Related papers: Classical and irregular Hodge numbers
We give a presentation of localized affine and degenerate affine Hecke algebras of arbitrary type in terms of weights of the polynomial subalgebra and varied Demazure-BGG type operators. We offer a definition of a graded algebra…
A classical result of A. Connes asserts that the Frechet algebra of smooth functions on a smooth compact manifold X provides, by a purely algebraic procedure, the de Rham cohomology of X. Namely the procedure uses Hochschild and cyclic…
Given a smooth compact complex surface together with a holomorphic line bundle on it, using the theory of Hodge modules, we compute the twisted Hodge groups/numbers of Hilbert schemes (or Douady spaces) of points on the surface with values…
The aim of this paper is to extend Gerstenhaber formal deformations of algebras to the case of Hom-Alternative and Hom-Malcev algebras. We construct deformation cohomology groups in low dimensions. Using a composition construction, we give…
We obtain a cohomological interpretation for Batyrev's stringy Hodge numbers in the full generality in which they are defined. In a previous paper, the second and third authors used motivic integration to define the stringy Hodge--Deligne…
We prove existence and uniqueness of complex Hodge structures on modular functors. The proof is based on the non-Abelian Hodge correspondence and Ocneanu rigidity. Given a modular functor, we explain how its Hodge numbers fit into a…
We give an explicit formula for the Hodge filtration on the $\mathscr{D}_X$-module $O_X(*Z)f^{1-\alpha}$ associated to the effective $\mathbb{Q}$-divisor $D=\alpha\cdot Z$, where $0<\alpha\le1$ and $Z=(f=0)$ is an irreducible hypersurface…
We investigate the relation between the Hodge theory of a smooth subcanonical $n$-dimensional projective variety $X$ and the deformation theory of the affine cone $A_X$ over $X$. We start by identifying $H^{n-1,1}_{\mathrm{prim}}(X)$ as a…
Let $f: X \to \mathbb{A}^1$ be a regular function on a smooth complex algebraic variety $X$. We formulate and prove an equivalence between the algebraic formal twisted de Rham complex of $f$ and the vanishing cycles with respect to $f$ as…
In this paper we prove the smoothness of the moduli space of Landau-Ginzburg models. We formulate and prove a Tian-Todorov theorem for the deformations of Landau-Ginzburg models, develop the necessary Hodge theory for varieties with…
We study several deformation functors associated to the normalization of a reduced curve singularity $(X,0) \subset (\c^n,0)$. The main new results are explicit formulas, in terms of classical invariants of (X,0), for the cotangent…
There is a natural descending filtration on the singular cohomology of a complex smooth projective variety called the coniveau filtration. The generalized Hodge conjecture would imply, rather trivially, that the coniveau filtration is…
Fres\'an, Sabbah, and Yu constructed motives $\mathrm{M}_{n+1}^k(\mathrm{Kl})$ over $\mathbb{Q}$ encoding symmetric power moments of Kloosterman sums in $n$ variables. When $n=1$, they use the irregular Hodge filtration on the exponential…
We prove by induction on dimension the Hodge conjecture for smooth complex projective varieties. Let $X$ be a smooth complex projective variety. Then $X$ is birational to a possibly singular projective hypersurface, hence to a smooth…
Recent discoveries make it possible to compute the K-theory of certain rings from their cyclic homology and certain versions of their cdh-cohomology. We extend the work of G. Corti\~nas et al. who calculated the K-theory of, in addition to…
We extend the Hodge atoms framework of Katzarkov--Kontsevich--Pantev--Yu to one-parameter conifold degenerations of Calabi--Yau threefolds. For a degeneration $\pi\colon X \to \Delta$ whose central fiber $X_0$ has $r$ ordinary double…
We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a…
We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study…
We prove an algebraic formula, conjectured by M. Kontsevich, for computing the monodromy of the vanishing cycles of a regular function on a smooth complex algebraic variety.
In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a…