Scalar Curvature Functions of Almost-K\"ahler Metrics

TL;DR

This paper introduces topological invariants for almost-Kähler metrics on symplectic manifolds, explores their properties, and solves a scalar curvature classification problem for these metrics.

Contribution

It defines new invariants $Z(M)$ and $Z(M, [[oldsymbol{\omega}]])$, demonstrates their variability, and establishes a scalar curvature realization result for almost-Kähler metrics.

Findings

Existence of a 6-dimensional manifold with multiple deformation classes of different $Z$-signs.

Classification of symplectic manifolds into three categories using $Z$ invariants.

Any function with negative and zero values can be realized as scalar curvature of an almost-Kähler metric.

Abstract

For a closed smooth manifold $M$ admitting a symplectic structure, we define a smooth topological invariant $Z(M)$ using almost-K\"ahler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce $Z(M, [[\omega]])$ depending on symplectic deformation equivalence class $[[\omega]]$. We first prove that there exists a 6-dimensional smooth manifold $M$ with more than one deformation equivalence classes with different signs of $Z(M, [[\omega]] )$. Using $Z$ invariants, we set up a Kazdan-Warner type problem of classifying symplectic manifolds into three categories. We finally prove that on every closed symplectic manifold $(M, \omega)$ of dimension $\geq 4$, any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-K\"ahler metric compatible with a symplectic form which is deformation equivalent to $\omega$.