Comparing Kahler cone and symplectic cone of one-point blowup of Enriques surface

TL;DR

This paper demonstrates that the one-point blowup of an Enriques surface admits non-Kähler symplectic forms, highlighting differences between Kahler and symplectic cones through algebraic geometric invariants.

Contribution

It provides the first example of a one-point blowup of an Enriques surface with non-Kähler symplectic forms, expanding understanding of Kahler and symplectic cone distinctions.

Findings

Existence of non-Kähler symplectic forms on the blowup

Quantitative comparison of algebraic invariants

Distinction between Kahler and symplectic cones

Abstract

We follow the study by Cascini-Panov on symplectic generic complex structures on Kahler surfaces with $p_g=0$, a question proposed by Tian-Jun Li, by demonstrating that the one-point blowup of an Enriques surface admits non-Kahler symplectic forms. This phenomenon relies on the abundance of elliptic fibrations on Enriques surfaces, characterized by various invariants from algebraic geometry. We also provide a quantitative comparison of these invariants to further give a detailed examination of the distinction between Kahler cone and symplectic cone.