The symplectic Thom conjecture

Peter Ozsváth and Zoltán Szabó

Review from Zentralblatt MATH:

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.

Reviewed by Alberto Parmeggiani

Keywords: symplectic manifold; Spin$_{\Bbb C}$ structure; Seiberg-Witten invariants; symplectic Thom conjecture

Classification (MSC2000): 53D35 57R57 53C55

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