Torus actions on small blow ups of CP^2

TL;DR

This paper investigates when symplectic blow-ups of the complex projective plane admit effective torus actions, establishing thresholds based on the number of blow-ups and their sizes, and linking geometric and combinatorial methods.

Contribution

It determines precise conditions under which these manifolds admit effective torus actions, combining symplectic geometry, combinatorics, and J-holomorphic curve techniques.

Findings

Effective torus actions exist for up to 3 blow-ups.

No such actions for 4 or more blow-ups under certain size constraints.

Links between symplectic geometry and combinatorial moment map images.

Abstract

A manifold obtained by k simultaneous symplectic blow-ups of CP^2 of equal sizes epsilon (where the size of CP^1 in CP^2 is one) admits an effective two-dimensional torus action if k <= 3. We show that it does not admit such an action if k >=4 and epsilon <= 1/(3k 2^{2k}). For the proof, we correspond between the geometry of a symplectic toric four-manifold and the combinatorics of its moment map image. We also use techniques from the theory of J-holomorphic curves.