Torus actions on small blow ups of CP^2

Details Torus actions on small blow ups of CP^2 Torus actions on small blow ups of CP^2

2008-09-09

arxiv.org/abs/0809.1620v1

Pacific Journal of Mathematics, vol. 244 (2010), no. 1, 133-154

Mathematics

Symplectic Geometry

Scientific paper

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.

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