Comparing Kähler 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 Kähler surfaces with , a question proposed by Li, by demonstrating that the one‐point blowup of an Enriques surface admits non‐Kähler symplectic forms. This phenomenon relies on the abundance of elliptic fibrations on Enriques surfaces, characterized by various invariants from the algebraic geometry. We also provide a quantitative comparison of these invariants to further give a detailed examination of the distinction between Kähler cone and symplectic cone.