Some remarks on the Kronheimer-Mrowka classes of algebraic surfaces

TL;DR

This paper proves that Kronheimer-Mrowka classes of simple algebraic surfaces are algebraic and closely related to the canonical class, establishing inequalities for minimal surfaces of general type without using gauge theory.

Contribution

It demonstrates that for simple simply connected algebraic surfaces, the Kronheimer-Mrowka classes are algebraic and linked to the canonical class, with specific inequalities for surfaces of general type.

Findings

Kronheimer-Mrowka classes are algebraic for simple algebraic surfaces.

For minimal surfaces of general type, $K_i^2 \,\le\, K_X^2$ with equality iff $K_i = \pm K_X$.

No gauge theory is used in the proofs.

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.