June minicourses on Hodge theory and geometry

Philip Engel, Compact moduli of K3 surfaces

For each \(d>0\), there is a 19-dimensional moduli space \(F_{2d}\) of K3 surfaces, with an ample line bundle of degree \(2d\). Choosing an ample divisor in a canonical way on each such K3 surface, the minimal model program provides a "KSBA" compactification of \(F_{2d}\). On the other hand, the Hodge theory of K3 surfaces implies that \(F_{2d}=\Gamma\backslash \mathbb{D}\) is a Type IV arithmetic quotient/orthogonal Shimura variety. In this capacity, it has a variety of compactifications: Baily-Borel, toroidal, semitoroidal. Can these two types of compactifications ever be identified?
Phil is kindly writing up notes for his talk.

Michael McBreen, Introduction to Microlocal Sheaves

Microlocal sheaf theory is the study of sheaves 'locally along the cotangent bundle' of a manifold. Many fundamental aspects of sheaves are best understood through this lens. This minicourse will give an introduction to the subject following Kashiwara and Schapira's classic book Sheaves on Manifolds. Although the book treats sheaves on smooth manifolds, we will focus on the complex analytic setting, in which many constructions admit purely algebraic interpretations.

Schedule

June 13 15:30 - 17:30 1.021 IRIS Adlershof Degenerations of K3 surfaces The first lecture will introduce K3 surfaces and their one-parameter degenerations, in particular, semistable (aka Kulikov) models. In analogy with how stable graphs encode degenerations of curves, we will describe a way to combinatorially encode the data of a degeneration, using integral-affine structures on the sphere.
June 16 15:30 - 17:30 1.021 IRIS Adlershof Degenerations of the period map, toroidal compactifications The second lecture will focus on the geometry of moduli spaces \(F_{2d}\) and their compactifications, from both the Hodge-theoretic and MMP perspectives. We will discuss degeneration of the period map, and give some explicit examples of (semi)toroidal compactifications, for \(F_{2}\) and for moduli of elliptic K3 surfaces.
June 19 15:30 - 17:30 1.221 IRIS Adlershof Recognizable divisors The third lecture will introduce the notion of a "recognizable divisor". These are divisors chosen on the generic polarized K3 surface whose KSBA compactifications are Hodge-theoretic. We will give examples for \(F_2\) and for moduli of elliptic K3 surfaces. Then we will discuss the general theory of recognizable divisors, and how it can be applied to compactify \(F_{2d}\).
June 20 13:15 - 14:45 3.006 Johannes von Neumann Gebäude Introduction to microlocal sheaves We introduce the basic tools of the trade : the singular support of a sheaf, the Fourier-Sato transform, Verdier specialization, Sato's microlocalisation and the microlocal hom sheaf.
June 21 13:15 - 14:45 3.007 Johannes von Neumann Gebäude General definitions and explicit constructions Given a manifold \(M\) and an open subset \(U\) of the cotangent bundle \(T^*M\), we define the category \(\mu Sh(U)\) of microlocal sheaves on \(U\), as well as the category \(\mu Sh_Z(U)\) of microlocal sheaves supported on any given subset \(Z\) of \(U\). We sketch the basic features of this category, and describe it as explicitly as possible using the constructions from Lecture 1.
June 26 14:30 - 16:30 1.221 IRIS Adlershof D-modules and Riemann-Hilbert We revisit Lectures 1 and 2 from the perspective of D-modules. We will discuss characteristic varieties, microdifferential operators and the Riemann-Hilbert correspondence.
June 27 15:30 - 17:30 1.021 IRIS Adlershof Advanced constructions We will discuss the behavior of microlocal sheaves under pushforwards, pullbacks and microlocal correspondences. We examine microlocal index theorems, which compute (local) Euler characteristics via intersection numbers of Lagrangian cycles in the cotangent bundles. We sketch the definition of microlocal sheaves on a general Weinstein manifold, and the relation with the Fukaya category.

Location

The lectures will be held at
Both buildings are near each other in the Adlershof campus of the Humboldt University of Berlin. From the city center, take the south/airport-bound S-Bahn trains 8, 9, 45, 46 or 85 to the S-Adlershof station. Then take the west-bound trams (any) or buses 162, 163, 164 for 2 stops to Magnusstraße (Magnusstr.), or walk for about 10 minutes.

A detailed local map