Dusa McDuff

The structure of rational and ruled symplectic 4-manifolds

This paper investigates the structure of compact symplectic 4-manifolds (V, w) which contain a symplectically embedded copy C of S2 with nonnegative self-intersection number. Such a pair (V, C, w) is called minimal if, in addition, the open manifold V C contains no exceptional curves (i.e., symplectically embedded 2-spheres with self-intersection -1) . We show that every such pair (V, C, w) covers a minimal pair (V, C, c() which may be obtained from V by blowing down a finite number of disjoint exceptional curves in V C. Further, the family of manifold pairs (V, C, w) under consideration is closed under blowing up and down. We next give a complete list of the possible minimal pairs. We show that V is symplectomorphic either to ? p2 with its standard form, or to an S2-bundle over a compact surface with a symplectic structure which is uniquely determined by its cohomology class. Moreover, this symplectomorphism may be chosen so that it takes C either to 2 a complex line or quadric in CP2, or, in the case when V is a bundle, to a fiber or section of the bundle. DEPARTMENT OF MATHEMATICS, STATE UNIVERSITY OF NEW YORK AT STONY BROOK, STONY BROOK, NEW YORK 11794-3651 E-mail address: dusa@math.sunysb.edu This content downloaded from 207.46.13.145 on Wed, 27 Apr 2016 05:10:07 UTC All use subject to http://about.jstor.org/terms

Symplectic manifolds with contact type boundaries

SummaryAn example of a 4-dimensional symplectic manifold with disconnected boundary of contact type is constructed. A collection of other results about symplectic manifolds with contact-type boundaries are derived using the theory ofJ-holomorphic spheres. In particular, the following theorem of Eliashberg-Floer-McDuff is proved: if a neighbourhood of the boundary of (V, ω) is symplectomorphic to a neighbourhood ofS2n−1 in standard Euclidean space, and if ω vanishes on all 2-spheres inV, thenV is diffeomorphic to the ballB2n.

Examples of symplectic structures.

SummaryIn this paper we construct symplectic forms

Blow ups and symplectic embeddings in dimension 4

\tilde \omega _k , k \geqq 0

Singularities and positivity of intersections of J-holomorphic curves

, on a compact manifold

Hofer{Zehnder capacity and length minimizing Hamiltonian paths

{\tilde Y}