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

TL;DR

This paper investigates the relationship between Poincaré type metrics with constant scalar curvature on the complement of a divisor in a Kähler manifold and the scalar curvature constraints on the divisor components, confirming conjectures in special cases.

Contribution

It establishes scalar curvature inequalities for Poincaré type metrics near divisor components, extending conjectures by Szekelyhidi to multiple components.

Findings

Scalar curvature of Poincaré type metrics is less than the mean scalar curvature on divisor components.

Results confirm conjectured inequalities for multiple divisor components.

Provides topological constraints related to the existence of constant scalar curvature metrics.

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.