A Kummer construction for Chern-Ricci flat balanced manifolds (Q6624135)

The study of the existence of special Hermitian metrics, such as Kähler-Einstein metrics, is an important topic in complex geometry. From the cohomological point of view, among the most studied metrics are the balanced metrics, which have many interesting properties and play a central role in open problems, such as the conjecture of \textit{A. Fino} and \textit{L. Vezzoni} [J. Geom. Phys. 91, 40--53 (2015; Zbl 1318.53049)] and the Gauduchon conjecture solved in [\textit{G. Székelyhidi} et al., Acta Math. 219, No. 1, 181--211 (2017; Zbl 1396.32010)].\N\NIn the present paper, via a gluing construction using Joyce's ALE metrics on the bubble (cf. [\textit{D. D. Joyce}, Compact manifolds with special holonomy. Oxford: Oxford University Press (2000; Zbl 1027.53052)]), the authors study the existence of Chern-Ricci flat balanced metrics on the crepant resolutions of certain non-Kähler Calabi-Yau singular manifolds endowed with a singular Chern-Ricci flat balanced metric. In particular, the authors consider compact orbifolds whose singular set consists of isolated singularities admitting crepant resolutions, and are endowed with a singular Chern-Ricci flat balanced metric. The interest in Chern-Ricci flat balanced metrics comes from Calabi-Yau geometry and is related to the problem of solving the Hull-Strominger system, i.e., the conformally balanced equation, coming from superstring theory. Apart from its physical significance, the problem generalizes the Calabi-Yau condition to the non-Kähler framework and is of great geometrical interest.\N\NThe proof method in this paper consists of two main steps. The first step is the gluing of the singular Chern-Ricci flat balanced metric with the rescaled Joyce's ALE metrics, and the second is an implicit function theorem deformation argument, where the deformation is a balanced deformation which was given in [\textit{J. Fu} et al., J. Differ. Geom. 90, No. 1, 81--129 (2012; Zbl 1264.32020)]. The authors' method has a complication hidden in the asymptotic behaviour of the standard Calabi-Yau metric on the small resolution of the standard conifold, but the authors give a partial result in Theorem 1.2 of the present paper.\N\NThe first result of the present paper consists in the construction of a balanced metric on the crepant resolution together with the construction of a global holomorphic volume form. By using the gluing construction, the authors construct a pre-gluing metric obtained from a rough cut-off procedure that provides an approximate solution to the problem, and from a perturbative argument to obtain a true solution. This gluing procedure shows that there is a canonical choice of balanced metric on the blow-up at a point of a balanced manifold, since any balanced metric can be glued to the Burns-Simanca metric while preserving the balanced condition. As in the paper, it is interesting to ask whether or not the construction by the gluing process can be adapted to the case of ordinary double points on threefolds in order to fit the authors' result into the conifold transition framework. The authors point out the problems that unfortunately arise when trying to repeat the deformation argument in this case.\N\NThe authors' result in this paper gives a fundamental step towards the solution of the problem proposed by \textit{M. Becker} et al. [Adv. Theor. Math. Phys. 13, No. 6, 1815--1845 (2009; Zbl 1200.81120)]. As the authors mention in the present paper, it would be interesting to know whether or not new non-Kähler solutions of the Hull-Strominger system can be constructed by a gluing process.