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PhD thesis

  • - - arχiv Formal loops and tangent Lie algebras -- Defended in June 2015.
    Essentially contains my first three papers.


  • - - arχiv [2015] Tangent Lie algebra of derived Artin stacks -- Journal für die reine und angewandte Mathematik (Crelles). DOI: 10.1515/crelle-2015-0065.
    Since the work of Mikhail Kapranov, it is known that the shifted tangent complex T[−1] of a smooth algebraic variety X is endowed with a weak Lie structure, the bracket being given by the Atiyah class. Moreover any complex of quasi-coherent sheaves on X is endowed with a weak Lie action of this tangent Lie algebra. We will generalize this result to (finite enough) derived Artin stacks, without any smoothness assumption. This in particular applies to (finite enough) singular schemes. This work uses tools of both derived algebraic geometry and ∞-category theory.
  • - -arχiv [2017] Higher dimensional formal loop spaces -- Annales Scientifiques de l'ENS, vol. 50 (4), pp. 609--663.
    If M is a symplectic manifold then the space of smooth loops C(S1,M) inherits of a quasi-symplectic form. We will focus in this work on an algebraic analogue of that result. Kapranov and Vasserot introduced and studied the formal loop space of a scheme X. It is an algebraic version of the space of smooth loops in a differentiable manifold. We generalize their construction to higher dimensional loops. To any scheme X -- not necessarily smooth -- we associate Ld(X), the space of loops of dimension d. We prove it has a structure of (derived) Tate scheme -- ie its tangent is a Tate module: it is infinite dimensional but behaves nicely enough regarding duality. We also define the bubble space Bd(X), a variation of the loop space. We prove that Bd(X) is endowed with a natural symplectic form as soon as X has one.
  • - - arχiv [2016] Tate objects in stable (∞,1)-categories -- to appear in Homology, Homotopy and Applications.
    Tate objects have been studied by many authors. They allow us to deal with infinite dimensional spaces by identifying some more structure. In this article, we set up the theory of Tate objects in stable (∞,1)-categories, while the literature only treats with exact categories. We will prove the main properties expected from Tate objects. This new setting includes several useful examples: Tate objects in the category of spectra for instance, or in the derived category of a derived algebraic object.


  • - - arχiv [2016] Formal glueing for non-linear flags -- joint with M. Porta and G. Vezzosi -- submitted.
    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 ∞-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.
  • - - arχiv [2017] Higher Kac–Moody algebras and moduli spaces of G-bundles -- joint with G. Faonte and M. Kapranov.
    We provide a generalization to the higher dimensional case of the construction of the current algebra g((z)), its Kac–Moody extension g̃ and of the classical results relating them to the theory of G-bundles over a curve. For a reductive algebraic group G with Lie algebra g, we define a dg-Lie algebra gn of n-dimensional currents in g. For any symmetric G-invariant polynomial P on g of degree n+1, we get a higher Kac-Moody algebra g̃n,P as a central extension of gn by the base field k. Further, for a smooth, projective variety X of dimension n≥2, we show that gn acts infinitesimally on the derived moduli space RBunrigG(X,x) of G-bundles over X trivialized at the neighborhood of a point x ∈ X. Finally, for a representation φ: G → GLr we construct an associated determinantal line bundle on RBunrigG(X,x) and prove that the action of gn extends to an action of g̃n,P on such bundle for P the (n+1)th Chern character of φ.
  • - - arχiv [2018] Gelfand-Fuchs cohomology in algebraic geometry and factorization algebras -- joint with M. Kapranov.
    Let X be a smooth affine variety over a field k of characteristic 0 and T(X) be the Lie algebra of regular vector fields on X. We compute the Lie algebra cohomology of T(X) with coefficients in k. The answer is given in topological terms relative to any embedding of k into complex numbers and is analogous to the classical Gelfand-Fuks computation for smooth vector fields on a C-infinity manifold. Unlike the C-infinity case, our setup is purely algebraic: no topology on T(X) is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts.