Related papers: A Vanishing Theorem for Varieties with Finitely Ma…

We continue our study of the Noether-Lefschetz loci in toric varieties and investigate deformation of pairs (V,X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a odd dimensional simplicial projective…

We show that the mod $p$ cohomology of a simple Shimura variety treated in Harris-Taylor's book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we…

We first prove that the Whitehead group of a torsion-free virtually solvable linear group vanishes. Next we make a reduction of the fibered isomorphism conjecture from virtually solvable groups to a class of virtually solvable Q-linear…

We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in $\C^*$ (or more generally, with coefficients in the complex points of a tori over $\C$) vanish, where the…

We study the residual Eisenstein cohomology of semisimple groups in the context of maximal parabolic subgroups which remain maximal over $\mathbb{R}$. Under certain general hypotheses, we show that these residual representations are…

In this paper, we clarify the mistakes made in the former article entitled "Vanishing Theorems on Toric Varieties in Positive Characteristic". On the other hand, we use the positive characteristic method to reprove the Bott vanishing…

We explore a relationship between the classical representation theory of a complex, semisimple Lie algebra \g and the resonance varieties R(V,K)\subset V^* attached to irreducible \g-modules V and submodules K\subset V\wedge V. In the…

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…

Let $E$ be a vector bundle and $L$ be a line bundle over a smooth projective variety $X$. In this article, we give a condition for the vanishing of Dolbeault cohomology groups of the form $H^{p,q}(X,\SSS^{\alpha}E\otimes \wedge^{\beta}…

A general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems is established. This is applied to the determination of the monodromy action on the first cohomology group of…

We prove several asymptotic vanishing theorems for Frobenius twists of ample vector bundles in positive characteristic. As an application, we prove a generalization of the Bott-Danilov-Steenbrink vanishing theorem for ample vector bundles…

We use a Mayer-Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality…

We prove some injectivity, torsion-free, and vanishing theorems for simple normal crossing pairs. Our results heavily depend on the theory of mixed Hodge structures on compact support cohomology groups. We also treat several basic…

We give a new proof of vanishing result of Esnault for the cohomology of constructible sheaves in the tower of ``mock'' Frobenius covers of projective space. The key idea is to use (a global form of) the perversity of nearby cycles.

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and $M\ne SO_0(2,2)/SO(2)\tm SO(2).$ Let E be any vector bundle over M, Then any E-valued $L^2$ harmonic 1-form…

We prove vanishing results of the cohomology groups of Aomoto complex over arbitrary coefficient ring for real hyperplane arrangements. The proof is using minimality of arrangements and descriptions of Aomoto complex in terms of chambers.…

We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber…

For all algebraic groups over non-Archimedean local fields, the bounded cohomology vanishes. This follows from the corresponding statement for automorphism groups of Bruhat--Tits buildings, which hinges on the solution to the flatmate…

We show that the hypercohomology of most character twists of perverse sheaves on a complex abelian variety vanishes in all non-zero degrees. As a consequence we obtain a vanishing theorem for constructible sheaves and a relative vanishing…

Let $X$ be a toric variety. We establish vanishing (and non-vanishing) results for the sheaves $R^if_*\Omega^p_{\tilde X}(\log E)$, where $f: \tilde{X} \to X$ is a strong log resolution of singularities with reduced exceptional divisor $E$.…