Perfect obstruction theories are a fundamental ingredient used to define invariants associated to moduli problems, such as virtual cycles in the Chow group and virtual structure sheaves in K-theory. However, several moduli spaces, such as the moduli space of simple perfect complexes and desingularizations of moduli stacks of semistable sheaves on Calabi-Yau threefolds, do not admit a perfect obstruction theory. In this talk we introduce the relaxed notion of an almost perfect obstruction theory on a Deligne-Mumford stack and show that it gives rise to a virtual struture sheaf in K-theory. This applies to many examples of interest, including the above, and enables us to define K-theoretic invariants and, in particular, K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. Based on joint work with Young-Hoon Kiem.

 

The seminar will take place online. Click here for the Zoom link.