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A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture. (English) Zbl 0974.53063

From the introduction: The Thom Conjecture asserts that any compact, embedded surface in \(\mathbb{C} P^2\) of degree \(d>0\) must have genus at least as large as the smooth algebraic curve of the same degree, namely \((d-1)(d-2)/2\). More generally, one can ask whether in any algebraic surface a smooth algebraic curve is of minimal genus in its homology class. There was one significant result in this direction. Using \(SU(2)\)-Donaldson invariants, P. Kronheimer showed in [Bull. Am. Math. Soc., New Ser. 29, 63-69 (1993; Zbl 0810.57014)] that this result is true for curves of positive self-intersection in a large class of simply connected surfaces with \(b^+_2>1\). Unfortunately, for technical reasons, this argument does not extend to cover the case of \(\mathbb{C} P^2\). It is the purpose of this paper to prove the general result that a smooth holomorphic curve of non-negative self-intersection in a compact Kähler manifold is genus minimizing.
For any closed, orientable Riemann surface \(C\) we denote its genus by \(g(C)\).
Theorem 1.1 (Generalized Thom Conjecture). Let \(X\) be a compact Kähler surface and let \(C \hookrightarrow X\) be a smooth holomorphic curve. Suppose that \(C\cdot C\geq 0\). Let \(C'\hookrightarrow X\) be a \(C^\infty\)-embedding of a smooth Riemann surface representing the same homology class as \(C\). Then \(g(C)\leq g(C')\).
In fact, there is a generalization of this result to symplectic manifolds.
Theorem 1.2. Let \(X\) be a compact symplectic four-manifold and let \(C\hookrightarrow X\) be a smooth symplectic curve with \(C\cdot C\geq 0\). (A symplectic curve is one for which the restriction of the symplectic form is everywhere non-zero.) Let \(C'\hookrightarrow X\) be a \(C^\infty\) embedding of a Riemann surface representing the same homology class as \(C\). Then \(g(C')\geq g(C)\).
The Thom Conjecture and very similar generalizations of it have been established independently by P. Kronheimer and T. Mrowka, see [Math. Res. Lett. 1, 797-808 (1994; Zbl 0851.57023)].

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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