## Journal of Differential Geometry

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

Sašo Strle

#### Abstract

In this paper we present a way of computing a lower bound for the 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 $b^{+}_{2}(X) = n$.

#### Article information

Source
J. Differential Geom., Volume 65, Number 3 (2003), 469-511.

Dates
First available in Project Euclid: 11 June 2015

https://projecteuclid.org/euclid.jdg/1434052757

Digital Object Identifier
doi:10.4310/jdg/1434052757

Mathematical Reviews number (MathSciNet)
MR2064429

Zentralblatt MATH identifier
1056.57022

#### Citation

Strle, Sašo. Bounds on genus and geometric intersections from cylindrical end moduli spaces. J. Differential Geom. 65 (2003), no. 3, 469--511. doi:10.4310/jdg/1434052757. https://projecteuclid.org/euclid.jdg/1434052757