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

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.