arXiv subjects

Vladimir Drinfeld

Publications and source records attributed to Vladimir Drinfeld.

34 records · Page 2Linked to original sources

arXiv2014

Fourier transform of algebraic measures

These are notes of a talk based on the work arXiv:1212.3630 joint with A. Aizenbud. Let V be a finite-dimensional vector space over a local field F of characteristic 0. Let f be a function on V of the form $f(x)= ψ(P(x))$, where P is a polynomial on V and $ψ$ is a nontrivial additive character of F. Then it is clear that the Fourier transform of f is well-defined as a distribution on $V^*$. Due to J.Bernstein, Hrushovski-Kazhdan, and Cluckers-Loeser, it is known that the Fourier transform is smooth on a non-empty Zariski-open conic subset of $V^*$. The goal of these notes is to sketch a proof of this result (and some related ones), which is very simple modulo resolution of singularities (the existing proofs use D-module theory in the Archimedean case and model theory in the non-Archimedian one).

math.AG↗

arXiv2014

On a theorem of Braden

We give a new proof of Braden's theorem ([Br]) about \emph{hyperbolic restrictions} of constructible sheaves/D-modules. The main geometric ingredient in the proof is a 1-parameter family that degenerates a given scheme Z equipped with a G_m-action to the product of the attractor and repeller loci.

math.AG↗

arXiv2014

The wave front set of the Fourier transform of algebraic measures

We study the Fourier transform of the absolute value of a polynomial on a finite-dimensional vector space over a local field of characteristic 0. We prove that this transform is smooth on an open dense set. We prove this result for the Archimedean and the non-Archimedean case in a uniform way. The Archimedean case was proved in [Ber]. The non-Archimedean case was proved in [HK] and [CL]. Our method is different from those described in [Ber,HK,CL]. It is based on Hironaka's desingularization theorem, unlike [Ber] which is based on the theory of D-modules and [HK,CL] which is based on model theory. Our method also gives bounds on the open dense set where the Fourier transform is smooth. These bounds are explicit in terms of resolution of singularities. We also prove the same result on the Fourier transform of other measures of algebraic origins.

math.AG↗

arXiv2013

On the notion of geometric realization

We explain why geometric realization commutes with Cartesian products and why the geometric realization of a simplicial set (resp. cyclic set) is equipped with an action of the group of orientation preserving homeomorphisms of the segment [0,1] (resp. the circle). Our approach is very similar to that of A. Besser and D. Grayson.

math.CT↗

arXiv2013

On the Langlands retraction

Given a root system in a vector space V, Robert Langlands defined in 1973 a canonical retraction of V onto the dominant chamber. In this note we give a short review of the material on this retraction (which is well known under the name of "Langlands' geometric lemmas"). The main purpose of this review is to provide a convenient reference for e-print arXiv:1112.2402 by D.Gaitsgory and me, in which the Langlands retraction is used to define a certain coarsening of the Harder-Narasimhan-Shatz stratification of the stack of G-bundles on a smooth projective curve.

math.RT↗

arXiv2012

Character sheaves on unipotent groups in characteristic p>0

These are slides for a talk given by the authors at the conference "Current developments and directions in the Langlands program" held in honor of Robert Langlands at the Northwestern University in May of 2008. The slides can be used as a short introduction to the theory of characters and character sheaves for unipotent groups in positive characteristic, developed by the authors in a series of articles written between 2006 and 2011. We give an overview of the main results of this theory along with a bit of motivation.

math.RT↗

arXiv2012

On some finiteness questions for algebraic stacks

We prove that under a certain mild hypothesis, the DG category of D-modules on a quasi-compact algebraic stack is compactly generated. We also show that under the same hypothesis, the functor of global sections on the DG category of quasi-coherent sheaves is continuous.

math.AG↗

arXiv2012

A duality formalism in the spirit of Grothendieck and Verdier

We study monoidal categories that enjoy a certain weakening of the rigidity property, namely, the existence of a dualizing object in the sense of Grothendieck and Verdier. We call them Grothendieck-Verdier categories. Notable examples include the derived category of constructible sheaves on a scheme (with respect to tensor product) as well as the derived and equivariant derived categories of constructible sheaves on an algebraic group (with respect to convolution). We show that the notions of pivotal category and ribbon category, which are well known in the setting of rigid monoidal categories, as well as certain standard results associated with these notions, have natural analogues in the world of Grothendieck-Verdier categories.

math.QA↗

arXiv2010

A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic

This article is based on lectures given by the authors in 2005 and 2006. Our first goal is to present an introduction to the orbit method with an emphasis on the character theory of finite nilpotent groups. The second goal (motivated by a recent work of G. Lusztig) is to explain several nontrivial aspects of character theory for finite groups of the form $G(F_{q^n})$, where $G$ is a unipotent algebraic group over a finite field $F_q$. In particular, we introduce the notion of a character sheaf for a unipotent group, and provide a toy model for the representation-theoretic notion of an L-packet.

math.RT↗

arXiv2010

On braided fusion categories I

This work is a detailed version of arXiv:0704.0195 [math.QA]. We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and self-contained exposition of the known results on braided fusion categories without assuming them pre-modular or non-degenerate. The guiding heuristic principle of our work is an analogy between braided fusion categories and Casimir Lie algebras.

math.QA↗

arXiv2008

DG quotients of DG categories

Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier's notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. More than a half of the text is devoted to an overview of "well known" definitions and results. As a result, the e-print is essentially self-contained.

math.KT↗

arXiv2007

Group-theoretical properties of nilpotent modular categories

We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects of C have integral Frobenius-Perron dimensions then C is group-theoretical. As a consequence, we obtain that semisimple quasi-Hopf algebras of prime power dimension are group-theoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasi-Lie bialgebras in terms of Manin pairs).

math.QA↗

arXiv2005

Opers

This article is an elementary introduction to opers without new results. We hope it can be useful for students.

math.AG↗

arXiv2004

Infinite-dimensional vector bundles in algebraic geometry (an introduction)

Raynaud and Gruson showed that there is a reasonable algebro-geometric notion of family of discrete (infinite-dimensional) vector spaces. The author introduces a notion of family of Tate spaces ("Tate" means "locally linearly compact") and claims that it is local. The definition takes in account that the K_{-1} of a ring is not necessarily zero. However, we prove that K_{-1} always vanishes after Nisnevich sheafification. As a discrete counterpart of families of Tate spaces, we introduce the notion of almost projective module. We discuss the notions of dimension torsor and determinant gerbe of a family of Tate spaces. The above technique has two different applications. First, we clarify the structure of the ind-scheme of formal loops of a smooth affine manifold Y. This allows to define a "refined" motivic integral of a differential form on Y with no zeros, which is an object of a triangulated category rather than an element of its K_0 group. Second, we show that almost projective modules and families of Tate spaces appear naturally in the study of the cohomology of a family of finite-dimensional vector bundles on a punctured smooth manifold. The canonical central extension that comes from this cohomology allows to interpret the "Uhlenbeck compactification" of the stack of vector bundles on the projective plane as the fine moduli space of a certain type of generalized vector bundles.

math.AG↗

arXiv2002

On the Grinberg - Kazhdan formal arc theorem

Let X be an algebraic variety over a field k, and L(X) be the scheme of formal arcs in X. Let f be an arc whose image is not contained in the singularities of X. Grinberg and Kazhdan proved that if k has characteristic 0 then the formal neighborhood of f in L(X) admits a decomposition into a product of an infinite-dimensional smooth piece and a piece isomorphic to the formal neighborhood of a closed point of a scheme of finite type. We give a short proof of this theorem without the characteristic 0 assumption.

math.AG↗

arXiv2001

On a conjecture of Kashiwara

Kashiwara conjectured that the hard Lefshetz theorem and the semisimplicity theorem hold for any semisimple perverse sheaf M on a variety over a field of characteristic 0. He also conjectured that if you apply to such M the nearby cycle functor corresponding to some function then the successive quotients of the monodromy filtration are semisimple. We prove that these conjectures would follow from de Jong's conjecture on representations modulo l of the arithmetic fundamental group of a variety over a finite field.

math.AG↗