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The geometry of coherent sheaves: From derived categories to Higgs bundles
April 24, 2023 at 15:00 – 17:30 CEST
GAUS-Workshop: “Recent developments in GIT”
14:00-15:00: Victoria Hoskins (Nijmegen, speaking remotely): An introduction to geometric invariant theory
Abstract: The aim of this survey talk is to give an introduction to geometric invariant theory in order to prepare the audience for the subsequent talks as requested by the organisers. I will start by explaining how group actions often appear in moduli problems and we will see how constructing algebra-geometric quotients is related to 19th century invariant theory. I will explain why the theory is simplest for non-reductive group actions and, in this case, I will explain how Mumford constructs quotients (of certain open ‘semistable’ subsets) using geometric invariant theory, as well as giving combinatorial and numerical criteria for semistability. If there is time, I will briefly mention some recent developments to extend GIT to certain non-reductive group actions.
15:20-16:20: Joshua Jackson (Sheffield): Advances in Non-reductive GIT and applications
Abstract: Following from the previous talk on reductive GIT, I will survey recent developments in extending this theory to non-reductive groups, with a particular focus on applications to moduli theory. Time permitting, I will then indicate how non-reductive GIT can be used in the study of sheaves, Higgs bundles, hypersurfaces, and singular curves.
16:40-17:40: Dario Weissmann (Essen): A stacky approach to identify the semi-stable locus of vector bundles
Abstract: I report on recent joint work with Xucheng Zhang focusing on our Theorem A for vector bundles in characteristic 0: The semi-stable locus in the stack of bundles over a smooth projective curve is the maximal open locus admitting a schematic good moduli space. This gives an intrinsic motivation for semi-stability of vector bundles. Historically, semi-stability appeared in the quest for a moduli space of bundles and the classical construction of this moduli space uses a non-canonical GIT-construction. Theorem A also provides us with natural examples of good moduli spaces which are only algebraic spaces and not schemes.