arXiv research

The paper solves a problem related to Higgs bundles and Hermitian metrics.

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

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

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

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…

Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.

Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,…

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…

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…

The paper extends orthogonal decomposition results to hermitian Higgs bundles.

In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily Kähler, we solve the Hermitian-Einstein equation on analytically stable Higgs bundles.

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,…

Paper solves tensor problem for holomorphic vector bundles.

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

Study on existence of harmonic metrics for non-Hermitian Yang-Mills bundles.

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

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

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…

We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the ba…

The paper characterizes curvature of quaternionic skew-Hermitian manifolds and constructs related geometric structures.

We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is desc…

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

Study on Gauduchon manifolds finds metrics for projectively flat bundles.

Functional-analytic method for stochastic parallel transport in bundles.

Study calculates curvatures in holomorphic fibrations using degenerate Hermitian forms.

We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be wr…

Proves stability of certain vector bundles on Kähler surfaces.

A-manifolds and A-bundles are manifolds and vector bundles modelled on a projective finitely generated module over a topological algebra A. In this paper we investigate the conditions under which an A-bundle is provided with an A-valued hermitian structure and a compatible connection, in case A is a commutative complet…

We review the theory of quaternionic Kahler and hyperkahler structures. Then we consider the tangent bundle of a Riemannian manifold M with a metric connection D (with torsion) and with its well estabilished canonical complex structure. With an extra almost Hermitian structure on M it is possible to find a quaternionic…

We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is se…

The paper connects bundle curvature to random zero currents.

In this paper, using Donaldson's heat flow, we show that the semi-stability of a Higgs bundle over a compact Kähler manifold implies the existence of approximate Hermitian-Einstein structure on the Higgs bundle.

The Horrocks-Mumford bundle $E$ is a famous stable complex vector bundle of rank 2 on 4-dimensional complex projective space. By construction, $E$ has a natural Hermitian metric $h_1$. On the other hand, stability implies the existence of a Hermitian-Einstein metric in $E$ which is unique up to a positive scalar. Now t…

We observe that the line bundle associated to the tame symbol of two invertible holomorphic functions also carries a fairly canonical hermitian metric, hence it represents a class in a Hermitian holomorphic Deligne cohomology group. We put forward an alternative definition of hermitian holomorphic structure on a gerbe …

Diagonal metrics solve Hermitian-Einstein equations for decomposed Higgs bundles.

Uniformizes compact Sasakian manifolds into circle bundles.

In this paper, we provide a systematic and constructive description of Vaisman structures on certain principal elliptic bundles over complex flag manifolds. From this description we explicitly classify homogeneous l.c.K. structures on compact homogeneous Hermitian manifolds using elements of representation theory of co…

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…

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.

Study proves correspondence for special bundles on complex surfaces.

Existence of twisted Hermitian-Einstein metrics on unstable vector bundles

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

Iterative method finds Hermitian-Einstein metrics on stable bundles.

In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic $(0,q)$-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we e…

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 …

Alternative metric defined on vector bundles, proving vanishing theorem.

We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.

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.