The Lax formulation of the hyper-Hermiticity condition in four dimensions is used to derive a potential that generalises Plebanski's second heavenly equation for hyper-Kahler 4-manifolds. A class of examples of hyper-Hermitian metrics which depend on two arbitrary functions of two complex variables is given. The twistor theory of four-dimensional hyper-Hermitian manifolds is formulated as a combination of the Nonlinear Graviton Construction with the Ward transform for anti-self-dual Maxwell fields.
In this paper, we extend a result of Gao Chen regarding the solvability of the twisted deformed Hermitian Yang-Mills equations on compact Kähler manifolds to allow for the twisting function to be non-constant and slightly negative in all dimensions. Using this result, we prove that the twisted supercritical dHYM equation on compact, projective manifolds can be solved provided certain numerical conditions are satisfied.
In this paper we establish partial structure results on the geometry of compact Hermitian manifolds of semipositive Griffiths curvature. We show that after appropriate arbitrary small deformation of the initial metric, the null spaces of the Chern-Ricci two-form generate a holomorphic, integrable distribution. This distribution induces an isometric, holomorphic, almost free action of a complex Lie group on the universal cover of the manifold. Our proof combines the strong maximum principle for the Hermitian Curvature Flow (HCF), new results on the interplay of the HCF and the torsion-twisted c…
In this paper, we study the semi-stable twisted holomorphic vector bundles over compact Gauduchon manifolds. By using Uhlenbeck--Yau's continuity method, we show that the existence of approximate Hermitian--Einstein structure and the semi-stability of twisted holomorphic vector bundles are equivalent over compact Gauduchon manifolds. As its application, we show that the Bogomolov type inequality is also valid for a semi-stable twisted holomorphic vector bundle.
The object of investigations are almost hypercomplex structures with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. There are studied both the basic classes of a classification of 4-dimensional indecomposable real Lie algebras depending on two parameters. Some geometric characteristics of the respective almost hypercomplex manifolds with Hermitian-Norden metrics are obtained.
There are studied Lie groups considered as almost hypercomplex Hermitian-Norden manifolds, which are integrable and have the lowest dimension four. It is established a correspondence of the derived Lie algebras of types of invariant hypercomplex structures and the explicit matrix representation of their Lie groups. There are constructed examples of the considered structure of different types on some known Lie groups.
Object of investigation are almost hypercomplex manifolds with Hermitian-Norden metrics of the lowest dimension. The considered manifolds are constructed on 4-dimensional Lie groups. It is established a relation between the classes of a classification of 4-dimensional indecomposable real Lie algebras and the classification of the manifolds under study. The basic geometrical characteristics of the constructed manifolds are studied in the frame of the mentioned classification of the Lie algebras.
We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic spinors isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms between other types of harmonic spinors and Bott-Chern, Aeppli and twisted cohomology.
Associated Nijenhuis Tensors on Manifolds with Almost Hypercomplex Structures and Metrics of Hermitian-Norden Type
引入了切丛中自同态的一个相伴的Nijenhuis张量.特别注意的张量几乎超复杂的结构和度量的Hermitian-Norden型。研究了六个相关联的Nijenhuis张量之间的关系以及它们的消失。给出了这些张量为零的几何解释，作为存在唯一的全斜对称挠率张量联络的充要条件。T. Friedrich和S. Ivanov在Asian J. Math.（2002）中对其他一些结构进行了讨论。最后，一个例子的4维流形所考虑的类型与消失相关联的Nijenhuis张量。
