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Variations of noncommutative Hodge structures in general and in distinguished cases

Banff International Research Station for Mathematical Innovation and Discovery

A variation of twistor structures in the sense of Simpson, Sabbah and Mochizuki is a 1-parameter family of flat connections on a complex vector bundle with (to be chosen) additional data and constraints. Some version on rank 2 bundles turns up in the DPW method for constructing CMC surfaces. Another version of arbitrary rank is equivalent to Simpson's harmonic bundles, which are a generalization and weakening of variation of Hodge structures. A generalization of variation of Hodge structures which is a not a weakening, can be encoded as an integrable variation of twistor structures. Closely related versions of this are tt^* geometry (Cecotti-Vafa), TERP structures (Hertling) and noncommutative Hodge structures (Katzarkov-Kontsevich-Pantev). In the talk, I will discuss these structures in general, and in distinguished cases which arise in the theory of isolated hypersurface singularities. A good way to control them is given by the theory of meromorphic connections with irregular poles and their Stokes structures. I will sketch some results and some open questions.

Mathematics; Differential geometry; Global analysis, analysis on manifolds

video/mp4

Author affiliation: Universitaet Manheim

2019-03-29

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69331

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/