We generalize to nearly Kähler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly Kähler $6$-manifolds that were established by Verbitsky. In particular, for a compact nearly Kähler manifold of any dimension, the (appropriately defined) Hodge numbers are related to the Betti numbers in the same way as on a compact Kähler manifold. In the $6$-dimensional case, Verbitsky was able to say slightly more using the induced $\mathrm{SU}(3)$ structure. We discuss potential extensions of this to twistor spaces over positive scalar curvature quaternionic-Kähler manifolds, which are a particular class of $(4n+2)$-dimensional nearly Kähler manifolds equipped with a special $\mathrm{SU}(n) \! \cdot \! \mathrm{U}(1)$ structure.
We prove uniqueness of a decomposition of 1 into indecomposable Hermitian idem potents in an order of a finite-dimensional Q-algebra with positive involution, by generalising a resul of Eichler on unique decomposition of lattices. We use this result to prove that polarised abelian varieties over any field admit a unique decomposition into indecomposable polarised abelian subva ricties, a result previously shown by Debarre over algebraically closed fields and Serre for arbitrary fields with different methods. We prove that an analogous uniqueness result holds true for arbitrary polarised integral Hodge structures, and derive a consequence for their automorphism groups.
In \cite{TY}, we consider the minimal paths in the path complex associated to a digraph $G$ under the strongly regular condition. In this paper, first, based on our new definitions of admissible pair and admissible relations among the set of minimal paths, we will give the definitions of cellular chain complex associated to $G$ and prove the well-definedness. Then we will study several properties of such cellular homologies. Finally, we give several interesting examples as well as some important observations.
In this paper we construct singular solutions to the critical Dirac equation on spheres.More precisely, first we construct solutions admitting two points singularities that we call Delaunay-type solutions because of their similarities with the Delaunay solutions constructed for the singularYamabe problem in [31, 34]. Then we construct another kind of singular solutions admitting a greatcircle as a singular set. These solutions are the building blocks for singular solutions on a generalSpin manifold
