Project 1
Proving wall-crossing for Calabi–Yau fourfolds
I am proving that for versions of Bridgeland stability conditions on the derived category of coherent sheaves, there are wall-crossing formulae relating the virtual fundamental classes counting semistable objects. One of the interesting outcomes is that this can be applied to Calabi–Yau dg-categories as long as assumptions are satisfied and to wall-crossing with certain nice insertions. I use this theory to prove multiple existing conjectures in the literature related to point and curve counting together with the results in 3. This is based on the work of Joyce in lower dimensions.
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.
The proof is intricate as 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 I check in my work.
On the left is represented the argument that proves that the produced invariants counting semistable torsion-free sheaves are well-defined. The second picture is one of the main applications of my work which has implications to 3-fold DT/PT correspondence. The figure represents the need to work with new families of stability conditions.