arXiv research

Hermitian-Einstein metrics linked to stability of bundles on orbifolds.

We study a singular Hermitian metric of a vector bundle. First, we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a Nadel-Nakano type v…

Optimal L2 extension theorem for holomorphic vector bundles with singular metrics.

New system solves curvature for ample vector bundles, proving Griffiths conjecture.

On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (Riemannain real vector bundle) with an arbitrary metric connection over a compact Hermitian manif…

In this paper, we introduce the notions of $α$-Hermitian-Einstein metric and $α$-stability for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include…

Study on hermitian Yang-Mills connections on blown-up manifolds.

The paper extends orthogonal decomposition results to hermitian Higgs bundles.

Paper solves tensor problem for holomorphic vector bundles.

Proves stability of certain vector bundles on Kähler surfaces.

Construct Hermitian-Einstein metrics on stable holomorphic vector bundles using dynamical methods.

Alternative metric defined on vector bundles, proving vanishing theorem.

Paper develops techniques for singular metrics on vector bundles.

Study on stable vector bundles over Gauduchon manifolds.

Existence of twisted Hermitian-Einstein metrics on unstable vector bundles

In this paper, we show that, for every Hermitian vector bundle over a compact Kaehler Einstein manifold, if the projection is biharmonic, then it is harmonic.

Let $E$ be a Hermitian vector bundle over a complete Kähler manifold $(X,ω)$, $\dim_{\mathbb{C}}X=n$, with a $d$(bounded) Kähler form $ω$, $d_{A}$ be a Hermitian connection on $E$. The goal of this article is to study the $L^{2}$-Hodge theory on the vector bundle $E$. We extend the results of Gromov's \cite{Gro} to the…

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular,…

Study perturbations of submodules in Drury-Arveson space, finding smooth vector bundles with Hermitian connections.

Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.

In \cite{BR1}, \cite{BR2}, a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line b…

Investigates $J$-equation on holomorphic vector bundles over Kähler manifolds.

Solves a Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.

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 Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tange…

In this paper, we study the curvature estimate of the Hermitian-Yang-Mills flow on holomorphic vector bundles. In one simple case, we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic subvariety determined by the Harder-Narasimhan-Seshadri filtration of the holomorphic …

Let $(E,h)$ be a holomorphic Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of $E$. If $E$ is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian.

The paper develops harmonic theory on vector bundles with singular metrics and extends results from complex geometry.

Proves curvature positivity of invariant direct images in complex geometry.

Develops noncommutative Cowen-Douglas theory for noncommuting operators.

Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.

Let $X$ be a smooth projective variety over $\mathbb C$. We prove that a twisted Higgs vector bundle $(\calE\, ,θ)$ on $X$ admits an Einstein--Hermitian connection if and only if $(\calE\, ,θ)$ is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved by S. Wang in \cite{Wa}. Our approach …

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that K_X \otimes [D] is ample. It turns out tha…

The paper defines positivity for singular metrics on vector bundles and proves related theorems.

We introduce $Z$-critical connections for holomorphic vector bundles and prove their existence under stability conditions.

Researchers solve the Calderón problem for fractional Dirac operators.

We use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every del-dellbar exact real form of the type (k,k) on an n-dimensional complex manifold X arises as a di…

Study on HYM connections on Kähler manifolds, calculating moduli space virtual dimension.

Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.

Study on Gauduchon manifolds finds metrics for projectively flat bundles.

We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern…

Functional-analytic method for stochastic parallel transport in bundles.

In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type was introduced from the moment map framework. In this paper we establish a Kobayashi-Hitchin correspondence, that is, the equivalence of the…

The paper proves a pointwise Gysin formula for vector bundles and applies it to show positivity of polynomials.

The paper develops quantitative estimates for holomorphic sections over bounded domains.

Study hermitian Yang-Mills connections on pullback bundles for holomorphic submersions.

Unique solution found for Demailly's equation on stable bundles.

Study calculates curvatures in holomorphic fibrations using degenerate Hermitian forms.

New equation approximates Kähler potentials using Hermitian-Yang-Mills metrics.