For a G-variety X with an open orbit, we define its boundary ∂X as the complement of the open orbit. The action sheaf SX is the subsheaf of the tangent sheaf made of vector fields tangent to ∂X. We prove, for a large family of smooth spherical varieties, the vanishing of the cohomology groups Hi(X,SX) for i > 0, extending results of F. Bien and M. Brion [BB96]. We apply these results to study t...

In this paper we study the tangent spaces of the smooth nested Hilbert scheme Hilb n,n−1 (A 2) of points in the plane, and give a general formula for computing the Euler characteristic of a T 2-equivariant locally free sheaf on Hilb n,n−1 (A 2). Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables q and t with non-negative integer coefficient...

Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve C; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the gro...

The classical n-variable Kloosterman sums over the finite field Fp give rise to a lisse Ql-sheaf Kln+1 on Gm,Fp = P 1 Fp − {0,∞}, which we call the Kloosterman sheaf. Let Lp(Gm,Fp ,Sym Kln+1, s) be the L-function of the k-fold symmetric product of Kln+1. We construct an explicit virtual scheme X of finite type over Spec Z such that the p-Euler factor of the zeta function of X coincides with Lp(...

We propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an l-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent b...

We propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an l-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent b...

Quot schemes of quotients a trivial bundle arbitrary rank on nonsingular projective surface X carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The associated generating series Euler characteristics was conjectured to be rational function in Oprea Pandharipande (in, Geom Topol. http://arxiv.org/abs/1903.08787 ) whe...

This article exhibits a particular encoding of logic circuits into a sheaf formalism. The central result of this article is that there exists strictly more information available to a circuit designer in this setting than exists in static truth tables, but less than exists in event-level simulation. This information is related to the timing behavior of the logic circuits, and thereby provides a ...

The goal of this note is to give an example for which Theorem 1.1 fails if we only relax the hypothesis that X is quasi-compact (Propositions 2.3 and 3.1). This example emerged from the author’s investigation on local cohomology of valuation rings [Dat16]. In particular, some results from [Dat16, Sections 6, 7] are reproduced below without citation. Any other outside result we use is accompanie...

A geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g. a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundle over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally ...