Embedding more than 8 symplectic balls in ℂP²

2026/05/22 by Sílvia Anjos, Jarek Kędra, Martin Pinsonnault · 1 voice
Mathematics · #math.SG

paper · pdf

arxiv published 2026/05/22 · arxiv updated 2026/06/29

Abstract

We prove that the space of symplectic embeddings of n≥ 1 standard balls into the standard complex projective plane ℂP², normalized so that a line has symplectic area 1, is homotopy equivalent to the configuration space of n points in ℂP², provided that the sum of the ball capacities is strictly less than 1. Our techniques further suggest that, for n=9, there are infinitely many homotopy types of spaces of symplectic ball embeddings, depending on the ball capacities. Moreover, for each n≥ 5, we exhibit capacities for which the embedding spaces are not simply connected, in contrast with the case n ≤ 4. As an application, we show that, for n≥ 9 equal balls of capacity c<1/n, the symplectomorphism group of the blow-up has the homotopy type of the stabilizer of n distinct points in ℂP².

Citations

• Configurations of 10 points and their incidence varieties
• J-tamed inflation via tame to compatible deformations
• Embeddings of symplectic balls into the complex projective plane
• Centralizers of Hamiltonian circle actions on rational ruled surfaces
• The curve cone of almost complex 4-manifolds
• Counting toric actions on symplectic four-manifolds
• Min-type Morse theory for configuration spaces of hard spheres
• Maximal compact tori in the Hamiltonian groups of 4-dimensional symplectic manifolds
• The topology of the space of symplectic balls in rational 4-manifolds
• Symplectomorphism groups and almost complex structures
• From symplectic deformation to isotopy

Discussions

• S\'ilvia Anjos, Jarek K\k{e}dra, Martin Pinsonnault: Embedding more than 8 symplectic balls in $\mathbb{C}\mathrm{P}^2$ https://arxiv.org/abs/2605.24161 https://arxiv.org/pdf/2605.24161 https://arxiv. [bsky, 0 points, 0 comments]

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