A workshop on D-modules and their applications in algebraic geometry (05.03.2025 to 07.03.2025 at Ruhr-Universität Bochum). 

The room is ID 03/653. Building ID can be found on the map of the campus here.

Abstracts and a prospective timetable are available below. A meet-and-greet session consisting in three 20-minute talks will be taking place on Wednesday evening. There will be a workshop dinner on Thursday night. 

If you wish to attend, please email christian.lehn(at)ruhr-uni-bochum.de timely. Registration is free of charge. Note that unfortunately, we are not able to offer funding for participants.

Abstracts

• Yajnaseni Dutta: Hodge modules related to Lagrangian fibrations of hyperkähler manifolds

Given a family of proper algebraic varieties arranged as the fibres of a smooth variety over a smooth base, the decomposition theorem captures how the singular cohomologies of these varieties in the fibres vary as they become more and more singular. In this setup, lot of symmetries, e.g. Hard Leschetz, Poincaré duality, that are enjoyed by smooth projective varieties, manifest themselves fibrewise. Cohomologies of smooth projective varieties also enjoy a very symmetric diamond shaped decomposition in subvector spaces (known as the Hodge decomposition). This kind of symmetry, albeit mysterious for general families, shows up very elegantly for a certain degenerate family of Abelian varieties; the Lagrangian fibrations of hyperkähler manifolds. Hodge module is a powerful tool for proving these in a rigorous, yet relatively lazy way. Furthermore, in some examples these abelian varieties generically arise as (intermediate) Jacobian of curves and cubic threefolds. Hodge module theoretic techniques also allow us to consider the relative sheaf of (intermediate) Jacobian, a gadget that helps us construct new Lagrangian fibrations from the old one.

The plan for the three talks will roughly be as follows. 1) Crash course on decomposition theorem via various examples after de Cataldo-Migliorini. 2) Lagrangian fibration of hyperkähler manifolds and symmetries after Matsuhita and Schnell 3) Intermediate Jacobians in family after D-Mattei-Shinder.

• Andreas Hohl:The Stokes phenomenon - From rainbows to D-modules

In the theory of linear complex differential equations, an essential distinction is that between two types of singular points: While regular singularities have been well understood for a long time, the classification of irregular ones is much more recent. A key ingredient in the latter is the Stokes phenomenon, which was originally discovered by Stokes while performing computations in optics. It allows us to adapt the concepts of monodromy, local systems and perverse sheaves to the irregular case.

In this lecture series, we will learn about topological perspectives on systems with possibly irregular singularities that have been developed
in the last 50 years, and we will use them to explain some explicit results on Fourier transforms, an integral transform that is ubiquitious
in mathematics and physics. In the first lecture, we survey some basics about D-modules, and we are going to see in particular how to classify them via so-called Riemann-Hilbert correspondences. In the second lecture, we will learn about the Stokes phenomenon in the context of irregular singularities, and in particular different ways of representing it geometrically. In the final lecture, the Fourier transform will come into play, and we will investigate the question of how the Stokes data behave under this transform.

• Thomas Krämer: Perverse Sheaves on Abelian Varieties

To any perverse sheaf on an abelian variety one may attach a linear algebraic group by applying Tannaka duality to the tensor category generated by its convolution powers. The arising groups play a fundamental role in the geometry and arithmetic of irregular varieties. We will give a self-contained introduction to the topic starting from generic vanishing and the geometry of Gauss maps, and then discuss some recent applications in two directions: (1) Singularities of theta divisors and the moduli of abelian varieties, and (2) big monodromy results in arithmetic geometry.

Meet-and-greet session

On Wednesday, March 5th, we will have three 20-minute talks by participants.

• Céline Fietz (Universitet Leiden): Categorical resolutions of A₂ singularities.
• Niklas Müller (Universität Duisburg-Essen): Inequalities of Miyaoko-Yau type.
• Constantin Podelski (HU Berlin): The Tannakian Schottky problem in genus five.

Prospective timetable

The room is ID 03/653 (in the building ID, on the floor 03 = "negative 3", in the room numbered 653). Building ID can be found on the map of the campus here.

Organisation

• Cécile Gachet (RU Bochum)
• Daniel Greb (Universität Duisburg-Essen)
• Christian Lehn (RU Bochum)