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Bertrand Toën

Publications and source records attributed to Bertrand Toën.

14 recordsLinked to original sources
arXiv2025

Foliations and stable maps

This paper is part of an ongoing series of works on the study of foliations on algebraic varieties via derived algebraic geometry. We focus here on the specific case of globally defined vector fields and the global behaviour of their algebraic integral curves. For a smooth and proper variety $X$ with a global vector field $ν$, we consider the induced vector field $ν_{g,n}$ on the derived stack of stable maps, of genus $g$ with $n$ marked points, to $X$. When $(g,n)$ is either $(0,2)$ or $(1,0)$, the derived stack of zeros of $ν_{g,n}$ defines a proper \emph{moduli of algebraic trajectories} of $ν$. When $(g,n)=(0,2)$ algebraic trajectories behave very much like rational algebraic paths from one zero of $ν$ to another, and in particular they can be composed. This composition is represented by the usual gluing maps in Gromov-Witten theory, and we use it give three categorical constructions, of different categorical levels, related, in a certain sense, by decategorification. In order to do this, in particular, we have to deal with virtual fundamental classes of non-quasi-smooth derived stacks. When $(g,n)=(1,0)$, zeros of $ν_{1,0}$ might be thought as algebraic analogues of periodic orbits of vector fields on smooth real manifolds. In particular, we propose a Zeta function counting the zeros of $ν_{1,0}$, that we like to think of as an algebraic version of Ruelle's dynamical Zeta function. We conclude the paper with a brief indication on how to extend these results to the case of general one dimensional foliation $F$, by considering the derived stack of $F$-equivariant stable maps.

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arXiv2023

Infinitesimal derived foliations

We introduce a notion of \emph{infinitesimal derived foliation}. We prove it is related to the classical notion of infinitesimal cohomology, and satisfies some formal integrability properties. We also provide some hints on how infinitesimal derived foliations compare to our previous notion of derived foliations.

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arXiv2021

Moduli of flat connections on smooth varieties

We study the moduli functor of flat bundles on smooth, possibly non-proper, algebraic variety $X$ (over a field of characteristic zero). For this we introduce the notion of \emph{formal boundary} of $X$, denoted by $\partial X$, which is a formal analogue of the boundary at infinity of the Betti topological space associated to $X$. We explain how to construct two derived moduli functors $Vect^{\nabla}(X)$ and $Vect^{\nabla}(\partial X)$, of flat bundles on $X$ and on $\partial X$, as well as a restriction map $R : Vect^{\nabla}(X) \rightarrow Vect^\nabla(\partial X)$ from the former to the later. This work contains two main results. First we prove that the morphism R comes equipped with a canonical shifted Lagrangian structure in the sense of [PTVV]. This first result can be understood as the de Rham analogue of the existence of Poisson structures on moduli of local systems previously studied by the authors. As a second statement, we prove that the geometric fibers of $R$ are representable by "quasi-algebraic spaces", a slight weakening of the notion of algebraic spaces.

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arXiv2021

A Universal HKR Theorem

In this work we study the failure of the HKR theorem over rings of positive and mixed characteristic. For this we construct a filtered circle interpolating between the usual topological circle and a formal version of it. By mapping to schemes we produce this way an interpolation, realized in practice by the existence of a natural filtration, from Hochschild and (a filtered version of) cyclic homology to derived de Rham cohomology. In particular, we show that this recovers the filtration of Antieau and Bhatt-Morrow-Scholze. The construction of our filtered circle is based on the theory of affine stacks and affinization introduced by the third author, together with some facts about schemes of Witt vectors.

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arXiv2020

Le problème de la schématisation de Grothendieck revisité

The objective of this work is to reconsider the schematization problem of [6], with a particular focus on the global case over Z. For this, we prove the conjecture [Conj. 2.3.6][15] which gives a formula for the homotopy groups of the schematization of a simply connected homotopy type. We deduce from this several results on the behaviour of the schematization functor, which we propose as a solution to the schematization problem.

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arXiv2020

Classes caractéristiques des schémas feuilletés

We study a notion of derived foliations on schemes and derived schemes of arbitrary characteristics. We introduce the Hodge filtration associated to a derived foliation, which functorialy filters derived de Rham cohomology. We use this filtration to study vanishing results of Chern classes of perfect complexes endowed with connexions along derived foliations. As an application, we prove a positive characteristic version of Bott's vanishing theorem and more generally of existence of residues for foliations with singularities due to Baum-Bott in characteristic zero.

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arXiv2020

Algebraic foliations and derived geometry II: the Grothendieck-Riemann-Roch theorem

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.

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arXiv2020

Algebraic foliations and derived geometry: the Riemann-Hilbert correspondence

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 $\geq 2$, 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.

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arXiv2019

Trace and Kunneth formulas for singularity categories and applications

We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. Moreover, we prove Künneth 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.

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arXiv2018

Structures symplectiques et de Poisson sur les champs en catégories

The purpose of this short note is to present two existence results concerning symplectic and lagrangian structures in the derived setting, in situations where the constructions of [Ca] and [PTVV] do not apply. For this we show that symplectic structures can be constructed out of Calabi-Yau structures on sheaves of dg-categories, or out of \emph{orientations} on sheaves of rigid dg-categories. These results follow from two main theorems: the HKR theorem and the cyclotomic aspect of traces in rigid infty-categories.

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arXiv2017

Géométrie non-commutative, formule des traces et conducteur de Bloch

This text is based on a talk by the first named author at the first congress of the SMF (Tours, 2016). We present Bloch's conductor formula, which is a conjectural formula describing the change of topology in a family of algebraic varieties when the parameter specialises to a critical value. The main objective of this paper is to describe a general approach to the resolution of Bloch's conjecture based on techniques from both non-commutative geometry and derived geometry.

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arXiv2014

Derived Algebraic Geometry

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.

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arXiv2011

Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

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, we consider X=S an algebraic K3-surface, $g\geq 0$, and $β$ a curve class, and we construct a derived stack whose truncation is the usual stack of pointed stable maps from curves of genus g to S hitting the class $β$, and such that the inclusion of the trunaction induces on a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of Okounkov-Maulik-Pandharipande-Thomas. 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 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.

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