Project 1

Proving wall-crossing for Calabi–Yau fourfolds

In this vast project, I have chosen to tackle the question of wall-crossing formulae for Calabi-Yau fourfolds, which were originally conjectured by Gross-Joyce-Tanaka. This work had two focal points: providing insight into the wall-crossing framework and being practical in applications. My first work on this subject focuses largely on the first point. Due to an oversight in my proof, it is limited to CY4 quivers and locally CY fourfolds. An approach that was shared with me by Nick Kuhn is capable of resolving the general case. It is expected to appear in a joint work with him, Liu, and Thimm.

One of the interesting outcomes of my research is that one can wall-cross with some nice invariants obtained from multiplicative insertions. This refines the original formulation of Gross-Joyce-Tanaka who only dealt with full virtual fundamental classes. The geometric set-up of such wall-crossing gives naturally rise to a representation theoretic structure I called additive deformations of vertex algebras. Combined with the general wall-crossing theorem, I will use these additive deformations to prove multiple existing conjectures in the literature related to curve and surface counting (see Project 3).

I use formal families of vertex algebras in a formal variable u to formulate wall-crossing of invariants. The first picture represents a physical perspective of the origin of such families. To obtain explicit formulae, one takes an appropriate coefficient of the expansion in $u$ as is portrayed in the second illustration.
One first needs to construct self-dual obstruction theories for Mochizuki's enhanced master spaces represented on the left. The proof sketched in the diagram on the right requires many compatibilities between these obstruction theories, which are violated in general.
The left picture represents the argument that leads to the proof that the invariants counting semistable torsion-free sheaves are well-defined. The second picture is one of the main applications of this project, which has implications to 3-fold DT/PT correspondence. The figure represents a new family of stability conditions, one is required to work with. In my follow-up work, I will therefore formulate wall-crossing in terms of some generalizations of Bridgeland stability conditions.

References