Bounds on genus and geometric intersections from cylindrical end moduli spaces

Details Bounds on genus and geometric intersections from cylindrical end moduli spaces Bounds on genus and geometric intersections from cylindrical end moduli spaces

2002-02-18

arxiv.org/abs/math/0202178v3

Mathematics

Differential Geometry

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Scientific paper

In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive self-intersection in a smooth four-manifold $X$ with second positive Betti number $b_2^+(X)=1$. We study the solutions of the Seiberg-Witten equations on the cylindrical end manifold which is the complement of the surface representing the class. The result can be formulated as a form of generalized adjunction inequality. The bounds obtained depend only on the rational homology type of the manifold, and include the Thom conjecture as a special case. We generalize this approach to derive lower bounds on the number of intersection points of $n$ algebraically disjoint surfaces of positive self-intersection in manifolds with $b_2^+(X)=n$.

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