Symplectic Classes on Elliptic Surfaces with positive Euler Number

Abstract

A key question for $4$-manifolds $M$ admitting symplectic structures is to determine which cohomology classes $\hat{I}\pm \in H^2(M,\mathbb{R})$ admit a symplectic representative. The collection of all such classes, the symplectic cone ${\mathcal{C}}_M$, is a basic smooth invariant of $M$. This paper describes the symplectic cone for elliptic surfaces with positive Euler number.

54 pages. Comments welcome! (This version differs from an earlier version by changing the focus from elliptic surfaces without multiple fibers to elliptic surfaces with positive Euler number.)