Symplectic generic complex structures on 4-manifolds with b_+ = 1

TL;DR

This paper investigates symplectic structures on certain Kähler surfaces with specific invariants, providing an example of a symplectic structure incompatible with any Kähler metric, highlighting differences between symplectic and Kähler geometries.

Contribution

It presents a novel example of a projective surface with a symplectic structure that cannot be realized as a Kähler metric, expanding understanding of symplectic structures on complex surfaces.

Findings

Existence of a symplectic structure not compatible with any Kähler metric on a projective surface

Illustration of differences between symplectic and Kähler structures on complex surfaces

Contribution to the classification of symplectic structures on 4-manifolds

Abstract

We study symplectic structures on K\"ahler surfaces with p_g = 0. We give an example of a projective surface which admits a symplectic structure which is not compatible with any K\"ahler metric.