arXiv research

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\,,…

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,…

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…

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…

The paper connects bundle curvature to random zero currents.

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 $(X,ω)$ be a compact Kähler manifold of complex dimension $n$ and $(L,h)$ be a holomorphic line bundle over $X$. The line bundle mean curvature flow was introduced in \cite{JY} in order to find deformed Hermitian-Yang-Mills metrics on $L$. In this paper, we consider the stability of the line bundle mean curvature f…

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

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 …

The paper establishes conditions for optimal sampling configurations on complex manifolds.

In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao an…

Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.

We show that normalized currents of integration along the common zeros of random $m$-tuples of sections of powers of $m$ singular Hermitian big line bundles on a compact Kähler manifold distribute asymptotically to the wedge product of the curvature currents of the metrics. If the Hermitian metrics are Hölder with sing…

The paper proves heat kernel asymptotics for high power line bundles on complex manifolds.

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Dona…

We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.

We study the distribution of the common zero sets of $m$-tuples of holomorphic sections of powers of $m$ singular Hermitian pseudo-effective line bundles on a compact Kähler manifold. As an application, we obtain sufficient conditions which ensure that the wedge product of the curvature currents of these line bundles c…

We find the entropy's infinite-size behavior in complex manifold sections.

Study of Fubini-Study forms on surfaces with punctures.

Let $X\rightarrow S$ be a smooth projective surjective morphism of relative dimension $n$, where $X$ and $S$ are integral schemes over $\mathbb C$. Let $L\rightarrow X$ be a relatively very ample line bundle. For every sufficiently large positive integer $m$, there is a canonical isomorphism of the Deligne pairing $\la…

Study gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.

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

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…

Master thesis proves Bergman kernel asymptotics for positive line bundles.

It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families of Dirac operators over the two halves. This curvatur…

Let $X$ be a compact Kähler manifold, $E\to X$ a Hermitian vector bundle and $L\to X$ an ample line bundle. We construct a non-linear heat flow corresponding to the almost Hermitian-Einstein equation introduced by N.C. Leung, and prove that the solution exists for a short time. We also construct a potential function $D…

Let $X$ be a non-compact geometrically finite hyperbolic 3-manifold without cusps of rank 1. The deformation space $\mc{H}$ of $X$ can be identified with the Teichmüller space $\mc{T}$ of the conformal boundary of $X$ as the graph of a section in $T^*\mc{T}$. We construct a Hermitian holomorphic line bundle $\mc{L}$ on…

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

The mirror of a projective toric manifold $X_Σ$ is given by a Landau-Ginzburg model $(Y,W)$. We introduce a class of Lagrangian submanifolds in $(Y,W)$ and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over $X_Σ$. Through this ge…

We discuss positive closed currents and Fubini-Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini-Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curva…

We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using …

In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given Kähler manifold. The goal of this paper is to give an $\varepsilon$-regularity theorem for the line bundle mea…

The abstract discusses embedding theorems for pseudo-Kähler manifolds.

The paper studies $m$-positive currents and line bundles on complex manifolds.

The paper studies singularities in a complex flow related to mean curvature.

The paper studies curvature properties of direct image bundles.

Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.

In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degener…

Study shows Bergman kernel quotient approaches one for punctured surfaces.

Starting from the description of Segre forms as direct images of (powers of) the first Chern form of the (anti)tautological line bundle on the projectivized bundle of a holomorphic hermitian vector bundle, we derive a version of the pointwise Kobayashi-Lübke inequality.

Study the Bochner-Schrödinger operator's trace in semiclassical limit.

In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of Chern-Ricci curvature will be preserved if the initial metric has non-positive bisectional curvature. As an application, we show that the canonical line bundle of a compact Hermitian manifold with nonpositive bisecti…

Abstract: Study of metrics on line bundles over complex varieties.

A differential operator introduced by A. Gray on the unit sphere bundle of a Kähler-Einstein manifold is studied. A lower bound for the first eigenvalue of the Laplacian for the Sasaki metric on the unit sphere bundle of a Kähler-Einstein manifold is derived. Some rigidity theorems classifying complex space forms among…

Let $L$ be a holomorphic line bundle over a compact Kähler manifold $X$. Motivated by mirror symmetry, we study the deformed Hermitian-Yang-Mills equation on $L$, which is the line bundle analogue of the special Lagrangian equation in the case that $X$ is Calabi-Yau. We show that this equation is the Euler-Lagrange equ…

The present article studies the class of Einstein-Hermitian harmonic maps of constant Kaehler angle from the projective line into quadrics. We provide a description of their moduli spaces up to image, and gauge-equivalence using the language of vector bundles and representation theory. It is shown that the dimension of…

Study shows convergence of Fubini-Study currents to equilibrium metrics on Kähler manifolds.

Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic $L^2$ torsion, which lies in the determinant line of the twisted $L^2$ Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite…