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 SUfalse(nfalse)$\mathrm{SU}(n)$, 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.

Harmonic metrics for the Hull-Strominger system and stability

“…Remark 4.3. More invariantly, in the formalism of Bott-Chern algebroids introduced in [17], the previous result can be stated as follows: for any x ∈ B there exists a Bott-Chern algebroid with underlying bundle P x and fixed Lie algebra bundle determined by the pairing (r 0 , r 1 ), (r ′ 0 , r ′ 1 ) := −αr 0 r ′ 0 + αr 1 r ′ 1 , such that any small Bott-Chern algebroid deformation admits a solution to the Hull-Strominger system and a harmonic metric for this solution.…”

“…We start by recalling some background on the Hull-Strominger system, including the recent development in [15] relating solutions to these equations to indefinite Hermitian-Einstein metrics on holomorphic Courant algebroids. We will use an abstract definition of the Hull-Strominger system, as considered in [17,Definition 2.4], which is valid in arbitrary dimensions. Our construction requires that the connection ∇ in the tangent bundle in the original formulation of the system is Hermitian-Yang-Mills, and hence in our discussion we will implicitly assume this condition (see Remark 2.2).…”

“…The holomorphic structure is then given by the Dolbeault operator 17,18]). The bundle Q = T 1,0 ⊕ ad P ⊕ T * 1,0 endowed with the Dolbeault operator (2.6) defines a holomorphic extension of the holomorphic Atiyah algebroid A P of P by the holomorphic cotangent bundle…”

Futaki Invariants and Yau's Conjecture on the Hull-Strominger system

“…The existence and uniqueness problem for the Hull-Strominger is currently widely open. The present work is motivated by a conjecture about the existence of solutions by S.-T. Yau [52], which provides an important test for the general methods recently developed in [32,46]. In order to understand this question, let us first explain three types of necessary conditions for (1.1).…”

“…In order to do this, we will exploit the special features of the solutions of the Hull-Strominger system with the ansatz (1.4). More precisely, we will be able to use generalized geometry and to apply the theory of metrics on holomorphic string algebroids introduced in [32,34].…”