Related papers: The construction problem for Hodge numbers modulo …

Using constructions of Voisin, we exhibit a smooth projective variety defined over a number field k and two complex embeddings of k, such that the two complex manifolds induced by these embeddings have non isomorphic cohomology algebras…

We show that the Hodge numbers of the moduli space of stable rank two sheaves with primitive determinant on a K3 surface coincide with the Hodge numbers of an appropriate Hilbert scheme of points on the K3 surface. The precise result is:…

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…

In this paper, we prove that the statement: ``The (Generalized) Hodge Conjecture holds for codimension-two cycles on a smooth projective variety $X$" is a birationally invariant statement, that is, if the statement is true for $X$, it is…

We introduce and study Hodge-de Rham numbers for compact almost complex 4-manifolds, generalizing the Hodge numbers of a complex surface. The main properties of these numbers in the case of complex surfaces are extended to this more general…

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in…

Based on results of Harding, Heunen, Lindenhovius and Navara, (2019), we give a connection between the category of AW*-algebras and their normal Jordan homomorphisms and a category COG of orthogemetries, which are structures that are…

Let $p$ be a prime number. Irreducible cyclic codes of length $p^2-1$ and dimension $2$ over the integers modulo $p^h$ are shown to have exactly two nonzero Hamming weights. The construction uses the Galois ring of characteristic $p^h$ and…

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration…

We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…

We survey recent results about the Torelli question for holomorphic-symplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup W, of the isometry group of the weight 2 Hodge structure,…

We investigate deformations of Milnor algebras of smooth homogeneous polynomials, and prove in particular that any smooth degree $d$ homogeneous polynomial in $n+1$ variables that is not of Sebastiani-Thom type is determined by the degree…

On a smooth projective variety over the complex numbers, there is the coniveau from the coniveau filtration, which is called geometric coniveau. On the same variety, there is another coniveau from the maximal sub-Hodge structure, which is…

In this article, we identify the existence of a divisibility relationship between the number of ring homomorphisms and surjective group homomorphisms. We demonstrate that for finite cyclic structures, the number of ring homomorphisms from…

The predictions of the Mirror Symmetry are extended in dimensions n>3 and are proven for projective complete intersections Calabi-Yau varieties. Precisely, we prove that the total collection of rational Gromov-Witten invariants of such…

We prove the Hard Lefschetz theorem and Hodge-Riemann relations for certain rings which resemble the cohomology rings of projectivizations of globally generated vector bundles over toric varieties. This proves new cases of the standard…

It is well known that numerical quantities arising from the theory of D-modules are related to invariants of singularities in birational geometry. This paper surveys a deeper relationship between the two areas, where the numerical…

This is a survey of vanishing and positivity theorems for Hodge modules, and their recent applications to birational and complex geometry, expanding on my lecture at the 2015 AMS Summer Institute.

Let $X$ be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products $Sym^{n}X$ when the cohomology of $X$ is given by exterior products of cohomology classes with odd degree.…

This is my dissertation about digraphs ordered by pp-constructability. We study in particular smooth digraphs, i.e., digraphs without sources or sinks, tournaments and semicomplete digraphs, orientations of paths and cycles, digraphs with…