Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class

TL;DR

This paper studies the Kodaira dimension of almost complex 4-manifolds with torsion first Chern class, showing it is either 0 or -infinity if the structure is tamed, and explores deformations in higher dimensions.

Contribution

It establishes the possible Kodaira dimensions for tamed almost complex 4-manifolds with torsion first Chern class and develops deformation theory for structures with pseudoholomorphically torsion canonical bundle.

Findings

Kodaira dimension is 0 or -infinity for tamed structures

Develops theory of pseudoholomorphic structures on vector bundles

Describes deformations of complex structures on K3 and Enriques surfaces

Abstract

In this paper we investigate the Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class. First, we prove that, if the almost complex structure is also tamed, the only possible values for the Kodaira dimension are $0$ or $-\infty$. This is done by developing the theory of pseudoholomorphic structures on vector bundles. In arbitrary dimension, we study infinitesimal deformations of structures with pseudoholomorphically torsion canonical bundle. We compute their tangent space and, under suitable assumptions, we prove an unobstructedness theorem in the spirit of Bogomolov--Tian--Todorov. Together, our results allow to fully describe non-integrable infinitesimal deformations of complex structures on $K3$ and Enriques surfaces in terms of their Kodaira dimension.