The symplectic Thom conjecture. (English) Zbl 0967.53052

Ozsváth, Peter; Szabó, Zoltán

Ann. Math. (2) 151, No. 1, 93-124 (2000).

In this paper, the authors prove the symplectic Thom conjecture in its full generality:
Theorem. An embedded symplectic surface in a closed, symplectic four-manifold is genus-minimizing in its homology class.
As a corollary, they also get the following result in the Kähler case.
Corollary. An embedded holomorphic curve in a Kähler surface is genus-minimizing in its homology class.
The theorem follows from a relation among Seiberg-Witten invariants, that holds in the case of embedded surfaces in four-manifolds whose self-intersection number is negative. Such a relation also yields a general adjunction inequality for embedded surface of negative self-intersection in four-manifolds.

Reviewer: Alberto Parmeggiani (Bologna)

MSC:

53D35	Global theory of symplectic and contact manifolds
57R57	Applications of global analysis to structures on manifolds
53C55	Global differential geometry of Hermitian and Kählerian manifolds

Keywords:

symplectic manifold; Spin\(_{\mathbb C}\) structure; Seiberg-Witten invariants; symplectic Thom conjecture