Blow ups and symplectic embeddings in dimension 4. (English) Zbl 0731.53035

The purpose of this paper is to prove the following main results:
(1) Let \(B_\lambda\) of the closed ball of radius \(\lambda\) in \(\mathbb R^4\). Then the space of symplectic embeddings of \(B_\lambda\) into the open unit ball in \(\mathbb R^4\) is connected.
(2) Blow up the complex projective plane \(\mathbb{CP}^2\) at a point \(x\) and denote the resultant manifold by \(\mathbb{CP}^2\#\overline{\mathbb{CP}}^2\). Let \(S_L\) be the image in \(\mathbb{CP}^2\#\overline{\mathbb{CP}}^2\) of a complex line in \(\mathbb{CP}^2\) which does not meet \(x\). Then a symplectic form on \(\mathbb{CP}^2\#\overline{\mathbb{CP}}^2\) which restricts to a nonvanishing form on \(S_L\) is characterized by its cohomology class up to a diffeomorphism which preserves \(S_L\).
By a blowing up process he proves that (1) is equivalent to the uniqueness up to diffeomorphisms of some symplectic structures on \(\mathbb{CP}^2\#\overline{\mathbb{CP}}^2\). Further, Gromov’s theory of pseudoholomorphic curves leads us to prove (2).


53D05 Symplectic manifolds (general theory)
57R17 Symplectic and contact topology in high or arbitrary dimension
57R40 Embeddings in differential topology
32Q65 Pseudoholomorphic curves
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