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

TL;DR

This paper establishes a deep connection between closed J-anti-invariant 2-forms and J-holomorphic curves in 4-manifolds, confirming a conjecture and exploring invariance properties, with implications for higher dimensions.

Contribution

It proves the zero set of closed J-anti-invariant 2-forms supports J-holomorphic subvarieties and shows the dimension of these forms is a birational invariant.

Findings

Zero set supports J-holomorphic subvarieties

Dimension is a birational invariant

Confirms a conjecture of Draghici-Li-Zhang

Abstract

We study the relation between $J$-anti-invariant $2$-forms and pseudoholomorphic 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.