


arχiv
[2018] Tangent Lie algebra of derived Artin stacks  Journal für die reine und angewandte Mathematik (Crelles), vol. 2018, issue 741, pp. 146.
DOI: 10.1515/crelle20150065.
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 quasicoherent 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. 609663.
If M is a symplectic manifold then the space of smooth loops C(S^{1},M) inherits of a quasisymplectic 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 L^{d}(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 B^{d}(X), a variation of the loop space.
We prove that B^{d}(X) is endowed with a natural symplectic form as soon as X has one.



arχiv
[2017] Tate objects in stable (∞,1)categories  Homology, Homotopy and Appl. 19 (2017), no. 2, pp. 373395.
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
[2019] Higher Kac–Moody algebras and moduli
spaces of Gbundles  joint with G. Faonte and M. Kapranov. Adv. Math. 346 (2019). pp. 389466.
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 Gbundles over a curve. For a reductive algebraic group G with Lie algebra g, we define a dgLie algebra g_{n} of ndimensional currents in g. For any symmetric Ginvariant polynomial P on g of degree n+1, we get a higher KacMoody algebra g̃_{n,P} as a central extension of g_{n} by the base field k. Further, for a smooth, projective variety X of dimension n≥2, we show that g_{n} acts infinitesimally on the derived moduli space RBun^{rig}_{G}(X,x) of Gbundles over X trivialized at the neighborhood of a point x ∈ X. Finally, for a representation φ: G → GL_{r} we construct an associated determinantal line bundle on RBun^{rig}_{G}(X,x) and prove that the action of g_{n} extends to an action of g̃_{n,P} on such bundle for P the (n+1)^{th} Chern character of φ.



arχiv
[2021] The tangent complex of Ktheory  J. de l'École Polytechnique. 8 (2021). pp. 895932.
We show that the relative algebraic Ktheory functor fully determines the absolute cyclic
homology over any field k of characteristic 0. More precisely, we prove that the tangent
complex of Ktheory, in terms of (abelian) deformation problems over k, is cyclic homology.
We also show that the LodayQuillenTsygan generalized trace comes as the tangent morphism of the canonical map BGL_{∞} → K mapping a vector bundle to its class in Ktheory.
The proof builds on results of Goodwillie, using Wodzicki’s excision for cyclic homology
and formal deformation theory à la LuriePridham.



arχiv
[2022] GelfandFuchs cohomology in algebraic geometry and factorization algebras  joint with M. Kapranov. To appear in JAMS.
DOI
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 GelfandFuks computation for smooth vector fields on a Cinfinity manifold. Unlike the Cinfinity 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 algebrogeometric and topological contexts.