No Arabic abstract
We study the (standard) cohomology $H^bullet_{st}(E)$ of a Courant algebroid $E$. We prove that if $E$ is transitive, the standard cohomology coincides with the naive cohomology $H_{naive}^bullet(E)$ as conjectured by Stienon and Xu. For a general Courant algebroid we define a spectral sequence converging to its standard cohomology. If $E$ is with split base, we prove that there exists a natural transgression homomorphism $T_3$ (with image in $H^3_{naive}(E)$) which, together with the naive cohomology, gives all $H^bullet_{st}(E)$. For generalized exact Courant algebroids, we give an explicit formula for $T_3$ depending only on the v{S}evera characteristic clas of $E$.
Let $A Rightarrow M$ be a Lie algebroid. In this short note, we prove that a pull-back of $A$ along a fibration with homologically $k$-connected fibers, shares the same deformation cohomology of $A$ up to degree $k$.
The solution of the Calabi Conjecture by Yau implies that every Kahler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $operatorname{SU}(n)$, 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 Yaus theorem to non-Kahler 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 the second cohomology is played by an affine space of `Aeppli classes naturally associated to $Q$ via secondary holomorphic characteristic classes introduced by Donaldson
We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we introduce the K-curvature and K-torsion of a Courant algebroid connection. These are conventional tensors on the body. Finally, we compute their Ricci and scalar curvature.
VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation groupoid/algebroid.