Blow ups and symplectic embeddings in dimension 4
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).
scientific article; zbMATH DE number 4209003