Some remarks on the Kronheimer-Mrowka classes of algebraic surfaces

Abstract

Define the Donaldson series of a simply connected 4-manifold by q(X) = \sum_d q_d(X)/d! Recently Kronheimer and Mroka have announced the result that the Donaldson series of so called simple 4-manifolds can be written as q(X) = e^{Q/2}\sum_{i=1}^p a_i e^{K_i} where $Q$ is the intersection form and the $K_i \in H^2(X,\Z)$ are the {\it Kronheimer-Mrowka classes}. We prove that for simple simply connected algebraic surfaces the $K_i$ are algebraic classes and that they are closely related to the canonical class $K_X$. For simple simply connected minimal surfaces of general type we prove $K_i^2 \le K_X^2$ with equality if and only if $K_i = \pm K_X$. Remark: although no gauge theory is used in this paper it should have a cross reference with the as yet non existent e-print service for low dimensional topology.