Title: The moduli of sheaves on a Calabi-Yau fourfold as a local derived critical locus

Abstract: It is well known that the moduli space M of sheaves on a Calabi-Yau threefold has a symmetric obstruction theory. In the framework of derived algebraic geometry, this is the classical (i.e. non-derived) truncation or “shadow” of a richer construction known as a (-1)-shifted symplectic structure. It is also known that, locally, any derived scheme or stack with a (-1)-shifted symplectic structure is a derived critical locus. For example, M is locally the critical locus of a function on the space of representations of a quiver. In this talk, we will discuss an analog of this result: that any (-2)-shifted symplectic derived stack is locally a derived Lagrangian intesection. In particular, so is the moduli of sheaves on a Calabi-Yau fourfold.

Translation: A somewhat complicated object can always be (locally) understood as the result of performing a (slightly) less complicated operation on a (slightly) less complicated object.

This is based on joint work with Yun Shi.

No prior knowledge of derived geometry will be assumed in this talk