Junior Geometry Seminar – Ashesh Bati

Abstract: Given a nodal variety, there are two natural ways to ‘remove’ the singularity: smoothing and resolving. In the case of (complex) dimension 3, we obtain the case of the conifold transition, which is the contraction of holomorphic curves to nodes, followed by a smoothing. These are ubiquitous in the theory of Calabi-Yau threefolds and are important examples in string theory and mirror symmetry. Categorical absorption is the decomposition of the bounded derived category of a singular variety into a smooth (perfect) component, and a highly ‘nonsingular’ component. The theory is particularly rich in the case of nodal varieties. 

In this talk, we will explore how important invariants of the varieties change under conifold transition, and how the configuration of (or relationships between) the nodes affect this change. This begins with discussing cohomology, then looking at a refinement of this in the bounded derived category of coherent sheaves. We will conclude with an interpretation of absorption of nodes in terms of relations between the nodes.