Related papers: The harmonic theory on a vector bundle with singul…

The aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades. Our purpose is to study several of the positivity notions studied for vector bundles with some…

In this paper, we show that if the holomorphic tangent bundle $TX$ of a compact K\"ahler manifold $X$ is uniformly weakly RC-positive, then $X$ is projective and rationally connected. This result is previously established by Xiaokui Yang…

We study conditions of H\"ormander's $L^2$-estimate and the Ohsawa-Takegoshi extension theorem. Introducing a twisted version of H\"ormander-type condition, we show a converse of H\"ormander $L^2$-estimate under some regularity assumptions…

In this paper, we introduce a new concept so called harmonic complex structure by using harmonic theory for vector bundle-valued differential forms. It is a new structure intermediates between complex structure and K\"ahler structure. From…

In this article we prove a general result on a nef vector bundle $E$ on a projective manifold $X$ of dimension $n$ depending on the vector space $H^{n,n} (X, E). $ It is also shown that $H^{n,n} (X, E)=0$ for an indecomposable nef rank 2…

We study the positivity properties of the curvature operator for holomorphic Hermitian vector bundles. We obtain new characterization of semi-positive curvature operators for $(n,q)$ and $(p,n)$-forms by L2-estimates. The characterization…

Two new classes of metrizable vector bundles have been presented in the papers [1] and [4]. The Lie algebroid generalized tangent bundle of a dual vector bundle is presented. This Lie algebroid is a new example of metrizable vector bundle.…

Using the framework of noncommutative Kahler structures, we generalise to the noncommutative setting the celebrated vanishing theorem of Kodaira for positive line bundles. The result is established under the assumption that the associated…

Let $E$ be a vector bundle over a smooth curve $C$, and $V$ a generating space of sections of $E$. We characterise Mumford linear stability of the associated projective model of $\mathbb{P} E^\vee$ in $\mathbb{P} V^\vee$ in terms of…

In this paper, we show that if a holomorphic vector bundle is slope polystable with respect to a K\"{a}hler class, then it admits a Hermitian-Yang-Mills metric with respect to a suitable K\"{a}hler current with singularities in higher…

In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem for Higgs bundles over compact complex manifolds. Let $ (E,\theta) $ be a Higgs bundle over a compact Hermitian manifold $(M,\omega_g) $. Suppose that there exists a…

Differential calculi are obtained for quantum homogeneous spaces by extending Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum…

In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact…

Let $X$ be a compact Riemann surface. Let $(E,\theta)$ be a stable Higgs bundle of degree $0$ on $X$. Let $h_{\det(E)}$ denote a flat metric of the determinant bundle $\det(E)$. For any $t>0$, there exists a unique harmonic metric $h_t$ of…

We establish necessary and sufficient conditions for the global hypoellipticity of $G$-invariant operators on homogeneous vector bundles. These criteria are established in terms of the corresponding matrix-valued symbols as developed by…

A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D-invariant subbundle (resp. if it is the D-invariant direct sum of stable subbundles). It has…

Let $E$ be a vector bundle and $L$ be a line bundle over a smooth projective variety $X$. In this article, we give a condition for the vanishing of Dolbeault cohomology groups of the form $H^{p,q}(X,\SSS^{\alpha}E\otimes \wedge^{\beta}…

Given a family $f:\mathcal X \to S$ of canonically polarized manifolds, the unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $\mathcal K_{\mathcal X/S}$. We use a global elliptic…

In heterotic string theories consistency requires the introduction of a non-trivial vector bundle. This bundle breaks the original ten-dimensional gauge groups $\text{E}_8\times\text{E}_8$ or $\text{SO}(32)$ for the supersymmetric heterotic…

We generalise Livens theorem, showing that Hamiltonian equation on the vector bundle $E^\ast\rightarrow M$, dual to a general algebroid $E\rightarrow M$, can be derived by means of a variational principle. The framework can be used to…