Communications in Analysis and Geometry

Volume 30 (2022)

Number 6

$J$-holomorphic curves from closed $J$-anti-invariant forms

Pages: 1196 – 1226

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n6.a1

Authors

Louis Bonthrone (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

Weiyi Zhang (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

Abstract

We study the relation between $J$-anti-invariant $2$-forms and pseudo-holomorphic curves in this paper. We show the zero set of a closed $J$-anti-invariant $2$-form on an almost complex $4$-manifold supports a $J$-holomorphic subvariety in the canonical class. This confirms a conjecture of Draghici–Li–Zhang. A higher dimensional analogue is established. We also show the dimension of closed $J$-anti-invariant $2$-forms on an almost complex $4$-manifold is a birational invariant, in the sense that it is invariant under degree one pseudoholomorphic maps.

Received 28 February 2019

Accepted 9 January 2020

Published 26 April 2023