Canonical metrics on holomorphic Courant algebroids

Abstract

The solution of the Calabi Conjecture by Yau implies that every Kähler Calabi–Yau manifold X $X$ admits a metric with holonomy contained in SU ( n ) $\mathrm{SU}(n)$ , and that these metrics are parameterized by the positive cone in H 1 , 1 ( X , R ) $H^{1,1}(X,\mathbb {R})$ . In this work, we give evidence of an extension of Yau's theorem to non-Kähler manifolds, where X $X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid Q $Q$ of Bott–Chern type. The equations that define our notion of best metric correspond to a mild generalization of the Hull–Strominger system, whereas the role of H 1 , 1 ( X , R ) $H^{1,1}(X,\mathbb {R})$ is played by an affine space of ‘Aeppli classes’ naturally associated to Q $Q$ via Bott–Chern secondary characteristic classes.