Papers by Yakov Varshavsky
arXiv (Cornell University), Aug 15, 2019
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Journal of Differential Geometry, 1998
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arXiv (Cornell University), Sep 23, 1999
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arXiv (Cornell University), May 13, 2002
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arXiv (Cornell University), May 4, 2023
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arXiv (Cornell University), Oct 5, 2020
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Algebraic Geometry
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The goal of this paper is to give a simple proof of Deligne’s conjecture (proven by Fujiwara) and... more The goal of this paper is to give a simple proof of Deligne’s conjecture (proven by Fujiwara) and to generalize it to the situation appearing in our joint project [KV] with David Kazhdan on the global Langlands correspondence over function fields. Our proof applies without any changes to more general situations like algebraic spaces or Deligne–Mumford stacks. Introduction Suppose we are given a correspondence X a1 ←− A a2 −→ X of schemes of finite type over a separably closed field k, an “l-adic sheaf” F ∈ D ctf(X,Ql) and a morphism u : a2!a ∗ 1F → F . If a1 is proper, then u gives rise to an endomorphism RΓc(u) : RΓc(X,F)→ RΓc(X,F). WhenX is proper, the general Lefschetz-Verdier trace formula [Il, Cor. 4.7] asserts that the trace Tr(RΓc(u)) equals the sum ∑ β∈π0(F ix(a)) LTβ(u), where Fix(a) := {y ∈ A | a1(y) = a2(y)} is the scheme of fixed points of a, and LTβ(u) is a so called “local term” of u at β. This result has two defects: it fails when X is not proper, and the “local terms...
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Cornell University - arXiv, Dec 14, 2020
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Israel Journal of Mathematics
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Advances in Mathematics
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arXiv: Algebraic Geometry, 2021
We prove that the trace of the Frobenius endofunctor of the category of automorphic sheaves with ... more We prove that the trace of the Frobenius endofunctor of the category of automorphic sheaves with nilpotent singular support maps isomorphically to the space of unramified automorphic functions, settling a conjecture from [AGKRRV1]. More generally, we show that traces of Frobenius-Hecke functors produce shtuka cohomologies.
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The goal of this note is to supply proofs of two theorems stated [AGKRRV], which deal with the Lo... more The goal of this note is to supply proofs of two theorems stated [AGKRRV], which deal with the Local Term map for the Frobenius endomorphism of Artin stacks of finite type over Fq. Namely, we show that the “true local terms” of the Frobenius endomorphism coincide with the “naive local terms” and that the “naive local terms” commute with !-pushforwards. The latter result is a categorical version of the classical Grothendieck–Lefschetz trace formula.
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arXiv: Representation Theory, 2015
The stable center conjecture asserts that the space of stable distributions in the Bernstein cent... more The stable center conjecture asserts that the space of stable distributions in the Bernstein center of a reductive p-adic is closed under convolution. It is closely related to the notion of an L-packet and endoscopy theory. We describe a categorical approach to the depth zero part of the conjecture. As an illustration of our method, we show that the Bernstein projector to the depth zero spectrum is stable.
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In this work we give a geometric proof of Hrushovski’s generalization of the LangWeil estimates o... more In this work we give a geometric proof of Hrushovski’s generalization of the LangWeil estimates on the number of points in the intersection of a correspondence with the graph of Frobenius.
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Let $G$ be a connected reductive group over $F=\mathbb F_q((t))$ splitting over $\overline{\mathb... more Let $G$ be a connected reductive group over $F=\mathbb F_q((t))$ splitting over $\overline{\mathbb F_q}((t))$. Following [KV], every tamely unramified Langlands parameter $\lambda:W_F\to{}^L G(\overline{\mathbb Q_l})$ in general position gives rise to a finite set $\Pi_{\lambda}$ of irreducible admissible representations of $G(F)$, called the $L$-packet. The goal of this work is to provide a geometric description of characters $\chi_{\pi}$ of all $\pi\in\Pi_{\lambda}$ in terms of homology of affine Springer fibers. As an application, we give a geometric proof of the stability of sum $\chi_{\lambda}^{st}:=\sum_{\pi\in\Pi_{\lambda}}\chi_{\pi}$. Furthermore, as in [KV] we show that the $\chi_{\lambda}^{st}$'s are compatible with inner twistings.
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G be a connected reductive group over an algebraically closed field k, set K := k((t)), let γ ∈ G... more G be a connected reductive group over an algebraically closed field k, set K := k((t)), let γ ∈ G(K) be a regular semisimple element, let Fl be the affine flag variety of G, and let Flγ ⊂ Fl be the affine Springer fiber at γ. For every element w of the affine Weyl group W̃ of G, we denote by Fl ⊂ Fl the corresponding affine Schubert variety, and set Fl γ := Flγ ∩Fl ≤w ⊂ Flγ . The main result of this paper asserts that if w1, . . . , wn ∈ W̃ are sufficiently regular, then the natural map Hi(∪ n j=1 Fl ≤wj γ ) → Hi(Flγ) is injective for every i ∈ Z. This result plays an important role in our work [BV]. To prove the result we show that every affine Schubert variety can be written as an intersection of closures of UB(K)-orbits, where B runs over Borel subgroups containing a fixed maximal torus T , and UB denotes the unipotent radical of B.
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Papers by Yakov Varshavsky