


arχiv
[2016] Formal glueing for nonlinear 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 pseudocoherent, perfect complexes, and Gbundles 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 flagrelated decomposition formula for the corresponding derived ∞categories of pseudocoherent 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 pseudocoherent complexes, which parallels Thomason’s localization results for perfect complexes.



arχiv
[2017] Higher Kac–Moody algebras and moduli
spaces of Gbundles  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 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
[2018] GelfandFuchs 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 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.