Canonical metrics on holomorphic Courant algebroids

Mario Garcia-Fernandez, Roberto Rubio, Carlos Shahbazi, Carl Tipler

Published 2018-03-05Version 1

Yau's solution of the Calabi Conjecture implies that every K\"ahler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\mathrm{SU}(n)$, and that these metrics are parametrized by the positive cone in $H^2(X,\mathbb{R})$. In this work we give evidence of an extension of Yau's theorem to non-K\"ahler manifolds, where $X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid. The equations that define our notion of 'best metric' are motivated by generalized geometry, and correspond to a mild generalization of the Hull-Strominger system.