rneans there is a smooth curve T. a cycle z E Cn'(rx X,) of codimension m, and points a. b E r so that f = ~(b)- i(n ). In a similar way. given ,- as above. we have a map on CHO(l)alg(the group of O-cycles on ï of degree O): For an- such z rve have (see [Lewis.lec.l2]) a commutatiw diagraiii

We can choose 2 so that [z]. is surjective. which if is surjective forces ivt ~2"1 (-Yt)= 1. So the weight of H2'-' (-Y,)is necessari1)- 1 for .Iacol>i Inversion to hold.

There is another special case of the General Hodge Coiijecture (i.~. not HodgePSP(X.Q)) which has not been discussed so far. Here tlir dimension of -Y is odd. sa? n = 2k - 1. It is an equaiity of trvo filtrations oii H'"'(s. Q):

The filtration FA.H2"-'(S. Q) is defined to be the maximal Hodg~Strur- turc in FL-i n $2k-i (X. Q). .And -1-is the filtration by conilt~au.defined b!. .\-'Hz"-l(S. Q) = {iino. 1 o. is the Gysin rnap H~'-"-'($-. Q)-> H~"' (..y-QI and = desing(Y). rvhere codimx Y = i). Sow suppose the General Hodge Conjecture - and this case in particiilar - is true. Suppose also that lY2"-' (-Y.Q) has weight 1. and niore part icularly that H2"-'(X. Q) = F?' Y H~"-'(x. Q). In this case H"-~ (S. Q) = Fk-l in; H2"-'(.Y. Q). Then by the finite dimensionality of the colioi~iology.tliere exists of pure dimension ri - (k- 1) = 1 + (TI - k) (aith k 5 11 ) so that

is surjective. By taking n - II general hyperplane sections of f' we get a smooth curve ï c so that

is surjective by the Weak Lefschetz Theorem. We also get the inclusion ~'(p)v H1(T) by duality. Consider a left inverse of it' H1(r) -t H' (p),as a morphism of Hodge structures. This Poincaré duality defines an elernent of H1(I')=S Hl (g-) -+ H1(T) 1:: H)and being a morphism of Hodge strucures it determines a class in

By the Lefschetz (1.1) Theorem. and by taking a suitable integral multi- ple. t his is the class induced by some 20 E CHL(Tx p). The (surjective) composite H1(r.Q) -HI(~. Q) 3HLk-l(X Q) is similaril- induced by some

where 2 E CfIk(rx X). We again have the commutative diagrani

And tire know [z].is surjective. so that .Jarobi Inkcrsion follows.

Ive wiish to find some examples of hypersurfaces S c P'"of dirnensioii n = '2k - 1) where wtHn(.Y) = 1 and where the General Hodg Conjecture holds. Let X be such a hypersurface. of degree d. From [Lewis. O. 121. the nu- merical condition

is equivalent to wt Hn(X)being 1. We want to know whet lier or not it is sufficient for (the special case of) Hoclge Conjecture. To investigate t his, we have the sufficient condit ion for where [ = [TJ.found in a paper of Lewis:

In this case we are assuming wtHn(S) = 1. so if 1 = [Fi is k - 1. tlieii .v'H"(.Y. Q) = Hn(S.Q) is .Y"-' H2"' (S.Q) = F$-' H*~-'( -Y.Q). OU^ sp- cial case of the Hodge Conjecture. Our investigation boils down to the question: "When does condition (r) imply 1 = k - 1 and condition (**) ?" Assume (+) holds. The cases are:

a k = 1 In this case (*) provides no restriction on d. but t hat hardly matters. for me know the Hodge Conjecture tiolds for curws (i.r. wl.li~re n = 1). (This is the Lefschetz (1.1) Tbeorem.) 0 C=2 Here.dS4. a k = 3 Here. d 5 3. 0 A. 2 4 Here. d 5 2. We may ignore this case. becaiisr the Geiieral Hodge Conjecture holds for al1 cases where d = 1 or 2.

U-e have the t wo cases k = 2.3 t o coirer. If k = 2. then d < 1.=+ 1 = 1 = k - 1. -4s for (*+). it 1iold.c:

f(n + 2 - [) + 1 - (y)

If k = 3. then d 5 3. So d is exactly il. making 1 = 2 = k - 1. .And (**; again holds: l(n +2 -1) + 1 - (d:')

What we have show is that condition (*) is equivalent to the version

of the Hodge Conjecture for odd-dimensional hypersufaces. INDEX OF NOTATION This section gives the first page on which notation appears. Page / Sotation X N' HP4 Ei H~dge~.~ L Prim' PrimP-q [ 1 cm @ .Y, -Y 1 - D 7 f (3 R R H 24 A' CP(U.Ag) CP4 -7-P C -1 i P r k 9 km 3C

BIBLIOGRAPHY

Andreotti. A..Frankel. T.: The Second Lefschetz Theoreiii oii Iiy y erplaiie sections. In: Global Analysis. pp. 1-20. Princeton Cniversi ty Press 196!)

Dcligne. P.: Equat ions différentielles à points singuliers rkgul iers. III: Lecture Sotes in Math. 163. Berlin-Heidelburg-Sew York: Spriiig~r1970

Griffiths. P.: On the periods of certain rational integrals: 1 aiid II. ;\nnal?; of Math. 90(3).460-041 (19691

Griffiths? P.. Harris. J.: Principles of Algebraic Geonietry. .lob LVile- and Sons 19:35

Katz. S.: The regularity theorem in algebraic geomrtry. .-\rt~s.C'o11gri.s Intern. Math. 1.437-44:3 (1970)

Landman. -4.: On the Picard-Lefschet z transformation for algebraic iiiati- ifolds acquiring general singularities. Tram AMS 181. 89-126 ( L!)3)

Lewis. James D.: .A Survey of the Hodge Conjecture. C'mt rr dr rrclierclirs mathématiques 1991

Zucker. Steven: Cieneralizecl Intermediate .lacobians aiid the Tlieorerii or1 Sornial Functions. In: Iin-entiones Mat 11. 33. 1%-2'1" (1 976) APPENDIX C\é can explicitly give the Abel-Jacobi niap for curves ou a surface i.r. for n = 2. n = 1. Consider a curve in O(X/P) whose hyperplane sections are tioniologous to O i.e. are O-cycles of degree O:

The Abel-Jacobi map takes tt to the functional

modulo periods

A*).

If ive have element v of HO(P.O(J) ). take V(t ) and iii\.rrr : get ,Er \vit li Q(&) = u(f).Piece the &'s together to get f = ilt& with @(O= V. To show that we do indeed have .lacobi intpersion in tliis case. LW caIi define a map (after choosing p in the connected .Y,)on the kt11 s>*ini~ietric product of -4", : S'(.Y,)i l(St)

This map. for k = g = geometric genus of S, = diniH1.O(.Y, ). is analyt ir. generically 1-1. and surjective. So O is surjective since its itiiage contairis that of this map.