Cohomologous symplectic forms with different Gromov widths

TL;DR

This paper demonstrates the existence of closed manifolds of dimension six or higher with cohomologous symplectic forms that have different Gromov widths, using advanced symplectic geometry techniques.

Contribution

It provides explicit examples of cohomologous symplectic forms with distinct Gromov widths and shows they can have different first Chern classes, addressing open questions.

Findings

Existence of cohomologous symplectic forms with different Gromov widths in dimension ≥6.

Cohomologous forms can have different first Chern classes.

Application of Li-Liu's theorem and Biran's ball packing theorem in the analysis.

Abstract

We study McDuff-Salamon's Problem 46 by showing that there exist closed manifolds of dimension $\geq 6$ admitting cohomologous symplectic forms with different Gromov widths. The examples are motivated by Ruan's early example of deformation inequivalent symplectic forms in dimension $6$ distinguished by Gromov-Witten invariants. To find cohomologous symplectic forms and compare their Gromov width, we make use of Li-Liu's theorem of symplectic cone for manifolds with $b_2^+=1$ and Biran's ball packing theorem in dimension $4$. Along the way, we also show that these cohomologous symplectic forms can have distinct first Chern classes, which answers another question by Salamon.