Hodge, is a method for studying the cohomology groups of a smooth manifold m using partial differential equations. However, in the case of genus0 maps to homogeneous varieties, it is. Indam program on serre conjectures and the padic local. The hodge conjecture for general prym varieties by indranil biswas school of maths, tifr, homi bhabha road, mumbai 400 005. Hodge decomposition university of california, berkeley. This is motivated largely by a recent approach of lawrencevenkatesh lv18 to proving analogs of the arithmetic shafarevich conjecture for families of varieties with. Given the paucity of evidence, why do people believe the hodge conjecture. Categories and noncommutative geometry statement of the conjecture kaledin, several papers on the proof math0511665, math0611623, 0708.
Serres conjecture for gl2 and to the geometric breuilmezard conjecture for potentially barsottitate representations. Find materials for this course in the pages linked along the left. Hodge conjecture implies that the locus where this happens is a denumerable union of algebraic subvarieties of s known. Hodge conjecture article about hodge conjecture by the. Ive read articles and watched youtube videos on the topology and algebraic geometry and i dont really understand it and i dont expect myself to. By contrast, the hodge conjecture for divisors on any smooth complex projective variety is known the lefschetz 1,1theorem. Logarithmic series and hodge integrals in the tautological.
The hodge conjecture for abelian varieties can thus be studied class by class. Mgis a gorenstein algebra with socle in codimension g. So the hc predicts that algebraic cycles jump up on special varieties of the. Subhodge structure means the subgroup of the integral lattice whose complexi. Logarithmic series and hodge integrals in the tautological ring. I think that the written paragraphs were irrelevant, but the assumption in the hodge conjecture that x is algebraic cannot be weakened. The hodge conjecture applied to the product of x and a suitable abelian variety predicts that h.
Say that a cohomology class on x is of colevel c coniveau c if it is the pushforward of a cohomology class on a c codimensional subvariety of x. Some geometric objects that are interesting to study and relevant to areas like engineering or physics can be defined algebraically, but not geometrically. Even though im able to understand what it says, and at first glance i do find it a very nice assertion, i cannot grasp yet why it is so relevant as to be. We furthermore conclude that the fvector of the independence. The conjecture was first formulated by british mathematician william hodge in 1941, though it received little attention before he presented it in an. We use the hodgeriemann relations to resolve a conjecture of heron, rota, and welsh that postulates the logconcavity of the coefficients of the characteristic polynomial of m. The characteristic polynomial and rotas conjecture overview of the proof of rotas conjecture whitney numbers of the second kind reeds conjecture let g be a connected nite graph. An automorphic variant of a conjecture of deligne jie lin abstract. On the hodge conjecture for products of certain surfaces. Jan 11, 2017 some progress was made by voisin and others using complex analytical tools and decomposition of the diagonal, in particular regarding complex tori, perhaps also hyperkahker varieties, say by ogrady. Hodge type conjecture for higher chow groups 949 conjecture see 21, 29, 32. Let x tbe a family of algebraic varieties parametrized by a. The key observation is that, given a riemannian metric on m, every cohomology class has a canonical representative, a differential form which vanishes under the laplacian operator of the metric. Lecture notes geometry of manifolds mathematics mit.
Even though im able to understand what it says, and at first glance i do find it a very nice assertion, i cannot grasp yet why it is so relevant as to be considered one of the biggest open problems in algebraic. Mar 20, 2011 hodge conjecture conjecture hackman remix label. Products of it with itself give candidates for counterexamples to the hodge conjecture which may be of interest. The hodge conjecture is not known even in the following nice examples. Devlin profiles each problem and offers insight into how it came about and its significance. We use the hodge riemann relations to resolve a conjecture of heron, rota, and welsh that postulates the logconcavity of the coefficients of the characteristic polynomial of m. Hodgetheoretic invariants for algebraic cycles mark green and phillip griffiths abstract. We prove the hard lefschetz theorem and the hodgeriemann relations for a commutative ring associated to an arbitrary matroid m. This is joint work with ana caraiani, matthew emerton, and toby gee. The hodge conjecture simply means that shapes fit inside shapes. Suppose one has a smooth family of complex projective varieties and a locally constant cohomology class in the fibres which is everywhere a hodge class and is algebraic at one fibre.
Pdf the general hodge conjecture for abelian varieties of cmtype is shown to be implied by the usual hodge conjecture for those up to codimension two. Journal of algebraic geometry 11 2002 here is the bundled source for this document. Hodge made an additional, stronger conjecture than the integral hodge conjecture. From the hodge conjecture for function elds, one expects that the galois group and hodgepink group are isomorphic, which we prove with the help of tannakian theory dmos82, prop. The evaluation is then a canonically normalized function on the socle. On this article, hodge conjecture, the original author as saying at the head of this paragraph assumes kahler manifold in order to be clear for the origin of hodge conjecure, which is the properties of kahler manifolds and fortunately on compact kahler manifolds there exist such harmonic forms. Ive read articles and watched youtube videos on the topology and algebraic geometry and i dont really understand it and i. We furthermore conclude that the fvector of the independence complex of a matroid forms. Then every hodge class on x is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of x. Hi all, in my engineering class we had to choose to do a product on any stem topic, and i chose the millenium problems. We also prove the hodge conjecture for arbitrary selfproducts of a k3 surface xsuch that the. Key to this approach is showing that a given family of hodge classes is absolute, extending delignes theorem for abelian varieties. If the hodge conjecture is known for a and b is another abelian variety such that gb ga then the hodge conjecture follows for b as well. The hodge conjecture regards the algebraicity of the hodge classes.
Paranjape imsc, cit campus, tharamani, chennai 600 1. Hodge theoretic invariants for algebraic cycles mark green and phillip griffiths abstract. This is a polynomial in t, called the chromatic polynomialof g. Period domains, algebraic cycles, and arithmetic 1020 june, 20 university of british columbia, vancouver a fourday summer school for graduate students and postdocs, followed by a sixday research conference on the. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. A gauge transformation is a gequivariant di eomorphism of p living over the identity, i. Pdf on the general hodge conjecture for abelian varieties. It can be considered an arithmetic analog of the hodge conjecture. The tate conjecture is the analogue where c \mathbfc is replaced by a finite field or number field, and the target category of pure hodge structures is replaced by the ladic representations.
Jandu, january 20, 2006, infinite bandwidth publishing edition, paperback in english. I cannot really answer this, but it is a key ingredient in a larger framework governing cycles in algebraic geometry. Assuming the generalized hodge conjecture and a conjecture of blochbeilinson, we show that. Hodge conjecture in the context of algebraic cycles, and discuss its relationship with the generalized bloch conjecture and the nilpotence conjecture. In fact, though not introduced in this article, zucker showed in 1977 that complex tori with nonanalytic rational cohomology of type. Let x be a nonsingular complex projective manifold. Gt to be the number of proper colorings of g using t colors. B, the hodge decompositions on x t vary continuously for t2b, and ltrations vary holomorphically. The hodge conjecture asserts that, for particularly nice types of spaces called projective algebraic varieties, the pieces called hodge cycles are actually rational. Thomas gives a proof to descending the hodge conjecture into showing that every n,nhodge class in a. Introduction to hodge theory 3 the decomposition 1. What progress has been made to date on the hodge conjecture.
The conjecture is a central problem in the theory of algebraic cycles. Logarithmic series and hodge integrals in the tautological ring 219 rings in the more general setting of stable maps. Hodgetype conjecture for higher chow groups 949 conjecture see 21, 29, 32. Nov 09, 2015 we prove the hard lefschetz theorem and the hodge riemann relations for a commutative ring associated to an arbitrary matroid m. Specifically, they are the riemann hypothesis, which lingers from hilberts list, yangmills theory and the mass gap hypothesis, the p versus np problem, the navierstokes equations, the poincaire conjecture, the birch and swinnertondyer conjecture, and the hodge conjecture. In this paper we prove the axschanuel conjecture in the more general setting of variations of pure hodge structures formulated recently by klingler kli17, conjecture 7. It is natural to hope analogous gorenstein properties hold for r. The hodge conjecture january 20, 2006 edition open library.
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