Twisted differentials and Lee classes of locally conformally symplectic complex surfaces

TL;DR

This paper investigates the Lee classes of locally conformally symplectic structures on compact complex surfaces of class VII, revealing their connectedness, bounds, and implications for complex geometry and bi-Hermitian structures.

Contribution

It provides a complete description of Lee classes on known class VII surfaces and introduces new obstructions to bi-Hermitian structures, linking bounds to holomorphic forms.

Findings

The set of Lee classes is connected.

Explicit bounds are established for hyperbolic Kato surfaces.

New obstructions to bi-Hermitian structures are identified.

Abstract

We study the set of deRham classes of Lee $1$-forms of the locally conformally symplectic (LCS) structures taming the complex structure of a compact complex surface in the Kodaira class VII, and show that the existence of non-trivial upper/lower bounds with respect to the degree function correspond respectively to the existence of certain negative/non-negative PSH functions on the universal cover. We use this to prove that the set of Lee deRham classes of taming LCS is connected, as well as to obtain an explicit negative upper bound for this set on the hyperbolic Kato surfaces. This leads to a complete description of the sets of Lee classes on the known examples of class VII complex surfaces, and to a new obstruction to the existence of bi-hermitian structures on the hyperbolic Kato surfaces of the intermediate type. Our results also reveal a link between bounds of the set of Lee classes and non-trivial logarithmic holomorphic $1$-forms with values in a flat holomorphic line bundle.