Volume 23, issue 4 (2019)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 23
Issue 4, 1621–2164
Issue 3, 1085–1619
Issue 2, 541–1084
Issue 1, 1–540

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Other MSP Journals
Holomorphic curves in exploded manifolds: virtual fundamental class

Brett Parker

Geometry & Topology 23 (2019) 1877–1960
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