Papers by Gabriele Vezzosi
arXiv (Cornell University), Feb 6, 2011
We show how a quasi-smooth derived enhancement of a Deligne-Mumford stack X naturally endows X wi... more We show how a quasi-smooth derived enhancement of a Deligne-Mumford stack X naturally endows X with a functorial perfect obstruction theory in the sense of Behrend-Fantechi. This result is then applied to moduli of maps and perfect complexes on a smooth complex projective variety. For moduli of maps, for X = S an algebraic K3-surface, g ∈ N, and β = 0 in H 2 (S, Z) a curve class, we construct a derived stack RM red g,n (S; β) whose truncation is the usual stack M g,n (S; β) of pointed stable maps from curves of genus g to S hitting the class β, and such that the inclusion M g (S; β) ֒→ RM red g (S; β) induces on M g (S; β) a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of Okounkov-Maulik-Pandharipande-Thomas [O-P2, M-P, M-P-T]. The approach we present here uses derived algebraic geometry and yields not only a full rigorous proof of the existence of a reduced obstruction theory-not relying on any result on semiregularity maps-but also a new global geometric interpretation. We give two further applications to moduli of complexes. For a K3-surface S we show that the stack of simple perfect complexes on S is smooth. This result was proved with different methods by Inaba ([In]) for the corresponding coarse moduli space. Finally, we construct a map from the derived stack of stable embeddings of curves (into a smooth complex projective variety X) to the derived stack of simple perfect complexes on X with vanishing negative Ext's, and show how this map induces a morphism of the corresponding obstruction theories when X is a Calabi-Yau threefold. An important ingredient of our construction is a perfect determinant map from the derived stack of perfect complexes to the derived stack of line bundles whose tangent morphism is, pointwise, Illusie's trace map for perfect complexes. We expect that this determinant map might be useful in other contexts as well.
arXiv (Cornell University), Apr 21, 2009
Ce travail accompagne [To-Ve4] et a pour objectif de fournir de plus amples détails sur la compar... more Ce travail accompagne [To-Ve4] et a pour objectif de fournir de plus amples détails sur la comparaison entre fonctions sur les espaces des lacets dérivés et théorie de de Rham. Pour une k-algèbre commutative A, lisse sur k de caractéristique nulle, nous montrons que deux objets S 1 ⊗ A et ǫ(A) se déterminent mutuellement, et ce fonctoriellement en A. L'objet S 1 ⊗ A est la k-algèbre simpliciale S 1-équivariante obtenue en tensorisant A par le groupe simplicial S 1 := BZ. L'objet ǫ(A) est l'algèbre de de Rham de A, munie de la différentielle de de Rham et considérée comme une ǫ-dg-algèbre (i.e. une algèbre dans une certaine catégorie monoïdale de k[ǫ]-dg-modules, où k[ǫ] := H * (S 1 , k)). Nous construisons uneéquivalence φ, entre la théorie homotopique des k-algèbres simpliciales S 1-équivariantes et celle des ǫ-dgalgèbres, et nous montrons l'existence d'uneéquivalence fonctorielle φ(S 1 ⊗ A) ∼ ǫ(A). Nous déduisons de cela la comparaison annoncée, identifiant les fonctions S 1-équivariantes sur LX, l'espace des lacets dérivé d'un k-schéma X lisse, et la cohomologie de de Rham algébrique de X/k. Nous déduisons aussi des versions fonctorielles et multiplicatives des théorèmes de décomposition de l'homologie de Hochschild (du type HKR), pour des k-schémas séparés quelconques.
arXiv (Cornell University), Oct 10, 2001
For a (semi-)model category M , we define a notion of a "homotopy" Grothendieck topology on M , a... more For a (semi-)model category M , we define a notion of a "homotopy" Grothendieck topology on M , as well as its associated model category of stacks. We use this to define a notion of geometric stack over a symmetric monoidal base model category; geometric stacks are the fundamental objects to "do algebraic geometry over model categories". We give two examples of applications of this formalism. The first one is the interpretation of DG-schemes as geometric stacks over the model category of complexes and the second one is a definition ofétale K-theory of E ∞-ring spectra. This first version is very preliminary and might be considered as a detailed research announcement. Some proofs, more details and more examples will be added in a forthcoming version.
arXiv (Cornell University), Apr 8, 2008
In this note we present a work in progress whose main purpose is to establish a categorified vers... more 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 O-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. Contents 1 Motivations and objectives 1 2 Categorification of homological algebra and dg-categories 6 3 Loop spaces in derived algebraic geometry 10 4 Construction of the Chern character 13 5 Final comments 15
arXiv (Cornell University), Dec 23, 2002
In [To-Ve2] we began the study of higher sheaf theory (i.e. stacks theory) on higher categories e... more In [To-Ve2] we began the study of higher sheaf theory (i.e. stacks theory) on higher categories endowed with a suitable notion of topology: precisely, we defined the notions of S-site and of model site, and the associated categories of stacks on them. This led us to a notion of model topos. In this paper we treat the analogous theory starting from (1-)Segal categories in place of S-categories and model categories. We introduce notions of Segal topologies, Segal sites and stacks over them, giving rise to a definition of Segal topos. We compare the notions of Segal topoi and of model topoi, showing that the two theories are equivalent in some sense. However, the existence of a nice Segal category of morphisms between Segal categories allows us to improve the treatment of topoi in this context. In particular we construct the 2-Segal category of Segal topoi and geometric morphisms, and we provide a Giraud-like statement charaterizing Segal topoi among Segal categories. As example of applications, we show how to reconstruct a topological space from the Segal topos of locally constant stacks on it, thus extending the main theorem of [To] to the case of un-based spaces. We also give some hints of how to define homotopy types of Segal sites: this approach gives a new point of view and some improvements on theétale homotopy theory of schemes, and more generally on the theory of homotopy types of Grothendieck sites as defined by Artin and Mazur.
arXiv (Cornell University), Mar 1, 2018
Ce texte est basé sur l'exposé du premier auteur au premier congrès de la SMF (Tours, 2016). On y... more Ce texte est basé sur l'exposé du premier auteur au premier congrès de la SMF (Tours, 2016). On y présente la formule du conducteur de Bloch, qui est une formule conjecturale décrivant le changement de topologie dans une famille de variétés algébriques lorsque le paramètre se spécialise en une valeur critique. L'objectif de ce texte est de présenter une approche généraleà la résolution de cette conjecture basée sur des techniques de géométrie non-commutative et de géométrie dérivée.
arXiv (Cornell University), Jan 16, 2020
This is the first in a series of papers about foliations in derived geometry. After introducing d... more This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations defined in terms of differential ideals in the algebra of forms. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety X is formally integrable at any point, and, if we suppose that its singular locus has codimension ≥ 2, its analytification is a locally integrable singular foliation on the associated complex manifold X h. We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on X, and prove a Riemann-Hilbert correspondence for them when X is proper. We discuss several examples and applications. Contents 2 BERTRAND TOËN AND GABRIELE VEZZOSI 4. The Riemann-Hilbert correspondence 41 4.1. The sheaf of flat functions 41 4.2. The Riemann-Hilbert correspondence 45 4.3. Examples 46 References 51
HAL (Le Centre pour la Communication Scientifique Directe), Mar 21, 2020
We present an-adic trace formula for smooth and proper admissible dg-categories over a base monoi... more We present an-adic trace formula for smooth and proper admissible dg-categories over a base monoidal dg-category. As an application, we prove (a version of) Bloch's conductor conjecture ([Bl, Conjecture, p. 423]) under the additional hypothesis that the inertia group acts with unipotent monodromy.
Compositio Mathematica, Mar 1, 2022
We present an ℓ-adic trace formula for saturated and admissible dg-categories over a base monoida... more We present an ℓ-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. As an application, we prove (a version of) Bloch's conductor conjecture ([Bl, Conjecture, p. 423]), under the additional hypothesis that the inertia group acts with unipotent monodromy.
Journal de l'École polytechnique, 2018
Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION-PAS... more Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION-PAS DE MODIFICATION 3.
arXiv (Cornell University), Jul 2, 2002
This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in h... more This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of ∞-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T , generalizing the model category structure on simplicial presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspodence between S-topologies on an S-category T , and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model topos introduced by C. Rezk, and we relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition ofétale K-theory of ring spectra, extending theétale K-theory of commutative rings.
Rendiconti del Seminario Matematico della Università di Padova, 1997
We study localization properties of some algebraic differential complexes associated to an arbitr... more We study localization properties of some algebraic differential complexes associated to an arbitrary commutative algebra which are higher order (in the sense of differential operators) analogues of the ordinary de Rham complex. These results should be considered, in the spirit of [ 11 ], as preliminaries to the study of the cohomological invariants provided by these higher de Rham complexes for singular varieties. Notations and Conventions. K: a commutative ring with unit; A : a commutative, associative K-algebra with unit; A-Mod : the category of A-modules; K Mod : the category of K-modules; DIFFA : the category whose objects are A-modules and whose morphisms are differential operators (Section 1), of any (finite) order, between them; Ens: the category of sets; [C, C] : the category of functors C ~ C, C being any category; Ob ( C) : the objects of C, C being any category; we will write C E e Ob(C) to mean that C is an object of C; (A, A )-B iMod : the category of (A, A)-bimodules, ...
ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
We prove a Hochschild-Kostant-Rosenberg decomposition theorem for smoothly compactifiable smooth ... more We prove a Hochschild-Kostant-Rosenberg decomposition theorem for smoothly compactifiable smooth schemes X in characteristic p when dim X p. The best known previous result of this kind, due to Yekutieli, required dim X < p. Yekutieli's result follows from the observation that the denominators appearing in the classical proof of HKR do not divide p when dim X < p. Our extension to dim X = p requires a homological fact: the Hochschild homology of a smooth proper scheme is self-dual.
We develop homotopical algebraic geometry ([To-Ve 1, To-Ve 2]) in the special context where the b... more We develop homotopical algebraic geometry ([To-Ve 1, To-Ve 2]) in the special context where the base symmetric monoidal model category is that of spectra S, i.e. what might be called, after Waldhausen, brave new algebraic geometry. We discuss various model topologies on the model category of commutative algebras in S, and their associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry). Two examples of geometric S-stacks are given: a global moduli space of associative ring spectrum structures, and the stack of elliptic curves endowed with the sheaf of topological modular forms.
These are expanded notes of some talks given during the fall 2002, about homotopical algebraic ge... more 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
arXiv: Algebraic Geometry, 2016
In this paper we prove formal glueing, along an arbitrary closed subscheme $Z$ of a scheme $X$, f... more In this paper we prove formal glueing, along an arbitrary closed subscheme $Z$ of a scheme $X$, for the stack of pseudo-coherent, perfect complexes, and $G$-bundles on $X$ (for $G$ a smooth affine algebraic group). By iterating this result, we get a decomposition of these stacks along an arbitrary nonlinear flag of subschemes in $X$. By taking points over the base field, we deduce from this both a formal glueing, and a flag-related decomposition formula for the corresponding derived $\infty$-categories of pseudo-coherent and perfect complexes. We finish the paper by highlighting some expected progress in the subject matter of this paper, that might be related to a Geometric Langlands program for higher dimensional varieties. In the Appendix we also prove a localization theorem for the stack of pseudo-coherent complexes, which parallels Thomason's localization results for perfect complexes.
arXiv: Algebraic Geometry, 2020
This is the first in a series of papers about foliations in derived geometry. After introducing d... more This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety X is formally integrable at any point, and, if we suppose that its singular locus has codimension ≥ 2, then the truncation of its analytification is a locally integrable singular foliation on the associated complex manifold X h. We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on X, and prove a Riemann-Hilbert correspondence for them when X is proper. We discuss several examples and applications.
arXiv: Algebraic Geometry, 2017
We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base mo... more We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. Moreover, we prove K\"unneth formulas for dg-category of singularities, and for inertia-invariant vanishing cycles. As an application, we prove a version of Bloch's Conductor Conjecture (stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.
arXiv: Algebraic Geometry, 2020
This is the second of series of papers on the study of foliations in the setting of derived algeb... more This is the second of series of papers on the study of foliations in the setting of derived algebraic geometry based on the central notion of derived foliation. We introduce sheaf-like coefficients for derived foliations, called quasi-coherent crystals, and construct a certain sheaf of dg-algebras of differential operators along a given derived foliation, with the property that quasi-coherent crystals can be interpreted as modules over this sheaf of differential operators. We use this interpretation in order to introduce the notion of good filtrations on quasi-coherent crystals, and define the notion of characteristic cycle. Finally, we prove a Grothendieck-Riemann-Roch (GRR) formula expressing that formation of characteristic cycles is compatible with push-forwards along proper and quasi-smooth morphisms. Several examples and applications are deduced from this, e.g. a GRR formula for D-modules on possibly singular schemes, and a foliated index formula for weakly Fredholm operators.
We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base mo... more We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. As an application, we prove (a version of) Bloch's conductor conjecture, under the additional hypothesis that the inertia group acts with unipotent monodromy.
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Papers by Gabriele Vezzosi