Abstract
Let (X, ω) be a symplectic rational surface. We study the space of tamed almost complex structures Jω using a fine decomposition via smooth rational curves and a relative version of the infinite dimensional Alexander- Pontrjagin duality. This decomposition provides new understandings of both the variation and the stability of the symplectomorphism group Symp(X, ω) when deforming ω In particular, we compute the rank of π1(Symp(X, ω)) with X(X) ≥ 7 in terms of the number Nω of (-2)-symplectic sphere classes.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 561-606 |
| Number of pages | 46 |
| Journal | Pacific Journal of Mathematics |
| Volume | 304 |
| Issue number | 2 |
| DOIs | |
| State | Published - 2020 |
Bibliographical note
Funding Information:We appreciate useful discussions with Silvia Anjos, Martin Pinsonnault, Weiwei Wu, Weiyi Zhang. For CP2#4CP2, when [ω] is in the face M4OBC, Anjos and Eden [2019] computed the rational homotopy groups using the toric method (see Remark 4.10). Results in Section 3A overlap with results in Section 4.1 in [Zhang 2017], which are in a slightly different context. We thank the referees for their detailed and constructive comments, which greatly improved the exposition of this article. The research is supported by NSF grant DMS-1611680.
Publisher Copyright:
© 2020 Mathematical Sciences Publishers.
Keywords
- Almost complex structure
- Lagrangian root system
- Rational symplectic manifold
- Symplectomorphism group