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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).

MSC:

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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