RGAS Sevilla

RGAS School on Singularities

8-12 January 2024

Instituto de Matemáticas de la Universidad de Sevilla (IMUS)

This winter school, organized by the Department of Algebra of the University of Seville, aims to bring together early career mathematicians and experts in singularities in both zero and positive characteristic. We will have three 6-hour lectures by Daniel Bath (KU Leuven), Eamon Quinlan-Gallego (U. Utah) and Ilya Smirnov (BCAM Bilbao). There will also be several research talks by young participants.

Courses

Daniel Bath

D-modules and (Twisted) Logarithmic Comparison Theorems: Beyond Free Divisors

A divisor is said to satisfy the Logarithmic Comparison Theorem (LCT) when the natural inclusion of the logarithmic subcomplex into the meromorphic (or rational) de Rham complex is a quasi-isomorphism. In other words, thanks to Grothendieck and Deligne, the logarithmic subcomplex computes the cohomology of the complement with constant coefficients. When the divisor is free (i.e. the syzygies of the Jacobian ideal are free) there is a D-module theoretic formulation of the LCT due to Calderón Moreno and Narváez Macarro. And when the divisor is free and satisfies certain homogeneity properties, the LCT holds by Castro Jiménez, Mond, and Narváez Macarro. But outside the free case this D-module formulation breaks down.

We will discuss a D-module theoretic construction valid for any divisor that implies the LCT and, in classical cases, equates to the LCT. We will do the same for twisted Logarithmic Comparison Theorems, where one aims to compute the cohomology of the complement with coefficients in an arbitrary rank one local system. This will let us give a new proof of that the LCT holds for hyperplane arrangements (a recently resolved conjecture of Terao) and give a complete characterization of when the LCT holds for isolated singularities, the latter completing a story started by Holland and Mond. Further applications, especially to Bernstein-Sato polynomials, will be discussed. Much of the new material is based on joint work by the speaker and Morihiko Saito.

Eamon Quinlan-Gallego

Test ideals and Bernstein-Sato polynomials in positive characteristic

Given a complex hypersurface V = {f = 0}, its Bernstein-Sato polynomial is a classical invariant that detects subtle properties of V, such as the jumping numbers for multiplier ideals. The goal of this course is to explain a characteristic-p analogue of this story.

We will begin by explaining the classical picture. After that, we will go through the basics of differential operators in positive characteristic, and use them to construct the so-called test ideals of Hara and Yoshida, which are known to provide characteristic-p analogues for multiplier ideals. We will then construct the Bernstein-Sato polynomial in positive characteristic, and explain how its connection to test ideals is much stronger than in characteristic zero. If time permits we will talk about extensions to arbitrary subvarieties and (Z/p^n)-coefficients. We will mention some open problems.

Ilya Smirnov

Hilbert-Kunz multiplicity as a measure of singularity

The driving force of algebra in characteristic p > 0 is the Frobenius endomorphism. The use of Frobenius in the study of singularities has its origin largely in the work of Kunz, who characterized in 1969 regular rings by the flatness of Frobenius. This opened a way and many new results followed over the years creating, for example, an entire field of F-invariants, i.e., the numerical invariants quantifying properties of Frobenius.

I will present the theorem of Kunz and will describe how it can be used to build singularity measures. As the main example, I will concentrate on the Hilbert-Kunz multiplicity and describe some of its useful properties, failures, and pressing questions.