A note on collapse, entropy, and vanishing of the Yamabe invariant of symplectic 4-manifolds

TL;DR

This paper investigates the geometric invariants of symplectic 4-manifolds, demonstrating their vanishing under certain conditions and extending known results relating Kodaira dimension and Yamabe invariant.

Contribution

It applies -structures and recent techniques to compute invariants and shows their vanishing for symplectic 4-manifolds with arbitrary fundamental groups, extending LeBrun's results.

Findings

Invariants vanish for symplectic 4-manifolds with any finitely presented fundamental group.

Extended LeBrun's relation between Kodaira dimension and Yamabe invariant to the symplectic setting.

Computed minimal entropy, volume, and Yamabe invariant using advanced geometric techniques.

Abstract

We make use of $\mathcal{F}$-structures and technology developed by Paternain - Petean to compute minimal entropy, minimal volume, and Yamabe invariant of symplectic 4-manifolds, as well as to study their collapse with sectional curvature bounded from below. \`A la Gompf, we show that these invariants vanish on symplectic 4-manifolds that realize any given finitely presented group as their fundamental group. We extend to the symplectic realm a result of LeBrun which relates the Kodaira dimension with the Yamabe invariant of compact complex surfaces.