Blowing up Chern-Ricci flat balanced metrics

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. Along the way, we study two Lichnerowicz-type operators originating from complex connections and investigate the relation between their kernel and holomorphic vector fields, with the aim of discussing the general constant Chern scalar curvature balanced case. Ultimately, we provide a variation of the main Theorem assuming the existence of a special $(n-2,n-2)$-form and we present several classes of examples in which all our results can be applied.