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Symplectic packings and algebraic geometry. (English) Zbl 0833.53028

This article brings substantial progress to the problem of symplectic packing: Can the volume of a symplectic manifold \(M\) be completely exhausted with embeddings of \(k\) disjoint standard balls of fixed radius. The authors use their previously established relation between the packing problem and symplectic blowing up and combine recent results on symplectic blowing up/down, symplectic branched covering and pseudoholomorphic curves to obtain their results: 1. For any positive integer \(p\) there is a full filling of the ball \(B^{2n}\) by \(k= p^n\) spheres (i.e. for infinitely many values of \(k\)). 2. For \(k\leq 9\) a complete exact solution for \(M= B^4\) is given. 3. For \(k\geq 10\) full fillings would follow from a conjecture of Nagata. 4. Several special results, e.g.: \(B^{2m}\) admits full filling of \(k^m\) standard equal symplectic balls, \(m\)-products of standard symplectic 2-spheres or 2- disks admit full fillings by \(m! k^m\) standard symplectic balls, etc.
Reviewer: C.Günther (Libby)

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R19 Algebraic topology on manifolds and differential topology
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 0833.53029
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References:

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