Metrics of Poincar\'e type with constant scalar curvature: a topological constraint

Abstract

Let D a divisor with simple normal crossings in a Kahler manifold X. The purpose of this short note is to show that the existence of a Poincare type metric with constant scalar curvature in on the complement of D implies for any component of the divisor that the scalar curvature of Poincare type metric outside of D is less than the mean scalar curvature attached to the component. We also explain how those results were already conjectured by G. Szekelyhidi when D is reduced to one component.