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The symplectic Thom conjecture. (English) Zbl 0967.53052

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.

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
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