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

We follow the study by Cascini–Panov on symplectic generic complex structures on Kähler surfaces with $p_g=0$, 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.