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Bounds on genus and geometric intersections from cylindrical end moduli spaces

Saso Strle
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Abstract

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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Saso Strle. 2003-01-22. Bounds on genus and geometric intersections from cylindrical end moduli spaces. https://arxiv.org/abs/math/0202178

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