Blowing up Chern-Ricci flat balanced metrics
Elia Fusi, Federico Giusti
TL;DR
The paper addresses constructing balanced metrics with constant $s^{ch}$ on blow-ups of compact balanced Chern-Ricci flat manifolds or orbifolds with isolated singularities, extending Arezzo–Pacard gluing to the balanced non-Kähler setting. It combines a Burns–Simanca gluing on the blow-up with a deformation in the balanced class via $oldsymbol{ heta}_\varphi^{n-1}=\boldsymbol{\theta}^{n-1}+i\partial\bar{\partial}\varphi$, reducing to a fixed-point problem after ensuring invertibility of a balanced-Lichnerowicz-type operator in weighted spaces and orthogonality to $\ker F_{\omega}$. A second deformation scheme uses a globally defined $(n-2,n-2)$-form $\tilde{\Omega}$ to obtain a kernel-free linearization through the operator $F_{\omega,\Omega}$, enabling additional gluing cases, including crepant resolutions via Joyce’s ALE metrics. The paper provides a broad set of examples (Iwasawa, Nakamura, torus fibrations) illustrating applicability and contributes new balanced Chern-Ricci flat and constant $s^{ch}$ metrics within prescribed balanced classes, enriching non-Kähler canonical geometry.
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.