On the notion of general type

Abstract

A smooth compact manifold M is said to have negative Yamabe invariant if all the scalar curvatures of unit-volume constant-scalar-curvature riemannian metrics on M are less than some (uniform) negative constant. If M happens to be the underlying 4-manifold of a non-singular complex-algebraic surface, Seiberg-Witten theory can be used to show that M has negative Yamabe invariant iff the surface is of general type in the sense of algebraic geometry. Based on this observation, it is suggested that one should define a 4-manifold to be of (riemannian) general type iff it has negative Yamabe invariant.

Keywords

• complex surface
• 4-manifold
• yamabe problem
• general type
• seiberg-witten theory