Canonical metrics on holomorphic Courant algebroids

Abstract

The solution of the Calabi Conjecture by Yau implies that every Kähler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\text{SU}(n)$, and that these metrics are parametrized by the positive cone in $H^{1,1}(X,\mathbb{R})$. In this work we give evidence of an extension of Yau's theorem to non-Kähler manifolds, where $X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid $Q$ 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 $H^{1,1}(X,\mathbb{R})$ is played by an affine space of 'Aeppli classes' naturally associated to $Q$ via Bott-Chern secondary characteristic classes.

55 pages, Lemma 2.2 fixed, presentation improved, appendix added, to appear in the Proceedings of the London Mathematical Society

References in corpus (9)

• Autour de la cohomologie de Bott-Chern
• Non-Kaehler Heterotic String Compactifications with non-zero fluxes and constant dilaton
• Generalized Kahler geometry
• Fu-Yau Hessian Equations
• Holomorphic string algebroids
• Gauge theory for string algebroids
• New curvature flows in complex geometry
• Generalized Calabi-Gray Geometry and Heterotic Superstrings
• The Fu-Yau equation in higher dimensions

Cited by in corpus (11)

• Finite deformations from a heterotic superpotential: holomorphic Chern--Simons and an $L_{\infty }$ algebra
• Gauge theory for string algebroids
• Heterotic Quantum Cohomology
• Generalising G${}_{\text{2}}$ geometry: involutivity, moment maps and moduli
• Geometric Partial Differential Equations from Unified String Theories
• The Heterotic-Ricci flow and its three-dimensional solitons
• Special Lagrangian cycles and Calabi-Yau transitions
• New curvature flows in complex geometry
• A flow of conformally balanced metrics with Kähler fixed points
• Balanced and Aeppli Parameters for the Heterotic Moduli
• Heterotic solitons on four-manifolds