An Integrability Theorem for Almost-K\"ahler Structures using J-anti-invariant Two-Forms on Four-Manifolds

TL;DR

This paper proves that on symplectic four-manifolds, the presence of three linearly independent closed J-anti-invariant two-forms guarantees the integrability of the almost complex structure, confirming a conjecture in the field.

Contribution

It establishes a new integrability criterion for almost complex structures using J-anti-invariant forms, solving a conjecture for almost-K"ahler four-manifolds.

Findings

Existence of three closed J-anti-invariant forms implies integrability.

Proves the Draghici-Li-Zhang conjecture in the almost-K"ahler setting.

Provides a new criterion for identifying K"ahler structures.

Abstract

We establish a new criterion for a compatible almost complex structure on a symplectic four-manifold to be integrable and hence K\"ahler. Our main theorem shows that the existence of three linearly independent closed J-anti-invariant two-forms implies the integrability of the almost complex structure. This proves the conjecture of Draghici-Li-Zhang in the almost-K\"ahler case