The closure of the symplectic cone of elliptic surfaces

TL;DR

This paper investigates the structure of the symplectic cone for elliptic surfaces, proving a conjecture for spin cases using inflation and diffeomorphisms, and identifying obstructions in non-spin cases.

Contribution

It proves the conjecture for the closure of the symplectic cone of spin elliptic surfaces E(2m) using inflation and diffeomorphism actions.

Findings

Confirmed the conjecture for spin elliptic surfaces E(2m).

Identified obstructions in the non-spin case.

Applied inflation techniques and diffeomorphism actions.

Abstract

The symplectic cone of a closed oriented 4-manifold is the set of cohomology classes represented by symplectic forms. A well-known conjecture describes this cone for every minimal Kaehler surface. We consider the case of the elliptic surfaces E(n) and focus on a slightly weaker conjecture for the closure of the symplectic cone. We prove this conjecture in the case of the spin surfaces E(2m) using inflation and the action of self-diffeomorphisms of the elliptic surface. An additional obstruction appears in the non-spin case.