Canonical metrics on holomorphic Courant algebroids

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 H1,1(X,R)$H^{1,1}(X,\mathbb {R})$. In this work, we give evidence of an e...

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Bibliographic Details
Published in:Proceedings of the London Mathematical Society 2022-09, Vol.125 (3), p.700-758
Main Authors: Garcia‐Fernandez, Mario, Rubio, Roberto, Shahbazi, Carlos, Tipler, Carl
Format: Article
Language:English
Subjects:
Differential Geometry
Mathematics
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Summary: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 H1,1(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 H1,1(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.
ISSN:0024-6115
1460-244X
DOI:10.1112/plms.12468