Papers by Yakov Varshavsky
arXiv (Cornell University), Aug 15, 2019
The goal of this paper is to provide a categorical framework that leads to the definition of shtu... more The goal of this paper is to provide a categorical framework that leads to the definition of shtukasà la Drinfeld and of excursion operatorsà la V. Lafforgue. We take as the point of departure the Hecke action of Rep(Ǧ) on the category Shv(BunG) of sheaves on BunG, and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.
Journal of Differential Geometry, 1998
In this paper we show that certain Shimura varieties, uniformized by the product of complex unit ... more In this paper we show that certain Shimura varieties, uniformized by the product of complex unit balls, can be p-adically uniformized by the product of Drinfeld upper half-spaces and their equivariant coverings. We also extend a p-adic uniformization to automorphic vector bundles. It is a continuation of our previous work [38] and contains all cases (up to a central modification) of a uniformization by known p-adic symmetric spaces. The idea of the proof is to show that an arithmetic quotient of the product of Drinfeld upper half-spaces cannot be anything else than a certain unitary Shimura variety. Moreover, we show that difficult theorems of Yau and Kottwitz appearing in [38] may be avoided.
arXiv (Cornell University), Sep 23, 1999
In this paper we recall the construction and basic properties of complex Shimura varieties and sh... more In this paper we recall the construction and basic properties of complex Shimura varieties and show that these properties actually characterize them. This characterization immediately implies the explicit form of Kazhdan's theorem on the conjugation of Shimura varieties. As a further corollary, we show that each Shimura variety corresponding to an adjoint group has a canonical model over its reflex field. We also indicate how this characterization implies the existence of a p-adic uniformization of certain unitary Shimura varieties. In the appendix we give a complete scheme-theoretic proof of Weil's descent theorem.
arXiv (Cornell University), May 13, 2002
In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) F-bu... more In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) F-bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected curve over a finite field, a split reductive group G, and an irreducible algebraic representation ω of (G) n /Z(G). Our spaces generalize moduli spaces of F-sheaves, studied by Drinfeld and Lafforgue, which correspond to the case G = GL r and ω is the tensor product of the standard representation and its dual. The importance of the moduli spaces of F-bundles is due to the belief that Langlands correspondence is realized in their cohomology.
arXiv (Cornell University), May 4, 2023
We prove that the adjoint equivariant derived category of a reductive group G is equivalent to th... more We prove that the adjoint equivariant derived category of a reductive group G is equivalent to the appropriately defined monoidal center of the torus-equivariant version of the Hecke category. We use this to give new proofs, independent of sheaf-theoretic set up, of the fact that the Drinfeld center of the abelian Hecke category is equivalent to the abelian category of unipotent character sheaves; and of a characterization of strongly-central sheaves on the torus.
arXiv (Cornell University), Oct 5, 2020
We define a new geometric object-the stack of local systems with restricted variation. We formula... more 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). We formulate a conjecture that makes precise the connection between the category of automorphic sheaves and the space of automorphic functions. D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY 3.6. Associated pairs and semi-simple local systems 3.7. Analysis of connected/irreducible components 4. Comparison with the Betti and de Rham versions of LocSys dR G (X) 4.1. Relation to the Rham version 4.2. A digression: ind-closed embeddings 4.3. Uniformization and the proof of Theorem 4.1.10 4.4. Algebraic proof of Proposition 4.3.5 4.5. The Betti version of LocSys G (X) 4.6. The coarse moduli space of Betti local systems 4.7. Relationship of the restricted and Betti versions 4.8. Comparison of LocSys restr G (X) vs LocSys Betti G (X) via the coarse moduli space 5. Geometric properties of LocSys restr G (X) 5.1. "Mock-properness" of red LocSys restr G (X) 5.2. A digression: ind-algebraic stacks 5.3. Mock-affineness and coarse moduli spaces 5.4. Coarse moduli spaces for connected components of LocSys restr G (X) 6. The formal coarse moduli space 6.1. The coarse moduli space in the Betti setting 6.2. Property W 6.3. Property A 6.4. A digression: the case of algebraic groups 6.5. The case of pro-algebraic groups 6.6. Proof of Theorem 6.5.7 6.7. Proof of Theorem 5.4.2 6.8. Proof of Theorem 6.7.8 7. Quasi-coherent sheaves on a formal affine scheme 7.1. Formal affine schemes: basic properties 7.2. Proof of Proposition 7.1.12 7.3. Mapping affine schemes into a formal affine scheme 7.4. Semi-rigidity and semi-passable prestacks 7.5. Duality for semi-passable prestacks 7.6. The functor of !-global sections 7.7. The functor of !-global sections on a formal affine scheme 7.8. Applications of 1-affineness 7.9. Compact generation of QCoh(Maps(Rep(G), H)) 7.10. Enhanced categorical trace 8. The spectral decomposition theorem 8.1. Actions of Rep(G) ⊗X-lisse 8.2. The coHom symmetric monoidal category 8.3. Maps vs coHom 8.4. Spectral decomposition vs actions 8.5. A rigidified version 9. Categories adapted for spectral decomposition 9.1. The Betti case 9.2. The heriditary property of being adapted 9.3. Proof of Theorem 9.2.2: identifying the essential image 9.4. Proof of Theorem 8.3.7, Betti and de Rham contexts 9.5. Proof of Theorem 8.3.7,étale context over a field of characteristic 0 9.6. Proof of Theorem 8.3.7,étale context over a field of positive characteristic 9.7. A simple proof of Theorem 5.4.2 9.8. Complements: de Rham and Betti spectral actions 10. Other examples of categories adapted for spectral decomposition 10.1. The case of Lie algebras GEOMETRIC LANGLANDS WITH NILPOTENT SINGULAR SUPPORT 10.2. The space of maps of Lie algebras 10.3. Proof of Theorem 10.1.3 10.4. Back to the Betti case 11. Ran version of Rep(G) and Beilinson's spectral projector 11.1. The category Rep(G)Ran 11.2. Relation to the lisse version 11.3. Rigidity 11.4. Self-duality 11.5. The progenitor of the projector 11.6. The progenitor as a colimit 11.7. Explicit construction of the Hecke isomorphisms 12. The spectral projector and localization 12.1. The progenitor for coHom 12.2. Abstract version of factorization homology 12.3. Proofs of Propositions 12.1.2 and 12.1.7 12.4. Applications to C Betti Ran 12.5. Applications to CRan 12.6. Identification of the diagonal 12.7. Localization on LocSys G (X) 12.8. Tensor products over Rep(G) vs. QCoh(LocSys restr G (X)) 13. Spectral projector and Hecke eigen-objects 13.1. Beilinson's spectral projector-abstract form 13.2. A multiplicativity property of the projector 13.3. The spectral (sub)category 13.4. Beilinson's spectral projector-the universal case 13.5. Beilinson's spectral projector-the general case 13.6. A version with parameters 13.7.
Algebraic Geometry
In this work, we give an algebro-geometric proof of Hrushovski's generalization of the Lang-Weil ... more In this work, we give an algebro-geometric proof of Hrushovski's generalization of the Lang-Weil estimates on the number of points in the intersection of a correspondence with the graph of the Frobenius map. This result has numerous applications to various areas of mathematics, including model theory, algebraic dynamics, group theory and arithmetic algebraic geometry.
Given a fixed integer n, we consider closed subgroups Ᏻ of GL n ޚ( p), where p is sufficiently ... more Given a fixed integer n, we consider closed subgroups Ᏻ of GL n ޚ( p), where p is sufficiently large in terms of n. Assuming that the identity component of the Zariski closure G of Ᏻ in GL n,ޑ p does not admit any nontrivial torus as quotient group, we give a condition on the (mod p) reduction of Ᏻ which guarantees that Ᏻ is of bounded index in GL n ޚ( p) ∩ G(ޑ p).
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...
Cornell University - arXiv, Dec 14, 2020
We identify the category Shv Nilp (BunG) of automorphic sheaves with nilpotent singular support w... more We identify the category Shv Nilp (BunG) of automorphic sheaves with nilpotent singular support with its own dual, and relate this structure to the Serre functor on Shv Nilp (BunG) and miraculous duality. D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY 4.2. Kernels defined by objects from Shv Nilp (BunG) 4.3. Pairings against sheaves with nilpotent singular support, revisited 4.4. A refined version of Theorem 1.3.7 4.5. A converse statement 4.6. A version for a universally Nilp-cotruncative quasi-compact open substack 4.7. Proof of Theorem 4.5.2 5. Serre functor on Shv Nilp (BunG) 5.1. The Serre functor 5.2. Serre vs pseudo-identity 5.3. Duality-adapted pairs, complements 5.4. Constraccessible pairs, complements 5.5. Serre pairs 5.6. Relation to the miraculous functor 5.7. Relation to the miraculous functor, continued 5.8. The (non)-Serre property of BunG Appendix A. Sheaves on stacks A.1. The basics A.2. Verdier-compatible stacks A.3. Base change maps A.4. Verdier self-duality A.5. Specifying singular support A.6. Categorical Künneth formulas Appendix B. Functors defined by kernels and miraculous duality B.1. Functors defined by kernels B.2. A discontinuous version B.3. Functors co-defined by kernels B.4. The miraculous functor B.5. Adjunctions for functors defined by kernels B.6. Adjunctions and Verdier duality B.7. Miraculous stacks B.8. A criterion for admitting an adjoint B.9. Proof of Theorem B.8.8 B.10. An example: the ULA property Appendix C. Sheaves on non quasi-compact algebraic stacks C.1. Cotruncative substacks C.2. The non-quasi-compact case: the "co"-category C.3. Verdier duality in the non-quasi-compact case C.4. Functors defined by kernels in the non-quasi-compact case C.5. Miraculous stacks-the non quasi compact case References
Israel Journal of Mathematics
We apply the dimension theory developed in [BKV] to establish some of Lusztig's conjectures [Lu2]... more We apply the dimension theory developed in [BKV] to establish some of Lusztig's conjectures [Lu2] on S-cells in affine Weyl groups. Contents 10 5. Proof of Proposition 4.5 13 6. Flatness conjecture 14 References 15
Advances in Mathematics
The goal of this work is to construct a perverse t-structure on the ∞-category of ℓ-adic L G-equi... more The goal of this work is to construct a perverse t-structure on the ∞-category of ℓ-adic L G-equivariant sheaves on the loop Lie algebra L g and to show that the affine Grothendieck-Springer sheaf S is perverse. Moreover, S is an intermediate extension of its restriction to the locus of "compact" elements with regular semi-simple reduction. Note that classical methods do not apply in our situation because L G and L g are infinite-dimensional ind-schemes. Date: October 13, 2020. 3.4. Codimension of strata 4. Geometry of the affine Grothendieck-Springer fibration 4.1. Generalities 4.2. The fibration over a regular stratum 4.3. The fibration over a general stratum 4.4. The (semi)-smallness Part 3. Sheaves on ∞-stacks and perverse t-structures 5. Categories of ℓ-adic sheaves on ∞-stacks 5.1. Limits and colimits of ∞-categories 5.2. Sheaves on qcqs algebraic spaces 5.3. Sheaves on ∞-stacks 5.4. Fp-locally closed embeddings 5.5. Infinity-stacks admitting gluing of sheaves 5.6. Endomorphisms of ω X 6. Perverse t-structures 6.1. Generalities 6.2. The case of algebraic spaces of finite type over k 6.3. The case of placid ∞-stacks 6.4. The case of placidly stratified ∞-stacks 6.5. Semi-small morphisms Part 4. The affine Springer theory 7. Application to the affine Springer theory 7.1. Main theorem 7.2. Perverse t-structure on [(L g) • / L G] Part 5. Appendices Appendix A. Categorical framework A.1. A variant of Simpson's construction A.2. The case of ∞-categories of sheaves A.3. Passing to pro-categories A.4. Extending of classes of morphisms A.5. Examples and complements A.6. Proof of Theorem A.2.8 Appendix B. Completion of proofs. B.1. Quotients of ind-schemes B.2. Proof of Theorem 4.1.9 and Theorem 4.3.3 B.3. Proof of Proposition 3.1.11 B.4. Proof of Theorem 3.4.7 List of main terms and symbols References
Let G be a connected reductive group over a local non-archimedean field F. The stable center conj... more Let G be a connected reductive group over a local non-archimedean field F. The stable center conjecture provides an intrinsic decomposition of the set of equivalence classes of smooth irreducible representations of G(F), which is only slightly coarser than the conjectural decomposition into L-packets. In this work we propose a way to verify this conjecture for depth zero representations. As an illustration of our method, we show that the Bernstein projector to the depth zero spectrum is stable.
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.
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.
The goal of this note it give a short geometric proof of a theorem of Hrushovski [Hr] asserting t... more The goal of this note it give a short geometric proof of a theorem of Hrushovski [Hr] asserting that an intersection of a correspondence with a graph of a sufficiently large power of Frobenius is non-empty. Contents Introduction 1 1. Locally invariant subsets 4 2. Main technical result 6 3. Geometric construction of Pink [Pi] 9 4. Formula for the intersection number 12 5. Proof of the main theorem 16 References 18
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.
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.
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.
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