Rational and ruled symplectic 4-manifolds

Summary

INTRODUCTION

This note describes the structure of compact symplectic 4-manifolds (V, ω) which contain a symplectically embedded copy C of S² with non-negative self-intersection number. (Such curves C are called “rational curves” by Gromov: see [G].) It turns out that there is a concept of minimality for symplectic 4-manifolds which mimics that for complex surfaces. Further, a minimal manifold (V, ω) which contains a rational curve C is either symplectomorphic to ℂP² with its usual Kähler structure τ, or is the total space of a “symplectic ruled surface” i.e. an S²-bundle over a Riemann surface M, with a symplectic form which is non-degenerate on the fibers. It follows that if a (possibly non-minimal) (V,ω) contains a rational curve C with C·C > 0, then (V, ω) may be blown down either to S² × S² with a product form or to (ℂP², τ), and hence is birationally equivalent to ℂP² in Guillemin and Steinberg's sense: see [GS]. (In analogy with the complex case, we will call such manifolds rational.) Moreover, if V contains a rational curve C with C·C = 0, then V may be blown down to a symplectic ruled surface. Thus, symplectic 4-manifolds which contain rational curves of non-negative self-intersection behave very much like rational or ruled complex surfaces.

It is natural to ask about the uniqueness of the symplectic structure on the manifolds under consideration: more precisely, if ω₀ and ω₁ are cohomologous symplectic forms on V which both admit rational curves of non-negative self-intersection, are they symplectomorphic?