Holomorphic curves in exploded manifolds: virtual fundamental class

Parker, Brett

doi:10.2140/gt.2019.23.1877

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Holomorphic curves in exploded manifolds: virtual fundamental class

Brett Parker

Geometry & Topology 23 (2019) 1877–1960

DOI: 10.2140/gt.2019.23.1877

Abstract

We define Gromov–Witten invariants of exploded manifolds. The technical heart of this paper is a construction of a virtual fundamental class $[K]$ of any Kuranishi category $K$ (which is a simplified, more general version of an embedded Kuranishi structure). We also show how to integrate differential forms over $[K]$ to obtain numerical invariants, and push forward such differential forms over suitable maps. We show that such invariants are independent of any choices, and are compatible with pullbacks, products and tropical completion of Kuranishi categories.

In the case of a compact symplectic manifold, this gives an alternative construction of Gromov–Witten invariants, including gravitational descendants.

Keywords

holomorphic curves, virtual fundamental class, exploded manifolds, Gromov–Witten invariants

Mathematical Subject Classification 2010

Primary: 53D45

References

Publication

Received: 29 June 2017

Revised: 10 September 2018

Accepted: 7 October 2018

Published: 17 June 2019

Proposed: Richard Thomas

Seconded: Jim Bryan, Dan Abramovich

Authors

Brett Parker
Monash University Melbourne, VIC Australia

Volume 23, issue 4 (2019)