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DEPARTSIEXT of Bi.4THE'vi.ATICAL SCIESC'es 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 Edmonton, Alberta FALL! 1997 National Library Bibliothèque nationale of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 WeiIington Street 395, nie Wellington OttawaON KIAON4 Ottawa ON K1A ON4 CaMda Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Libraly of Canada to Bibliothèque nationale du Canada de reproduce, loan, distriiute or sell reproduire, prêter' distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la fome de microfiche/film, de reproduction sur papier ou sur format éiectronique. 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.
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