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. Categories and noncommutative geometry statement of the conjecture kaledin, several papers on the proof math0511665, math0611623, 0708. Q, in other words for rational points on the curve. 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.
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. The hodge conjecture for abelian varieties can thus be studied class by class. Hodgetheoretic invariants for algebraic cycles mark green and phillip griffiths abstract. Then the tate conjecture for divisors on x is equivalent to the birchswinnertondyer conjecture for the jacobian variety of f.
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. Hodge decomposition university of california, berkeley. Ive read articles and watched youtube videos on the topology and algebraic geometry and i dont really understand it and i. The hodge conjecture for general prym varieties by indranil biswas school of maths, tifr, homi bhabha road, mumbai 400 005. 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. Hodge theoretic invariants for algebraic cycles mark green and phillip griffiths abstract. Pdf on the general hodge conjecture for abelian varieties. The hodge conjecture regards the algebraicity of the hodge classes. B, the hodge decompositions on x t vary continuously for t2b, and ltrations vary holomorphically. I cannot really answer this, but it is a key ingredient in a larger framework governing cycles in algebraic geometry. Mar 20, 2011 hodge conjecture conjecture hackman remix label. 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 ring 219 rings in the more general setting of stable maps. Logarithmic series and hodge integrals in the tautological ring. 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. Subhodge structure means the subgroup of the integral lattice whose complexi. 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. 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.
Mgis a gorenstein algebra with socle in codimension g. The conjecture was first formulated by british mathematician william hodge in 1941, though it received little attention before he presented it in an. Devlin profiles each problem and offers insight into how it came about and its significance. 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. 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.
However, in the case of genus0 maps to homogeneous varieties, it is. This is motivated largely by a recent approach of lawrencevenkatesh lv18 to proving analogs of the arithmetic shafarevich conjecture for families of varieties with. The evaluation is then a canonically normalized function on the socle. The hodge conjecture is not known even in the following nice examples. Lecture notes geometry of manifolds mathematics mit. Gt to be the number of proper colorings of g using t colors. 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. In mathematics, the hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a nonsingular complex algebraic variety to its subvarieties.
Find materials for this course in the pages linked along the left. Paranjape imsc, cit campus, tharamani, chennai 600 1. Hodge type conjecture for higher chow groups 949 conjecture see 21, 29, 32. Algebraicity of hodge loci for variations of hodge structure. Let x tbe a family of algebraic varieties parametrized by a. 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 conjecture is a central problem in the theory of algebraic cycles. We furthermore conclude that the fvector of the independence. Indam program on serre conjectures and the padic local. In fact, though not introduced in this article, zucker showed in 1977 that complex tori with nonanalytic rational cohomology of type. 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.
Given the paucity of evidence, why do people believe the hodge conjecture. I think that the written paragraphs were irrelevant, but the assumption in the hodge conjecture that x is algebraic cannot be weakened. Jandu, january 20, 2006, infinite bandwidth publishing edition, paperback in english. Hodge conjecture in the context of algebraic cycles, and discuss its relationship with the generalized bloch conjecture and the nilpotence conjecture. We prove the hard lefschetz theorem and the hodgeriemann relations for a commutative ring associated to an arbitrary matroid m. It is natural to hope analogous gorenstein properties hold for r. 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. What progress has been made to date on the hodge conjecture. However, some objects are even more complicated and cannot be built wi.
Logarithmic series and hodge integrals in the tautological. 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. On the hodge conjecture for products of certain surfaces. Hodge conjecture implies that the locus where this happens is a denumerable union of algebraic subvarieties of s known. We furthermore conclude that the fvector of the independence complex of a matroid forms. An automorphic variant of a conjecture of deligne jie lin abstract. This is joint work with ana caraiani, matthew emerton, and toby gee. The hodge conjecture january 20, 2006 edition open library. 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.
Some geometric objects that are interesting to study and relevant to areas like engineering or physics can be defined algebraically, but not geometrically. We also prove the hodge conjecture for arbitrary selfproducts of a k3 surface xsuch that the. A gauge transformation is a gequivariant di eomorphism of p living over the identity, i. 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. Hodge conjecture article about hodge conjecture by the. Introduction to hodge theory 3 the decomposition 1. Key to this approach is showing that a given family of hodge classes is absolute, extending delignes theorem for abelian varieties. So the hc predicts that algebraic cycles jump up on special varieties of the. This is a polynomial in t, called the chromatic polynomialof g. Let x be a nonsingular complex projective manifold.
The hodge conjecture applied to the product of x and a suitable abelian variety predicts that h. Journal of algebraic geometry 11 2002 here is the bundled source for this document. It can be considered an arithmetic analog of the hodge conjecture. The hodge conjecture simply means that shapes fit inside shapes. 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. Then every hodge class on x is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of x. 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. Hodge made an additional, stronger conjecture than the integral hodge conjecture.
Hodge, is a method for studying the cohomology groups of a smooth manifold m using partial differential equations. The hodge conjecture asserts that, for particularly nice types of spaces called projective algebraic varieties, the pieces called hodge cycles are actually rational. Three lectures on the hodge conjecture deduce from this that for 2k hodge conjecture is true for all x of dimension conjecture 1. Assuming the generalized hodge conjecture and a conjecture of blochbeilinson, we show that. Products of it with itself give candidates for counterexamples to the hodge conjecture which may be of interest. Nov 09, 2015 we prove the hard lefschetz theorem and the hodge riemann relations for a commutative ring associated to an arbitrary matroid m. Hi all, in my engineering class we had to choose to do a product on any stem topic, and i chose the millenium problems. Hodgetype conjecture for higher chow groups 949 conjecture see 21, 29, 32. Thomas gives a proof to descending the hodge conjecture into showing that every n,nhodge class in a. Serres conjecture for gl2 and to the geometric breuilmezard conjecture for potentially barsottitate representations. Hodge conjecture, in algebraic geometry, assertion that for certain nice spaces projective algebraic varieties, their complicated shapes can be covered approximated by a collection of simpler geometric pieces called algebraic cycles. In this paper we introduce an automorphic variant of the deligne conjecture for tensor product of two motives over a quadratic imaginary eld.
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