Artan Sheshmani

Artan Sheshmani (Harvard) посетил Лабораторию в ноябре 2017 г.

17 ноября 2017 года он выступил с докладом "Nested Hilbert schemes, local Donaldson-Thomas theory, Vafa-Witten / Seiberg-Witten correspondence" на еженедельном семинаре Лаборатории.

Abstract: We report on the recent rigorous and general construction of the deformation-obstruction theories and virtual fundamental classes of nested (flag) Hilbert scheme of one dimensional subschemes of a smooth projective algebraic surface. This construction will provide one with a general framework to compute a large class of already known invariants, such as Poincare invariants of Okonek et al, or the reduced local invariants of Kool and Thomas in the context of their local surface theory. We show how to compute the generating series of deformation invariants associated to the nested Hilbert schemes, and via exploiting the properties of vertex operators, prove that in some cases they are given by modular forms. We finally establish a connection between the Vafa-Witten invariants of local-surface threefolds (recently analyzed Tanaka and Thomas) and such nested Hilbert schemes. This construction (via applying Mochizuki's wall- crossing techniques) enables one to obtain a relations between the generating series of Seiberg-Witten invariants of the surface, the Vafa-Witten invariants and some modular forms. This is joint work with Amin Gholampour and Shing-Tung Yau following arXiv:1701.08902 and arXiv:1701.08899.

Также Артан Шешмани принял участие в работе международного семинара по бирациональной геометрии, который проходил в Лаборатории в период с 22 по 24 ноября 2017 года, и семинара для студентов и аспирантов факультета математики "Гиперкэлерова суббота" на факультете математике 25 ноября, где выступил с докладом PT/DT duality on nodal K3-fibrations

Аннотация: We study Pandharipande-Thomas (PT) stable pair theory on smooth K3 fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on K3 surfaces and Noether-Lefschetz numbers of the fibration. We further investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the K3-fibration. In the case that the fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas (DT) invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers. This talk is based on arXiv1308.4722.