arXiv subjects

Kendric Schefers

Publications and source records attributed to Kendric Schefers.

4 recordsLinked to original sources

arXiv2023

A microlocal Feigin-Tsygan-Preygel theorem

Let $\boldsymbol{Z}$ be a derived global complete intersection over $\mathbb{C}$. We compute the periodic cyclic homology of the category of ind-coherent sheaves with prescribed singular support on $\boldsymbol{Z}$ in terms of the microlocal homology, a family of chain theories living between cohomology and Borel-Moore homology. Our result is a microlocal generalization of both the Feigin-Tsygan theorem identifying $\mathrm{HP}_{\bullet}^{\mathbb{C}}(\mathsf{QCoh}(\boldsymbol{Z}))$ with the $2$-periodized cohomology of $\boldsymbol{Z}$ and A. Preygel's theorem identifying $\mathrm{HP}_{\bullet}^{\mathbb{C}}(\mathsf{IndCoh}(\boldsymbol{Z}))$ with the $2$-periodized Borel-Moore homology of $\boldsymbol{Z}$. Our proof strategy makes extensive use of categories of matrix factorizations, which we treat using Preygel's formalism. This paper contains generalizations of several known results in the subject which we prove in this formalism, and which may be of independent interest to the reader.

math.AG↗

arXiv2023

Derived $V$-filtrations and the Kontsevich-Sabbah-Saito theorem

Let $f: X \to \mathbb{A}^1$ be a regular function on a smooth complex algebraic variety $X$. We formulate and prove an equivalence between the algebraic formal twisted de Rham complex of $f$ and the vanishing cycles with respect to $f$ as objects in the category of sheaves valued in the derived $\infty$-category of modules over $\widehat{\mathscr{E}}_{\mathbb{C},0}^{\mathrm{alg}}$, the ring of germs of algebraic formal microdifferential operators. This is a direct generalization of Kontsevich's conjecture, proven in work by Sabbah and then Sabbah--Saito, of an algebraic formula computing vanishing cohomology. The novelty in our approach is the introduction of a canonical $V$-filtration on the derived $\infty$-category of regular holonomic $\mathscr{D}_{\mathbb{C},0}$-modules, and the use of various techniques from the theory of higher categories and higher algebra in the context of the subject of microdifferential calculus.

math.AG↗

arXiv2023

Liquid functional calculus

We develop an elementary formalism of functional calculus for entire holomorphic functions in the setting of Clausen and Scholze's $p$-liquid vector spaces.

math.AG↗

arXiv2022

Microlocal homology

Let $Z$ be an l.c.i. scheme over $\mathbb{C}$. In this paper, we introduce a Kashiwara--Schapira-style functor of derived microlocalization, which we use to define a perverse sheaf $\mu_{Z}$ on the $-1$-shifted cotangent bundle, $T^*[-1]Z$. The sheaf $\mu_{Z}$ is designed to be a refinement of the microlocal homology of $Z$: a family of invariants introduced by Nadler that interpolates between the singular cohomology and Borel--Moore homology of $Z$. Our main result is an equivalence between $\mu_{Z}$ and the DT sheaf $\varphi_{T^*[-1]Z}$ on $T^*[-1]Z$. This provides an alternative construction for the DT sheaf in the case of a shifted cotangent bundle. The main step of our argument, which may be of independent interest, is a local computation -- closely related to one obtained recently by Kinjo using different methods -- providing a description of the classical microlocalization functor in terms of vanishing cycles.

math.AG↗