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The geometry of coherent sheaves: From derived categories to Higgs bundles
February 6 at 15:00 – 17:30 CET
GAUS-Workshop: “Invariants and curve counting”
15:00-16:00: Luca Battistella (Frankfurt):
Logarithmic and orbifold Gromov-Witten invariants
Abstract: Logarithmic Gromov-Witten theory can be thought of as the study of curves in open manifolds, or, in other words, curves with tangency conditions to a boundary divisor. When the divisor is smooth, several techniques have been developed to compute the invariants, most notably orbifold stable maps. When the divisor is normal crossings, on the other hand, the logarithmic theory remains hardly accessible. The strategy of rank reduction, i.e. looking at the components of the boundary one at a time, is more directly applicable to other theories than the logarithmic one (as shown in Nabijou-Ranganathan and B.-Nabijou-Tseng-You) due to tropical obstructions. Inspired by one of the distinguishing features of the logarithmic theory – namely, birational invariance [Abramovich-Wise] – in joint work with Nabijou and Ranganathan we show that, when the genus is zero, tropical obstructions can be disposed of by blowing up the target sufficiently. The slogan is that the logarithmic theory is the limit orbifold theory under birational modifications along the boundary divisor. If time permits I will discuss work in progress towards understanding negative contact.
16:20-17:20: Georg Oberdieck (Stockholm):
Pandharipande-Thomas theory of elliptic threefolds and Jacobi forms
Abstract: Pandharipande-Thomas theory is the study of the intersection theory of the moduli space of stable pairs of a threefold. The intersection numbers, called Pandharipande-Thomas invariants, may be viewed as counting curves on the threefold subject to given incidence conditions. In this talk we explore the properties of the generating series of Pandharipande-Thomas invariants of elliptically fibered threefolds. There will be two main conjectures: Quasi-Jacobi Property and Holomorphic Anomaly Equations. Together these essentially determine the modular properties of the generating series. The conjectures are motivated by the case of Calabi-Yau threefolds where by mirror symmetry computations Huang-Katz-Klemm conjectured that the series of PT invariants are Jacobi forms. I discuss several examples, in particular the equivariant geometry of K3xA^1. Here the conjectures lead to explicit new formulas for the invariants. Based on joint work with Maximilian Schimpf.