Speaker : Artan Sheshmani (Harvard University & Institute for Mathematics at Aarhus University)
Title : Virtual Reductions and Intersection Theory on Moduli Space of Sheaves on Calabi-Yau 4 Folds
Time : 2021-12-23 (Thu) 09:30 - 10:30
Place : Auditorium, 6 Floor, Institute of Mathematics (NTU Campus)

I overview some of the earlier work with collaborators, Diaconscu and Yau, on reduction of virtual fundamental classes on moduli spaces of sheaves in Calabi Yau 4 folds to virtual cycles of moduli spaces of sheaves on threefolds which embed in the 4 fold as suitable divisors. In particular I discuss some examples of such virtual reductions which turn out to be useful in the study of Donaldson-Thomas (DT) invariants of torsion sheaves with support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface.

We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in arXiv:1701.08899 and arXiv:1701.08902. We then make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.

If time permits, I will elaborate on extensions of this project to the case where we study particular framed sheaves on cotangent bundles of surfaces and show how a version of degeneration technique and our results, can help relating invariants on the 4 fold to Vafa-Witten invariants of the surface. Finally, if time permits, I will provide hints about how to generalize these results to more general case of sheaves with support on general 4 manifolds (non-algebraic surfaces) and hidden structures which one can discover in their associated partition functions.