Blowing up Chern-Ricci flat balanced metrics

TL;DR

This paper extends the construction of constant Chern scalar curvature balanced metrics to blow-ups and crepant resolutions of Chern-Ricci flat orbifolds, providing new examples and methods in complex geometry.

Contribution

It introduces a method to produce constant Chern scalar curvature balanced metrics on blow-ups and crepant resolutions of orbifolds, expanding the class of known solutions.

Findings

Existence of constant Chern scalar curvature balanced metrics on blow-ups.

Existence of Chern-Ricci flat balanced metrics on crepant resolutions.

Application to several classes of complex orbifolds.

Abstract

Given a compact Chern-Ricci flat balanced orbifold, we show that its blow-up at a finite family of smooth points admits constant Chern scalar curvature balanced metrics, extending Arezzo-Pacard's construction to the balanced setting. Moreover, if the orbifold has isolated singularities and admits crepant resolutions, we show that they always carry Chern-Ricci flat balanced metrics, without any further hypothesis. In addition, we discuss the general constant Chern scalar curvature balanced case and discuss another version of the main Theorem assuming the existence of a special (n-2, n-2)-form. We also present several classes of examples in which our results can be applied.