Related papers: Hodge structures on abelian varieties of type III

We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without nontrivial…

In this paper, we establish an innovative framework in logarithmic Hodge theory for toroidal varieties, introducing weighted toroidal structures and developing a systematic obstruction theory for Hodge classes. Building upon recent advances…

Let A be an abelian fourfold. We prove the Standard Conjecture of Hodge type for A. By combining this result with a theorem of Clozel we deduce that numerical equivalence on A coincides with l-adic homological equivalence on A for…

We study the Mumford--Tate conjecture for hyperk\"{a}hler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional…

In this paper we prove the Mumford-Tate conjecture in degree 2 for the product of an abelian surface $A$ and a K3 surface $X$ over a finitely generated field $K \subset \mathbb{C}$. The Mumford-Tate conjecture is a precise way of saying…

We consider a recently proposed two-dimensional Abelian model for a Hodge theory, which is neither a Witten type nor a Schwarz type topological theory. It is argued that this model is not a good candidate for a Hodge theory since, on-shell,…

We prove Bloch's conjecture for correspondences on powers of complex abelian varieties, that are "generically defined". As an application we establish vanishing results for (skew-)symmetric cycles on powers of abelian varieties and we…

For each prime number $p$ and each integer $g \geqslant 5$, we construct infinitely many abelian varieties of dimension $g$ over $\overline{\mathbb{F}}_p$ satisfying the standard conjecture of Hodge type. The main tool is a recent theorem…

Tate's theorem (Invent. Math. 1966)implies that the Tate conjecture holds for any abelian variety over a finite field whose Q_l-algebra of Tate classes is generated by those of degree 1. We construct families of abelian varieties over…

For any subfield K of the complex numbers which is not contained in an imaginary quadratic number field, we construct conjugate varieties whose algebras of K-rational (p,p)-classes are not isomorphic. This compares to the Hodge conjecture…

We discuss variations of Hodge structures on abelian varieties that arise from intersecting translates of theta divisors with a special focus on the case of abelian varieties of dimension 4

We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the…

We describe in some details an idea of M. Kontsevich how one can try to find a counterexample to the Hodge conjecture using tropical geometry.

We prove the conjectures of Hodge and Tate for any six-dimensional hyper-K\"ahler variety that is deformation equivalent to a generalized Kummer variety.

The goal is to verify the Hodge conjecture (and some related conjectures) for certain moduli spaces. It is shown that the (generalized) Hodge conjecture holds for the projective moduli spaces of vector bundles over an abelian or K3 surface…

We survey the Mumford construction of degenerating abelian varieties, with a focus on the analytic version of the construction, and its relation to toric geometry. Moreover, we study the geometry and Hodge theory of multivariable…

With any hyper-K\"ahler variety $K$ of generalized Kummer type is associated via Hodge theory a K3 surface $S_K$. We show how they are related geometrically through a moduli space of sheaves on $S_K$. As a consequence, building…

We prove the numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension, and also in some related cases.

The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of…

This paper determines all the possible endomorphism algebras for polarizable Q-Hodge structures of type (n,0,...,0,n). This generalizes the classification of the possible endomorphism algebras of abelian varieties by Albert and Shimura. As…