Taming the pseudoholomorphic beasts in ℝ × (S1 × S2)

Gerig, Chris

doi:10.2140/gt.2020.24.1791

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Taming the pseudoholomorphic beasts in $\mathbb{R} \times (S^1 \times S^2)$

Chris Gerig

Geometry & Topology 24 (2020) 1791–1839

DOI: 10.2140/gt.2020.24.1791

Abstract

For a closed oriented smooth $4$ –manifold $X$ with $b_{+}^{2} (X) > 0$ , the Seiberg–Witten invariants are well-defined. Taubes’ “ $SW = Gr$ ” theorem asserts that if $X$ carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes’ Gromov invariants. In the absence of a symplectic form, there are still nontrivial closed self-dual $2$ –forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describe well-defined integral counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic $2$ –forms, and it is shown that they recover the Seiberg–Witten invariants over $ℤ ∕ 2 ℤ$ . This is an extension of “ $SW = Gr$ ” to nonsymplectic $4$ –manifolds.

The main result of this paper asserts the following. Given a suitable near-symplectic form $ω$ and tubular neighborhood $𝒩$ of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism $(X - 𝒩, ω)$ which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form “near-symplectic” Gromov invariants as a function of spin-c structures on $X$ .

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Keywords

near-symplectic, Seiberg–Witten, Gromov, pseudoholomorphic, ECH

Mathematical Subject Classification 2010

Primary: 53D42

Secondary: 57R57

References

Publication

Received: 16 October 2018

Revised: 14 August 2019

Accepted: 9 November 2019

Published: 10 November 2020

Proposed: Simon Donaldson

Seconded: Ciprian Manolescu, Leonid Polterovich

Authors

Chris Gerig
Mathematics Department University of California, Berkeley Berkeley, CA United States

Volume 24, issue 4 (2020)