Tim Hosgood: Geometry-independent constructions of bundles and sheaves
Time: Wed 2023-01-25 13.15
Location: KTH, 3418
Participating: Tim Hosgood, SU
The interplay between complex analysis and complex algebraic geometry dates back to the birth of the subjects, with definitions and constructions being passed back and forth between the two. One fundamental example is that of a coherent sheaf, which first arose in complex analysis but eventually became one of the many finiteness conditions that form the initial study of SGA 6, for example. It is a very important fact, however, that objects with the same definition behave very differently from one another when in the algebraic world or when in the analytic world — coherent algebraic sheaves and coherent analytic sheaves do not satisfy the same properties as one another.
The main aim of this talk is to outline a category-theoretic construction that recovers vector bundles, coherent sheaves, twisting cochains, perfect complexes, and simplicial vector bundles (objects which will be introduced in the talk), all in both the algebraic and analytic setting, as well as many others besides. The hope is that this can be used to resolve open questions concerning the relations between some of these objects in the complex-analytic case. This is a work-in-progress, joint with Mahmoud Zeinalian.