Cohomologous symplectic forms with different Gromov widths

TL;DR

The paper resolves McDuff-Salamon's Problem 46 in dimensions $\ge 6$ by constructing closed manifolds with cohomologous symplectic forms that have different Gromov widths, and, in some cases, different first Chern classes. The strategy combines Li-Liu's symplectic cone theory for $b_2^+=1$, Biran's 4D ball-packings, and uniruledness to bound and distinguish widths, then uses diffeomorphism classifications (Wall, Jupp) to transport symplectic data to product manifolds $X\times S^2$ and $X'\times S^2$. The core construction starts with a non-diffeomorphic pair of homeomorphic 4-manifolds $X,X'$ and builds two cohomologous forms on $X\times S^2$ whose widths differ, with a further argument (Liu-Ohta-Ono) showing the first Chern classes must differ in this setting. Consequences include explicit examples such as $(\mathbb{C}P^2\#k\overline{\mathbb{C}P^2})\times S^2$ for $k\ge2$, and the method generalizes to higher dimensions via iterative products, providing negative answers to both problems in the stated regime.

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.

Theorems & Definitions (18)

• Theorem 1.3
• Corollary 1.4
• Theorem 2.1: McDuffrationalruledMcduffimmersed
• Theorem 2.2: LiLiu01
• Theorem 2.3: biranpacking
• Corollary 2.4
• Remark 2.5
• Remark 2.6
• Definition 2.7
• Theorem 2.8: Gromov