June Minicourses on Hodge Theory and Geometry
Compact moduli of K3 surfaces and microlocal sheaves, Humboldt Universitat Berlin
Courses
Philip Engel, Compact moduli of K3 surfaces
For each d > 0, there is a 19-dimensional moduli space F2d 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 F2d. Hodge theory also realizes F2d as a Type IV arithmetic quotient with Baily-Borel, toroidal, and semitoroidal compactifications. These lectures ask when the two perspectives can be identified. Phil's notes are available here.
Michael McBreen, Introduction to microlocal sheaves
Microlocal sheaf theory studies sheaves locally along the cotangent bundle of a manifold. The minicourse introduced the subject following Kashiwara and Schapira's Sheaves on Manifolds, with emphasis on the complex analytic setting where many constructions have algebraic interpretations.
Schedule
| Date | Time | Room | Topic | Description |
|---|---|---|---|---|
| June 13 | 15:30-17:30 | 1.021 IRIS Adlershof | Degenerations of K3 surfaces | K3 surfaces, one-parameter degenerations, semistable/Kulikov models, and integral-affine structures on the sphere. |
| June 16 | 15:30-17:30 | 1.021 IRIS Adlershof | Degenerations of the period map, toroidal compactifications | Geometry of moduli spaces F2d, Hodge-theoretic and MMP compactifications, and examples for F2 and elliptic K3 surfaces. |
| June 19 | 15:30-17:30 | 1.221 IRIS Adlershof | Recognizable divisors | Divisors on generic polarized K3 surfaces whose KSBA compactifications are Hodge-theoretic. |
| June 20 | 13:15-14:45 | 3.006 Johannes von Neumann Gebaude | Introduction to microlocal sheaves | Singular support, Fourier-Sato transform, Verdier specialization, Sato microlocalization, and the microlocal hom sheaf. |
| June 21 | 13:15-14:45 | 3.007 Johannes von Neumann Gebaude | General definitions and explicit constructions | Categories of microlocal sheaves on open subsets of cotangent bundles and on specified supports. |
| June 26 | 14:30-16:30 | 1.221 IRIS Adlershof | D-modules and Riemann-Hilbert | Characteristic varieties, microdifferential operators, and the Riemann-Hilbert correspondence. |
| June 27 | 15:30-17:30 | 1.021 IRIS Adlershof | Advanced constructions | Pushforwards, pullbacks, microlocal correspondences, microlocal index theorems, and microlocal sheaves on Weinstein manifolds. |
Location
The lectures were held at:
- Johannes von Neumann Gebaude, Rudower Chaussee 25.
- IRIS Adlershof, Zum Grossen Windkanal 2.
Both buildings are near each other on the Adlershof campus of Humboldt Universitat Berlin. A detailed local map is available here.