# RGAS School on Singularities

Date: Mon, Jan 8 - Fri, Jan 12 2024

Location: Instituto de Matemáticas de la Universidad de Sevilla (IMUS)

Speakers: Daniel Bath (KU Leuven), Eamon Quinlan-Gallego (U. Utah), Ilya Smirnov (BCAM Bilbao) and Guillem Blanco (KU Leuven)

This winter school of the Algebraic Geometry and Singularities Network aims to bring together early career mathematicians and experts in singularities in both zero and positive characteristic. We will have four 5-hour lectures by Daniel Bath (KU Leuven), Guillem Blanco (KU Leuven), Eamon Quinlan-Gallego (U. Utah) and Ilya Smirnov (BCAM Bilbao). There will also be short research talks and, eventually, a poster session, both presented by young participants.

**D-modules and (Twisted) Logarithmic Comparison Theorems: Beyond Free Divisors - Daniel Bath (KU Leuven)**

A divisor is said to satisfy the Logarithmic Comparison Theorem (LCT) when the natural inclusion of 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 Calderon-Moreno and Narvaez-Macarro. And when the divisor is free and satisfies certain homogeneity properties, the LCT holds by Castro-Jimenez, Mond, and Narvaez-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 an arbitrary rank one local system. This will let us give a new proof 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.

**Bernstein-Sato polynomials and invariants of singularities - Guillem Blanco (KU Leuven)**

The Bernstein-Sato polynomial is an invariant of singularities lying at the heart of D-modules theory but that has its origin in the analytic continuation of certain integrals. In this course we will explore the relation between the Bernstein-Sato polynomial and different types of zeta functions as well as other invariants of singularities. For the case of isolated singularities, we will present its relation with the Gauss-Manin connection and periods of integrals in the Milnor fiber.

**Test ideals and Bernstein-Sato polynomials in positive characteristic - Eamon Quinlan-Gallego (U. Utah)**

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.

**Hilbert-Kunz multiplicity as a measure of singularity - Ilya Smirnov (BCAM Bilbao)**

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.

40 students and researchers from national and international institutions are expected to participate. All the information about the event can be found on the web page:

https://sites.google.com/view/sevillargas/rgas-sevilla

## Organizers:

Instituto de Matemáticas de la Universidad de Sevilla (IMUS)

## Confirmed speakers:

Daniel Bath (KU Leuven) “D-modules and (Twisted) Logarithmic Comparison Theorems: Beyond Free Divisors”

Eamon Quinlan-Gallego (U. Utah) “Test ideals and Bernstein-Sato polynomials in positive characteristic”

Ilya Smirnov (BCAM Bilbao) “Hilbert-Kunz multiplicity as a measure of singularity”

Guillem Blanco (KU Leuven) “Berstein-Sato polynomials and invariants of singularities”

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