The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane

TL;DR

This paper computes the rational homotopy Lie algebra of symplectomorphism groups for 3-point blow-ups of the projective plane, revealing cases of infinite dimensionality and linking topology to toric structures.

Contribution

It provides the first explicit computation of the rational homotopy Lie algebra for these symplectomorphism groups, showing dependence on blow-up sizes and topology generation by toric structures.

Findings

Rational homotopy Lie algebra can be infinite dimensional depending on blow-up sizes.

Topology of symplectomorphism groups is generated by toric structures.

Method involves studying compatible almost complex structures and inflation techniques.

Abstract

By a result of Kedra and Pinsonnault, we know that the topology of groups of symplectomorphisms of symplectic 4-manifolds is complicated in general. However, in all known (very specific) examples, the rational cohomology rings of symplectomorphism groups are finitely generated. In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane (with an arbitrary symplectic form) and show that in some cases, depending on the sizes of the blow-ups, it is infinite dimensional. Moreover, we explain how the topology is generated by the toric structures one can put on the manifold. Our method involve the study of the space of almost complex structures compatible with the symplectic structure and it depends on the inflation technique of Lalonde-McDuff.