Related papers: Vanishing theorems

We describe the standard and Leray filtrations on the cohomology groups with compact supports of a quasi projective variety with coefficients in a constructible complex using flags of hyperplane sections on a partial compactification of a…

We establish a positive characteristic analogue of intersection cohomology theory for variations of Hodge structure. It includes: a) the de Rham-Higgs comparison theorem for the intersection de Rham complex; b) the $E_1$-degeneration…

We introduce preordered semi-orthogonal decompositions (psod-s) of dg-categories. We show that homotopy limits of dg-categories equipped with compatible psod-s carry a natural psod. This gives a way to glue semi-orthogonal decompositions…

We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite $\Gamma_1(p^\infty)$-level (defined with respect to a Borel subgroup) vanishes above the middle degree, under the assumption that the group of…

We provide a unified proof of all known examples of locally compact groups that enjoy the Howe-Moore property, namely, the vanishing at infinity of all matrix coefficients of the group unitary representations that are without non-zero…

We introduce the notion of infinitesimal variations of mixed Hodge structures and invariants associated to them. We describe these invariants in the case of a pair $(X,Y)$ with $X$ a Fano 3-fold and $Y$ a smooth anticanonical K3 surface and…

Several algebraic criteria, reflecting displacement properties of transformation groups, have been used in the past years to prove vanishing of bounded cohomology and stable commutator length. Recently, the authors introduced the property…

A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed characteristic.) It…

We solved the long-standing problem of describing the cohomology ring of semiample hypersurfaces in complete simplicial toric varieties. Also, the monomial-divisor mirror map is generalized to a map between the whole Picard group and the…

We prove an equivariant version of the local splitting theorem for tame Poisson structures and Poisson actions of compact Lie groups. As a consequence, we obtain an equivariant linearization result for Poisson structures whose transverse…

Using techniques of projective geometry, we give elementary proofs of two theorems concerning Hagge configurations.

We show that compact K\"ahler manifolds have the rational cohomology ring of complex projective space provided a weighted sum of the lowest three eigenvalues of the K\"ahler curvature operator is positive. This follows from a more general…

This article contains a new argument which proves vanishing of the first cohomology for negative vector bundles over a complex projective variety if the rank of the bundle is smaller than the dimension of the base. Similar argument is…

We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several…

Andreotti-Vesentini, Ohsawa, Gromov, Koll\'ar, among others, have observed that Hodge theory could be extended to (non compact) K\"ahler complete manifolds, within the L^2 framework. Also, many vanishing theorems on projective or K\"ahler…

Musta\c{t}\u{a} a and Popa introduce the notion of Hodge ideals for an effective $\mathbb{Q}$-divisor $D$ and prove a vanishing theorem for Hodge ideals, which generalizes Nadel vanishing for multiplier ideals. However, their proof needs an…

A characteristic property of cohomology with compact support is the long exact sequence that connects the compactly supported cohomology groups of a space, an open subspace and its complement. Given an arbitrary cohomology theory of…

Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. Fix $n\geq 2$, and an integer $d$. A pair $(E,\phi)$ over $X$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $X$ and a section…

We generalise the construction of Rouquier complexes to the setting of singular Soergel bimodules by taking minimal complexes of the restriction of Rouquier complexes. We show that they retain many of the properties of ordinary Rouquier…

The Kodaira-Nakano Vanishing Theorem has been generalized to the relative setting by A. Sommese. We prove a version of this theorem for non-compact manifolds. As an apllication, we prove that the cohomology of a fiber of a symplectic…