arXiv subjects

Guglielmo Nocera

Publications and source records attributed to Guglielmo Nocera.

5 recordsLinked to original sources

arXiv2025

Isotopy invariance and stratified $\mathbb{E}_2$-structure of the Ran Grassmannian

Let $G$ be a complex reductive group. A folklore result asserts the existence of an $\mathbb{E}_2$-algebra structure on the Ran Grassmannian of $G$ over $\mathbb{A}^1_{\mathbb{C}}$, seen as a topological space with the complex-analytic topology. The aim of this paper is to prove this theorem, by establishing a homotopy invariance result: namely, an inclusion of open balls $D' \subset D$ in $\mathbb{C}$ induces a homotopy equivalence between the respective Beilinson--Drinfeld Grassmannians $\mathrm{Gr}_{G, {D'}^n} \hookrightarrow \mathrm{Gr}_{G, D^n}$, for any positive integer $n$. We use a purely algebraic approach, showing that automorphisms of a complex smooth algebraic curve $X$ can be lifted to automorphisms of the associated Beilinson--Drinfeld Grassmannian. As a consequence, we obtain a stronger version of the usual homotopy invariance result: namely, the homotopies can be promoted to equivariant stratified isotopies, where "equivariant" refers to the action of the arc group $\mathrm{L}^+G$ and "stratified" refers to the stratification induced by the Schubert stratification of $\mathrm{Gr}_G$ and the incidence stratification of $\mathbb{C}^n$.

math.AT↗

arXiv2025

A model for the assembly map of bordism-invariant functors

We study oplax colimits of stable categories, of hermitian categories and of Poincaré categories in nice cases. This allows us to produce a categorical model of the assembly map of a bordism-invariant functor of Poincaré categories which is also a Verdier projection, whose kernel we explicitly describe. As a direct application, we generalize the Shaneson splitting for bordism-invariant functors of Poincaré categories proved by Calmès-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle to allow for twists. We also show our methods can tackle their general twisted Shaneson splitting of Poincaré-Verdier localizing invariants which specifies to a twisted Bass-Heller-Swan decomposition for the underlying stable categories, generalizing part of recent work of Kirstein-Kremer.

math.KT↗

arXiv2025

A model for the E3 fusion-convolution product of constructible sheaves on the affine Grassmannian

Let $G$ be a complex reductive group. The spherical Hecke category of $G$ can be presented as the category of $G_{\mathcal O}$-equivariant constructible sheaves on the affine Grassmannian $\mathrm{Gr}_G$. This category admits a convolution product, extending the convolution product of equivariant perverse sheaves. In this paper, we upgrade the mentioned convolution product to a left t-exact $\mathbb E_3$-monoidal structure in $\infty$-categories. The construction is intrinsic to the automorphic side. Our main tools are the Beilinson--Drinfeld Grassmannian, Lurie's characterization of $\mathbb E_k$-algebras via the topological Ran space, the homotopy theory of stratified spaces, and the formalism of correspondences.

math.AG↗

arXiv2023

The derived Brauer map via twisted sheaves

Let $X$ be a quasicompact quasiseparated scheme. The collection of derived Azumaya algebras in the sense of Toën forms a group, which contains the classical Brauer group of $X$ and which we call $Br^\dagger(X)$ following Lurie. Toën introduced a map $ϕ:Br^\dagger(X)\to H^2_{et}(X,\mathbb G_m)$ which extends the classical Brauer map, but instead of being injective, it is surjective. In this paper we study the restriction of $ϕ$ to a subgroup $Br(X)\subset Br^\dagger(X)$, which we call the "derived Brauer group", on which $ϕ$ becomes an isomorphism $Br(X)\simeq H^2_{et}(X,\mathbb G_m)$. This map may be interpreted as a derived version of the classical Brauer map which offers a way to "fill the gap" between the classical Brauer group and the cohomogical Brauer group. The group $Br(X)$ was introduced by Lurie by making use of the theory of prestable $\infty$-categories. There, the mentioned isomorphism of abelian groups was deduced from an equivalence of $\infty$-categories between the "Brauer space" of invertible presentable prestable $\mathcal O_X$-linear categories, and the space $Map(X,K(\mathbb G_m,2))$. We offer an alternative proof of this equivalence of $\infty$-categories, characterizing the functor from the left to the right via gerbes of connective trivializations, and its inverse via connective twisted sheaves. We also prove that this equivalence carries a symmetric monoidal structure, thus proving a conjecture of Binda an Porta.

math.AG↗

arXiv2023

Whitney stratifications are conically smooth

The notion of conically smooth structure on a stratified space was introduced by Ayala, Francis and Tanaka. This is a very well behaved analogue of a differential structure in the context of stratified topological spaces, satisfying good properties such as the existence of resolutions of singularities and handlebody decompositions. In this paper we prove Ayala, Francis and Tanaka's conjecture that any Whitney stratified space admits a canonical conically smooth structure. We thus establish a connection between the theory of conically smooth spaces and the classical examples of stratified spaces from differential topology.

math.DG↗