On the relative/local principle for (refined) Gromov--Witten theory

Andrea Brini (University of Sheffield)

Tuesday 22nd October 16:00-17:00 Maths 311B

Abstract

The Gromov--Witten theory of a smooth projective variety (compact Kahler manifold) X deals, roughly speaking, with the enumeration of the number of complex curves of given genus and degree passing through a given collection of cycles in X, and it has grown to become a pretty unique area of interplay of very different mathematical communities of mathematicians, including algebraic geometers, stringy theoretical physicists, and integrable systems specialists. An additional layer of sophistication can be introduced by considering the enumerative problem "relative to a fixed smooth divisor D" in X, thereby substantially increasing both interest and difficulty of the enumerative count. An appealing recent result of van Garrel-Graber-Ruddat overcomes these difficulties by positing that the relative problem can be understood as the ordinary GW theory of total space of O_X(−D). In this talk I discuss generalisations of this correspondence to the case when X or D aren’t smooth, as well as a refined correspondence for the all-genus theory when (X, D) is a log Calabi-Yau surface with maximal boundary. This is a joint work with P. Bousseau (ETH) and M. van Garrel (Warwick).

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