Torus actions on small blowups of ℂℙ²

Liat Kessler

Vol. 244 (2010), No. 1, 133–154

DOI: 10.2140/pjm.2010.244.133

Abstract

A manifold obtained by k simultaneous symplectic blowups of ℂℙ² of equal sizes 𝜖 (where the size of ℂℙ¹ ⊂ ℂℙ² 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 𝜖 ≤ 1∕(3k2^2k). For the proof, we show a correspondence 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.

Keywords

torus action, symplectic manifold, dimension four, symplectic blowup, J-holomorphic curve, moment map, Delzant polygon

Mathematical Subject Classification 2000

Primary: 53D20, 53D35, 53D45

Secondary: 57S15

Milestones

Received: 8 September 2008

Accepted: 25 March 2009

Published: 17 November 2009