Gabriele Vezzosi

Homotopical algebraic geometry. II. Geometric stacks and applications.

This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category, such as derivations, etale and smooth maps, flat and projective modules, etc. We then use the theory of stacks over model categories introduced in \cite{hagI} in order to define a general notion of geometric stack over a base symmetric monoidal model category C, and prove that this notion satisfies the expected properties. The rest of the paper consists in specializing C to several different contexts. First of all, when C=k-Mod is the category of modules over a ring k, with the trivial model structure, we show that our notion gives back the algebraic n-stacks of C. Simpson. Then we set C=sk-Mod, the model category of simplicial k-modules, and obtain this way a notion of geometric derived stacks which are the main geometric objects of Derived Algebraic Geometry. We give several examples of derived version of classical moduli stacks, as for example the derived stack of local systems on a space, of algebra structures over an operad, of flat bundles on a projective complex manifold, etc. Finally, we present the cases where C=(k) is the model category of unbounded complexes of modules over a char 0 ring k, and C=Sp^{\Sigma} the model category of symmetric spectra. In these two contexts, called respectively Complicial and Brave New Algebraic Geometry, we give some examples of geometric stacks such as the stack of associative dg-algebras, the stack of dg-categories, and a geometric stack constructed using topological modular forms.

Shifted Poisson structures and deformation quantization

This paper is a sequel to [PTVV]. We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and poly-vector 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.

From Hag To Dag: Derived Moduli Stacks

These are expanded notes of some talks given during the fall 2002, about homotopical algebraic geometry with special emphasis on its applications to derived algebraic geometry and derived deformation theory. We use the general framework developed in [HAG-I], and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and study their geometric properties. We start by defining the model category of D-stacks, with respect to an extension of the etale topology to the category of commutative differential graded algebras, and we show that its homotopy category contains interesting objects, such as schemes, algebraic stacks, higher algebraic stacks, dg-schemes, etc. We define the notion of geometric D-stacks and present some related geometric constructions (O-modules, perfect complexes, K-theory, derived tangent stacks, cotangent complexes, various notions of smoothness, etc.). Finally, we define and study the derived moduli problems classifying local systems on a topological space, vector bundles on a smooth projective variety, and A ∞-categorical structures. We state geometricity and smoothness results for these examples. The proofs of the results presented in this paper will be mainly given in [HAG-II].

Higher algebraic K-theory of group actions with finite stabilizers

We prove a decomposition theorem for the equivariantK-theory of actions of ane group schemes G of nite type over a eld on regular separated noetherian algebraic spaces, under the hypothesis that the actions have nite geometric stabilizers and satisfy a rationality condition together with a technical condition which holds e.g. for G abelian or smooth. We reduce the problem to the case of a GLn-action and nally to a split torus action.

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.

EXISTENCE OF VECTOR BUNDLES AND GLOBAL RESOLUTIONS FOR SINGULAR SURFACES

We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on sep- arated normal surfaces any coherent sheaf is the quotient of a vector bundle. As a consequence, for such surfaces the Quillen K-theory of vector bundles co- incides with the Waldhausen K-theory of perfect complexes. Examples show that, on nonseparated schemes, usually many coherent sheaves are not quo- tients of vector bundles.

A remark on the Hochschild-Kostant-Rosenberg theorem in characteristic p

We prove a Hochschild-Kostant-Rosenberg decomposition theorem for smooth proper schemes

A model structure on relative dg-Lie algebroids

X

The \(\ell \)-adic trace formula for dg-categories and Bloch’s conductor conjecture

in characteristic

Infinitesimal Stokes’ Formula for Higher-Order de Rham Complexes

p