Journal of Differential Geometry

Bounds on genus and geometric intersections from cylindrical end moduli spaces

Sašo Strle

Full-text: Open access


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

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

First available in Project Euclid: 11 June 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]


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.

Export citation