Strictly regular and cscK metrics

TL;DR

This paper proves the existence of complex manifolds with strictly partially regular and cscK metrics, allowing scalar curvature to be zero, positive, or negative in higher dimensions, advancing understanding of Kähler geometry.

Contribution

It demonstrates the existence of manifolds with strictly partially regular cscK metrics in all dimensions n≥2, with scalar curvature flexibility for n≥3.

Findings

Existence of such manifolds in all dimensions n≥2

Construction of metrics with scalar curvature zero, positive, or negative for n≥3

Advancement in understanding of Kähler metrics with partial regularity

Abstract

A Kaehler metric $g$ with integral Kaehler form is said to be partially regular if the partial Bergman kernel associated to mg is a positive constant for all integer m sufficiently large. The aim of this paper is to prove that for all n\geq 2 there exists an n-complex dimensional manifold equipped with strictly partially regular and cscK metric g. Further, for n\geq 3, the (constant) scalar curvature of g can be chosen to be zero, positive or negative.