In this note we describe aspects of the cohomology of coherent sheaves on a complete toric variety X over a field k and, more generally, the local cohomology, with supports in a monomial ideal, of a finitely generated module over a polynomial ring S. This leads to an efficient way of computing such cohomology, for which we give explicit algorithms. The problem is finiteness. The ith local cohom...

We relate the cohomology of the Orlik-Solomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The Orlik-Solomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the complement, the cohomology of which is isomorphic to that of the complement with coefficients in ...

In the theory of deformation of Okamoto-Painlevé pair (S, Y ), a local cohomology group H D (ΘS(− logD)) plays an important role. In this paper, we estimate the local cohomology group of pair (S, Y ) for several types, and obtain the following results. For a pair (S, Y ) corresponding to the space of initial conditions of the Painlevé equations, we show that the local cohomology group H D (ΘS(−...

In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL.2;C/. The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twist...

Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham c...

Let $R = bigoplus_{n in mathbb{N}_{0}} R_{n}$ be a standardgraded ring, $M$ be a finitely generated graded $R$-module and $J$be a homogenous ideal of $R$. In this paper we study the gradedstructure of the $i$-th local cohomology module of $M$ defined by apair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. Moreprecisely, we discuss finiteness property and vanishing of thegraded components $H^...

Let R be a polynomial ring in finitely many variables over the integers, and fix an ideal a of R. We prove that for all but finitely prime integers p, the Bockstein homomorphisms on local cohomology, H a (R/pR) −→ H k+1 a (R/pR), are zero. This provides strong evidence for Lyubeznik’s conjecture which states that the modules H a (R) have a finite number of associated prime ideals.

Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian scheme of Y , homological invariants derived from logarithmic differential forms along Y , and their relationship with the property that Y be a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local...

Let V be a holomorphic vector bundle on a complex manifold M , with a flat connection ∇. We shall make the following three assumptions: (1) M is an open subset of a bigger complex manifold M . (2) The boundary D = M \M is a divisor with normal crossing singularities. (3) The connection ∇ is unipotent along D. Under these assumptions, V has a canonical extension to a vector bundle V on M . Since...