Searcharxiv⌕ Search

arXiv subjects

G. Vezzosi

Publications and source records attributed to G. Vezzosi.

8 recordsLinked to original sources
arXiv2020

Motivic Realizations of Singularity Categories and Vanishing Cycles

In this paper we establish a precise comparison between vanishing cycles and the singularity category of Landau-Ginzburg models over a complete discrete valuation ring. By using noncommutative motives, we first construct a motivic $\ell$-adic realization functor for dg-categories. Our main result, then asserts that, given a Landau-Ginzburg model over a complete discrete valuation ring with potential induced by a uniformizer, the $\ell$-adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and $\infty$-categorical version of Orlov's comparison theorem between the derived category of singularities and the derived category of matrix factorizations for a Landau-Ginzburg model over a noetherian regular local ring.

math.AG↗
arXiv2017

Shifted Poisson Structures and Deformation Quantization

This paper is the sequel to [PTVV] (IHES Vol. 117, 2013). We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and polyvector fields. We then introduce the formalism of formal derived stacks and prove formal localization and gluing results. These allow us to define shifted Poisson structures on general derived Artin stacks, and prove that the non-degenerate Poisson structures correspond exactly to shifted symplectic forms. Shifted deformation quantization for a derived Artin stack endowed with a shifted Poisson structure is discussed in the last section. This paves the way for shifted deformation quantization of many interesting derived moduli spaces, like those studied in [PTVV] and probably many others.

math.AG↗
arXiv2016

The $\ell$-adic trace formula for dg-categories and Bloch's conductor conjecture

Building on recent results by A. Blanc, M. Robalo and the authors, we present an $\ell$-adic trace formula for smooth and proper dg-categories over a base $\mathbb{E}_{\infty}$-algebra $B$. We also give a variant when $B$ is only an $\mathbb{E}_2$-algebra. As an application of this trace formula, we propose a strategy of proof of Bloch's conductor conjecture. This is a research announcement and detailed proofs will appear elsewhere.

math.AG↗
arXiv2013

Shifted Symplectic Structures

This is the first of a series of papers about \emph{quantization} in the context of \emph{derived algebraic geometry}. In this first part, we introduce the notion of \emph{$n$-shifted symplectic structures}, a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks. We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-shifted symplectic structures. Our main existence theorem states that for any derived Artin stack $F$ equipped with an $n$-shifted symplectic structure, the derived mapping stack $\textbf{Map}(X,F)$ is equipped with a canonical $(n-d)$-shifted symplectic structure as soon a $X$ satisfies a Calabi-Yau condition in dimension $d$. These two results imply the existence of many examples of derived moduli stacks equipped with $n$-shifted symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We also show that Lagrangian intersections carry canonical (-1)-shifted symplectic structures.

math.AG↗
arXiv2011

Caractères de Chern, traces équivariantes et géométrie algébrique dérivée

The purpose of this work is to provide details about the construction of the Chern character for categorical sheaves mentioned in our previous work "Chern character, loop spaces and derived algebraic geometry". For this, we introduce and study the notion of rigid symmetric monoidal \infty-category. We show how trace maps can be constructed in this higher categorical setting, and using a recent work of Hopkins-Lurie we prove the existence of a "cyclic trace", which is the main ingredient in the construction of the Chern character. Our Chern character is then constructed for any pair (T,A), consisting of a \infty-topos T and a stack of rigid symmetric monoidal \infty-categories A on T. We propose two main applications of this construction. First of all, we show how to recover the Chern character of perfect complexes on schemes, with values in cyclic homology, and show how it can be extended to an interesting new Chern character for Artin stacks. Our second application provides invariants of famillies of dg-categories. A consequence of the existence of these invariants is the construction of a Gauss-Manin connexion on the cyclic homology complex of such a familly, generalizing previous constructions by Getzler and of Dolgushev-Tamarkin-Tsygan. Another consequence is the construction of the "character sheaf" associated to a representation of an algebraic group into a dg-category, which is a categorification of the character function of a linear representation. Finally, for a familly of saturated dg-categories we construct a "secondary Chern character", taking values in a new cohomology theory called "secondary cyclic homology".

math.AG↗
arXiv2009

S^1-Equivariant simplicial algebras and de Rham theory

This is a companion paper our previous submission "\infty-categories monoidales rigides et caracteres de Chern", in which we give a comparison between functions on the derived loop space of a smooth scheme of caracteristic zero, and its algebraic de Rham cohomology. As a consequence we obtain functorial and multiplicative versions of the HKR decomposition theorem relating Hochschild homology and Hodge cohomology.

math.AG↗
arXiv2008

A note on Chern character, loop spaces and derived algebraic geometry

In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere.

math.AG↗