The closure of the symplectic cone of elliptic surfaces (Q462511)

Let \(M\) be a simply-connected elliptic surface \(E(n)\) without multiple fibres, and \(F\) be the fibre class of an elliptic fibration on \(M\). The positive cone \(\mathcal{P}\) is the set of elements in \(H^{2}(M;\mathbb{R})\) whose square is positive. It contains subcones \(\mathcal{P}^A\triangleq \{\omega\in \mathcal{P}|{ } \omega\cdot A>0 \}\), for any \(A\neq0\in H^{2}(M;\mathbb{R})\); \(\mathcal{P}^{>}\triangleq \{\omega\in \mathcal{P}|{ } \omega^2>(\omega\cdot PD(F))^2 \}\); \(\mathcal{P}^{A>}=\mathcal{P}^{>}\cap \mathcal{P}^A\), and the symplectic cone \(\mathcal{C}_M\subseteq \mathcal{P}\) (i.e., cone of symplectic forms). NEWLINENEWLINENEWLINEUsing Seiberg-Witten theory [\textit{R. E. Gompf} and \textit{A. I. Stipsicz}, 4-manifolds and Kirby calculus. Providence, RI: American Mathematical Society (1999; Zbl 0933.57020); \textit{C. H. Taubes}, Math. Res. Lett. 2, No. 2, 221--238 (1995; Zbl 0854.57020)], we know that \(\mathcal{C}_M\subset \left(\mathcal{P}^{c_1(M)}\cup \mathcal{P}^{-c_1(M)}\right). \)NEWLINENEWLINEA question of T.-J. Li asked whether \(\left(\mathcal{P}^{c_1(M)}\cup \mathcal{P}^{-c_1(M)}\right)\subset \mathcal{C}_M, \) i.e., whether every class of positive square whose cup product with the first Chern class of \(M\) is non-zero can be represented by a symplectic form. The author of this paper proves the following result. For any integer \(m\geq 2\), then: {\parindent=0.5cm\begin{itemize}\item[{\(\bullet\)}] if \(M\) is the spin surface \(E(2m)\), then \(\left(\mathcal{P}^{c_1(M)}\cup \mathcal{P}^{-c_1(M)}\right)\subset \overline{\mathcal{C}}_M\); \item[{\(\bullet\)}] if \(M\) is the non-spin surface \(E(2m-1)\), then \(\left(\mathcal{P}^{c_1(M)>}\cup \mathcal{P}^{-c_1(M)>}\right)\subset \overline{\mathcal{C}}_M\),NEWLINENEWLINE\end{itemize}} where \(\overline{\mathcal{C}}_M\) is the closure of the symplectic cone in \(H^{2}(M;\mathbb{R})\).