University of Sydney Algebra Seminar
Sandor Kovacs
Friday 21 Apr, 12-1pm, Place: Carslaw 173
Liftable local cohomology, Du Bois singularities, and KSB stability
This is a report on joint work with János Kollár. Mumford developed Geometric Invariant Theory (GIT) in order to construct and study the moduli spaces of curves (of general type). Due to various serious difficulties it took another 50 years after Mumford to construct the moduli spaces of surfaces and higher dimensional varieties (of general type). One of the major difficulties lay in finding the "right" notion of stability, because it turned out that GIT did not provide the same powerful tool in higher dimensions as it did for curves. Today we know the correct stability criterion, known as Kollár-Shepherd-Barron, or KSB stability. This has a rather technical definition, including the class of singularities allowed on KSB stable varieties. In this talk, instead of going through this technical definition I will discuss a very useful property of these singularities, called liftable local cohomology. I will explain what this is and how it is related to KSB stability. I will also explain recent results that show that this is a powerful notion that allows us to prove fundamental results regarding KSB stable families and the resulting moduli spaces.