Related papers: On a Dynamical Mordell-Lang Conjecture for Coheren…

This article contains a proof of the basic lemma. This lemma, discovered by Beilinson, yields a motivic proof of the Andreotti-Frankel theorem for affine varieties. Next, it is shown that the category of Cohomologically Constructible…

Let F : P^n --> P^n be a morphism of degree d > 1 defined over C. The dynamical Mordell--Lang conjecture says that the intersection of an orbit O_F(P) and a subvariety X of P^n is usually finite. We consider the number of linear…

The goal of this paper is to motivate a boundedness conjecture on nearby slopes of $\ell$-adic sheaves in positive characteristic, and to prove it for smooth curves. For a constructible $\ell$-adic sheaf, we prove the finiteness of the set…

In this paper, we prove the converse of the dynamical Mordell--Lang conjecture in positive characteristic: For every subset $S \subseteq \mathbb{N}_0$ which is a union of finitely many arithmetic progressions along with finitely many…

We survey old and new results on the existence of moduli spaces of semistable coherent sheaves both in algebraic and in complex geometry.

We define a new geometric object--the stack of local systems with restricted variation. We formulate a version of the categorical geometric Langlands conjecture that makes sense for any constructible sheaf theory (such as l-adic sheaves).…

A conjecture of Mumford predicts a complete set of relations between the generators of the cohomology ring of the moduli space of rank 2 semi-stable sheaves with fixed odd degree determinant on a smooth, projective curve of genus at least…

The Mordell--Lang conjecture for abelian varieties states that the intersection of an algebraic subvariety $X$ with a subgroup of finite rank is contained in a finite union of cosets contained in $X$. In this article, we prove a uniform…

We show that there exists a fine moduli space for torsion-free sheaves on a projective surface, which have a "good framing" on a big and nef divisor. This moduli space is a quasi-projective scheme. This is accomplished by showing that such…

We show that the cohomology table of any coherent sheaf on projective space is a convergent--but possibly infinite--sum of positive real multiples of the cohomology tables of what we call supernatural sheaves.

We present a uniform theory of constructible sheaves on arbitrary schemes with coefficients in topological or even condensed rings. This is accomplished by defining lisse sheaves to be the dualizable objects in the derived infinity-category…

The aim of these notes is to generalize Laumon's construction [18] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite…

We define Strebel differentials for stable complex curves, prove the existence and uniqueness theorem that generalizes Strebel's theorem for smooth curves, prove that Strebel differentials form a continuous family over the moduli space of…

We construct moduli stacks of stable sheaves for surfaces fibered over marked nodal curves by using expanded degenerations. These moduli stacks carry a virtual class and therefore give rise to enumerative invariants. In the case of a…

Let \(E\) be a finite-dimensional real vector space. We study invertible objects in the monoidal category of constructible sheaves on \(E\), endowed with the convolution product \(\star\). We show that the inverse of an invertible…

In this paper we give a new and simplified proof of the variational Hodge conjecture for complete intersection cycles on a hypersurface in projective space.

We expand the toolbox of (co)homological methods in computational topology by applying the concept of persistence to sheaf cohomology. Since sheaves (of modules) combine topological information with algebraic information, they allow for…

A conjecture of Morel asserts that the sheaf of A^1-connected components of a simplicial sheaf X is A^1-invariant. A conjecture of Asok-Morel asserts that A^1-connected components of smooth k-schemes coincide with their A^1-chain-connected…

This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse $t$-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends…

This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular…