Portal de Periódicos da CAPES

SW ⇒ Gr: From the Seiberg-Witten equations to pseudo-holomorphic curves

1996; American Mathematical Society; Volume: 9; Issue: 3 Linguagem: Inglês

10.1090/s0894-0347-96-00211-1

1088-6834

Clifford Henry Taubes,

Geometry and complex manifolds

TAUBES(Note that SW as a map from Spin is diffeomorphism invariant, but that the identification in (0.2) is not.The effect of a diffeomorphism on (0.3) depends on the behavior of c 1 (K).)A submanifold Σ of a symplectic manifold is called symplectic when the restriction of the symplectic form to T Σ is non-degenerate.With the preceding understood, the following is a corollary to Theorem 1.3 (with Proposition 7.1): Theorem 0.1.Let X be a compact, oriented, 4-dimensional symplectic manifold with b + 2 > 1.Let e = 0 ∈ H 2 (X; Z) be a class with SW(e) = 0. Then the Poincaré dual to e is represented by the fundamental class of an embedded, symplectic curve.And, this curve has genus g = 1 + e • e.In the preceding, • denotes the cup product pairing from H 2 (X; Z) to Z. Thus, the number e • e is the value of the cup of e with itself on X's fundamental class.(Remark that Simon Donaldson [Do] announced last year a theorem to the effect that classes e ∈ H 2 (X; Z) with very large positive pairing with ω have embedded, symplectic representatives.)Various other consequences of Theorem 1.3 and Proposition 7.1 were discussed in [T1].In particular, consider: Theorem 0.2.Let X be a compact, oriented, 4-manifold with b + 2 > 1 and with a symplectic form ω. Then (1) The Poincaré dual of c 1 (K) is represented by the fundamental class of an embedded, symplectic curve.(2) Let e ∈ H 2 (X; Z) denote a homology class which is represented by an embedded sphere with self-intersection number -1.Then e is represented by a symplectically embedded 2-sphere and c 1 (K), e = ±1.(3) If c 1 (K) has negative square, then X can be blown down along a symplectic sphere of self-intersection -1.(4) Suppose that X cannot be blown down along a symplectic sphere of selfintersection -1.Then the signature of the intersection form of X is no smaller than -4 3 (1b 1 ) -2 3 b 2 .(The b i 's are the Betti-numbers of X.) (5) If c 1 (K) has square zero and X has no symplectically embedded 2-spheres with self-intersection -1, then c 1 (K) is Poincaré dual to a disjoint union of embedded, symplectic tori with zero self-intersection number.In fact, any class in H 2 (X; Z) with non-zero Seiberg-Witten invariant is represented by disjoint, symplectically embedded tori with square zero.(6) Symplectic manifolds have "simple type" in that only the dimension zero Seiberg-Witten invariants are non-zero.That is, SW(e) = 0 if c 1 (K)•e-e•e = 0.Note that Assertion (2) above is a refinement due to Daniel Ruberman of a somewhat weaker assertion in [T1].Also, Assertion (6) can be proved using the afore-mentioned announced results of Donaldson.Note that symplectic submanifolds are closely related to pseudo-holomorphic submanifolds.The definition of the latter requires the introduction of an almost complex structure J for X. (This is an endomorphism whose square is minus the identity.)A submanifold Σ is called pseudo-holomorphic (after Gromov [Gr]) when J maps T Σ to itself.An almost complex structure is said to be ω-compatible when