Volume 23, issue 4 (2019)

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Holomorphic curves in exploded manifolds: virtual fundamental class

Brett Parker

Geometry & Topology 23 (2019) 1877–1960

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.

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holomorphic curves, virtual fundamental class, exploded manifolds, Gromov–Witten invariants
Mathematical Subject Classification 2010
Primary: 53D45
Received: 29 June 2017
Revised: 10 September 2018
Accepted: 7 October 2018
Published: 17 June 2019
Proposed: Richard Thomas
Seconded: Jim Bryan, Dan Abramovich
Brett Parker
Monash University
Melbourne, VIC