An algebraic model for smoothing Calabi-Yau varieties
Kwokwai Chan - Chinese University of Hong Kong
We are interested in smoothing of a degenerate Calabi-Yau variety or a pair (degenerate CY, sheaf). I will explain an algebraic framework for solving such smoothability problems. The idea is to glue local dg Lie algebras (or dg Batalin-Vilkovisky algebras), coming from suitable local models, to get a global object. The key observation is that while this object is only an almost dg Lie algebra (or pre-dg Lie algebra), it is sufficient to prove unobstructedness of the associated Maurer-Cartan equation (a kind of Bogomolov-Tian-Todorov theorem) under suitable assumptions, so the former can be regarded as a singular version of the Kodaira-Spencer DGLA. Our framework applies to degenerate CY varieties previously studied by Kawamata-Namikawa and Gross-Siebert, as well as a more general class of varieties called toroidal crossing spaces (by the recent work of Felten-Filip-Ruddat). This talk is based on joint works with Conan Leung, Ziming Ma and Y.-H. Suen.