Special non-Kähler metrics on resolutions

Given a mildly singular complex varieties admitting certain special singular hermitian metrics, e.g., Calabi-Yau metrics, it is a natural problem to study the existence of similar metrics on resolutions of such spaces which should degenerate back to the singular metric. However, it is often the case that on resolutions we cannot hope to solve exactely the same equation due to some “cohomological” constraint (e.g., the resolution is not longer Kähler). Somewhat motivated by the Hull-Strominger system, in this talk I will describe aspects of this problem for constructions of Chern-Ricci flat balanced metrics across such resolutions via gluing techniques. The results are based on joint works with Federico Giusti.