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Derived equivalences of varieties and Torelli-type questions for derived categories and Chow theory

Banff International Research Station for Mathematical Innovation and Discovery

This is a report on joint work with Martin Olsson. I will review the basic results on equivalences of the derived categories of coherent sheaves on smooth projective varieties and discuss some attempts to use them to produce Torelli theorems in positive characteristic. Derived equivalences between two varieties always give correspondences between the two varieties in many cohomology theories. In particular, in characteristic 0, one can link them to Hodge theory and rephrase Torelli theorems in terms of a package made of the derived category and the Chow theory. This leads one to wonder if a similar thing happens in positive characteristic. For K3 surfaces, we have a complete answer to this question (that I will explain). Among other things, this has a strong relationship to the Tate conjecture. I will finish with some open questions.

Mathematics; Category theory; homological algebra; Algebraic structures

video/mp4

Author affiliation: University of Washington

2016-12-20

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60055

Unreviewed

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