Talk page
Title:
From categories to curve counts I: automatic split generation and non-degeneracy
Speaker:
Abstract:
We will explain two structural results in mirror symmetry. The first is about the endgame of proving homological mirror symmetry for a particular Calabi-Yau mirror pair (X,Y). Namely, suppose that you have found full subcategories of Fuk(X) and $D^b$Coh(Y), and proved that they are quasi-equivalent; and on the algebraic geometry side, suppose that the subcategory split-generates, and furthermore that Y has 'maximally unipotent monodromy'. Our result says that the full subcategory of Fuk(X) then necessarily split-generates too, by Abouzaid's criterion: so one obtains the full HMS equivalence. The second result is a byproduct of the proof of the first: it says that, under the same hypotheses, if X and Y are homologically mirror, then Fuk(X) is automatically `non-degenerate' (in the sense that the open-closed map hits the unit); this will be crucial in the second talk in the series. Joint work with Tim Perutz.
Link:
Workshop: