Distinguished Lectures in Geometry
|Speaker:||Mohammed Abouzaid (Columbia University & Stanford University)|
|Title:||Gromov-Witten theory and generalised cohomology III: Global charts in Gromov-Witten theory|
|Time:||2023-06-26 (Mon.) 10:00 - 12:00|
|Place:||Auditorium, 6 Floor, Institute of Mathematics (NTU Campus)|
Gromov-Witten theory has played a central point in organising enumerative invariants in algebraic and symplectic geometry, as well as an entry point for relating Floer theoretic invariants to geometric ones. One disadvantage of the classical formulation of Gromov-Witten invariants is that they are in general formulated as rational counts. Work of Givental and Lee earlier this century gave, in the algebro-geometric context, a lift of these invariants to (complex) K-theory, thus removing the need to introduce denominators.
In this lecture series, I will describe a parallel development on the symplectic side, as well as generalisations to other cohomology theories, and point out some applications to symplectic topology. These lectures will largely describe joint work with Mark McLean and Ivan Smith, some of which is forthcoming.
Lecture 3: Global charts in Gromov-Witten theory
It is not difficult to provide local descriptions of moduli spaces of holomorphic curves in terms of finite group actions on manifolds and vector bundles on them. The heavy part of the theory of virtual fundamental chains amounts to patching these local descriptions into a global object. Recent work has identified a completely different approach, where one uses the idea behind to Kodaira embedding theorem to produce, for each genus and energy level, a single manifold (with an action of a unitary group), equipped with a vector bundle, from which the moduli space is obtained as the 0-locus of a smooth section. I will describe this construction, and use it to show that a version of the classical Kontsevich-Manin axioms holds.
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