Holomorphic Vector Bundles Research Articles

A classification of homogeneous operators in the Cowen–Douglas class

An explicit construction of all the homogeneous holomorphic Hermitian vector bundles over the unit disc D is given. It is shown that every such vector bundle is a direct sum of irreducible ones. Among these irreducible homogeneous holomorphic Hermitian vector bundles over D, the ones corresponding to operators in the Cowen–Douglas class Bn(D) are identified. The classification of homogeneous operators in Bn(D) is completed using an explicit realization of these operators. We also show how the homogeneous operators in Bn(D) split into similarity classes.

Integration of Cocycles and Lefschetz Number Formulae for Differential Operators

Let ${\mathcal E}$ be a holomorphic vector bundle on a complex manifold $X$ such that $\dim_{{\mathbb C}}X=n$. Given any continuous, basic Hochschild $2n$-cocycle $\psi_{2n}$ of the algebra ${\rm Diff}_n$ of formal holomorphic differential operators, one obtains a $2n$-form $f_{{\mathcal E},\psi_{2n}}(\mathcal D)$ from any holomorphic differential operator ${\mathcal D}$ on ${\mathcal E}$. We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405-448; J. Noncommut. Geom. 3 (2009), 27-45] to show that $\int_X f_{{\mathcal E},\psi_{2n}}({\mathcal D})$ gives the Lefschetz number of $\mathcal D$ upto a constant independent of $X$ and ${\mathcal E}$. In addition, we obtain a "local" result generalizing the above statement. When $\psi_{2n}$ is the cocycle from [Duke Math. J. 127 (2005), 487-517], we obtain a new proof as well as a generalization of the Lefschetz number theorem of Engeli-Felder. We also obtain an analogous "local" result pertaining to B. Shoikhet's construction of the holomorphic noncommutative residue of a differential operator for trivial vector bundles on complex parallelizable manifolds. This enables us to give a rigorous construction of the holomorphic noncommutative residue of $\mathcal D$ defined by B. Shoikhet when ${\mathcal E}$ is an arbitrary vector bundle on an arbitrary compact complex manifold $X$. Our local result immediately yields a proof of a generalization of Conjecture 3.3 of [Geom. Funct. Anal. 11 (2001), 1096-1124].

On the double commutant of Cowen–Douglas operators

Let T be a Cowen–Douglas operator. In this paper, we study the von Neumann algebra V ⁎ ( T ) consisting of operators commuting with both T and T ⁎ from a geometric viewpoint. We identify operators in V ⁎ ( T ) with connection-preserving bundle maps on E ( T ) , the holomorphic Hermitian vector bundle associated to T. By studying such bundle maps, the structure of V ⁎ ( T ) as well as information on reducing subspaces of T can be determined.

Laplacien hypoelliptique et cohomologie de Bott–Chern

Let p : M → S be a proper submersion of complex manifolds, and let F be a holomorphic vector bundle on M. When R ⋅ p ⁎ F is locally free, we establish a Riemann–Roch–Grothendieck theorem in Bott–Chern cohomology. When M is equipped with a ∂ ¯ ∂ -closed ( 1 , 1 ) form inducing a Hermitian metric along the fibres, the proof is obtained by using elliptic superconnections. In the general case, we construct an exotic version of the hypoelliptic superconnections which we introduced in previous work.

Moving poles of meromorphic linear systems on ℙ1(ℂ) in the complex plane

Let E0be a holomorphic vector bundle over P1(C) and †0be a meromorphic connection of E0. We introduce the notion of an integrable connection that describes the movement of the poles of †0in the complex plane with integrability preserved. We show the that such a deformation exists under sufficiently weak conditions on the deformation space. We also show that if the vector bundle E0 is trivial, then the solutions of the corresponding nonlinear equations extend meromorphically to the deformation space.

Uniqueness of balanced metrics on holomorphic vector bundles

Let E → M be a holomorphic vector bundle over a compact Kähler manifold ( M , ω ) . We prove that if E admits a ω -balanced metric (in X. Wang’s terminology (Wang, 2005 [3])) then it is unique. This result together with Biliotti and Ghigi (2008) [14] implies the existence and uniqueness of ω -balanced metrics of certain direct sums of irreducible homogeneous vector bundles over rational homogeneous varieties. We finally apply our result to show the rigidity of ω -balanced Kähler maps into Grassmannians.

Harmonic Functions and Instanton Moduli Spaces on the Multi-Taub–NUT Space

Explicit construction of the basic SU(2) anti-instantons over the multi-Taub–NUT geometry via the classical conformal rescaling method is exhibited. These anti-instantons satisfy the so-called weak holonomy condition at infinity with respect to the trivial flat connection and decay rapidly. The resulting unit energy anti-instantons have trivial holonomy at infinity.We also fully describe their unframed moduli space and find that it is a five dimensional space admitting a singular disk-fibration over \({\mathbb{R}^3}\) .On the way, we work out in detail the twistor space of the multi-Taub–NUT geometry together with its real structure and transform our anti-instantons into holomorphic vector bundles over the twistor space. In this picture we are able to demonstrate that our construction is complete in the sense that we have constructed a full connected component of the moduli space of solutions of the above type.We also prove that anti-instantons with arbitrary high integer energy exist on the multi-Taub–NUT space.

Moduli spaces of vector bundles over a Klein surface

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (Ho and Liu in Commun Anal Geom 16(3):617–679, 2008).

Horizontal Forms of Chern Type on Complex Finsler Bundles

The aim of this paper is to construct horizontal Chern forms of a holomorphic vector bundle using complex Finsler structures. Also, some properties of these forms are studied.

On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces

We study holomorphic (n+ 1)-chains En → En−1 → · · · → E0 consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on n real parameters was introduced in [1] and moduli spaces were constructed in [22, 24]. In this paper we study the variation of the moduli spaces with respect to the stability parameters. In particular we characterize a parameter region where the moduli spaces are birationally equivalent. A detailed study is given for the case of 3-chains, generalizing that of 2-chains (triples) in [6]. Our work is motivated by the study of the topology of moduli spaces of Higgs bundles and their relation to representations of the fundamental group of the surface.

Holomorphic vector bundles on Sasakian threefolds and parabolic vector bundles

A bijective correspondence is constructed between the parabolic vector bundles with rational parabolic weights on a Riemann surface X and the holomorphic vector bundles on quasiregular Sasakian threefolds over X. This bijective correspondence preserves the stable objects.

Exploring a new peak in the heterotic landscape

We study the existence of realistic heterotic vacua on a new Abelian surface fibered Calabi-Yau threefold X with Z_8 x Z_8 fundamental group. Our main result is a no-go theorem, which says that (under mild assumptions) there is no stable holomorphic vector bundle on X satisfying the constraints required by global consistency of the heterotic vacuum and phenomenology. To prove the theorem we explore in some detail the Fourier-Mukai transform of vector bundles on Abelian surface fibrations.

Numerical Hermitian Yang-Mills connections and vector bundle stability in heterotic theories

A numerical algorithm is presented for explicitly computing the gauge connection on slope-stable holomorphic vector bundles on Calabi-Yau manifolds. To illustrate this algorithm, we calculate the connections on stable monad bundles defined on the K3 twofold and Quintic threefold. An error measure is introduced to determine how closely our algorithmic connection approximates a solution to the Hermitian Yang-Mills equations. We then extend our results by investigating the behavior of non slope-stable bundles. In a variety of examples, it is shown that the failure of these bundles to satisfy the Hermitian Yang-Mills equations, including field-strength singularities, can be accurately reproduced numerically. These results make it possible to numerically determine whether or not a vector bundle is slope-stable, thus providing an important new tool in the exploration of heterotic vacua.

Yukawa textures from heterotic stability walls

A holomorphic vector bundle on a Calabi-Yau threefold, X, with h^{1,1}(X)>1 can have regions of its Kahler cone where it is slope-stable, that is, where the four-dimensional theory is N=1 supersymmetric, bounded by "walls of stability". On these walls the bundle becomes poly-stable, decomposing into a direct sum, and the low energy gauge group is enhanced by at least one anomalous U(1) gauge factor. In this paper, we show that these additional symmetries can strongly constrain the superpotential in the stable region, leading to non-trivial textures of Yukawa interactions and restrictions on allowed masses for vector-like pairs of matter multiplets. The Yukawa textures exhibit a hierarchy; large couplings arise on the stability wall and some suppressed interactions "grow back" off the wall, where the extended U(1) symmetries are spontaneously broken. A number of explicit examples are presented involving both one and two stability walls, with different decompositions of the bundle structure group. A three family standard-like model with no vector-like pairs is given as an example of a class of SU(4) bundles that has a naturally heavy third quark/lepton family. Finally, we present the complete set of Yukawa textures that can arise for any holomorphic bundle with one stability wall where the structure group breaks into two factors.

Singular Hermitian–Einstein Monopoles on the Product of a Circle and a Riemann Surface

In this paper, the moduli space of singular unitary Hermitian–Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bundle. The singularities of this automorphism correspond to the singularities of the singular monopole. We then consider the complex geometry of the moduli space; in particular, we compute dimensions, both from the complex geometric and the gauge theoretic point of view.

Formula omitted]-estimates and existence theorems for the [formula omitted]-operator on complete Kähler manifolds

In this paper we establish some L p -estimates and existence theorems for the ∂ ¯ -operator on a holomorphic Hermitian vector bundle over a complete Kähler manifold. As applications, we prove some vanishing theorems of the L p -Dolbeault cohomology on complete Kähler manifolds with suitable geometric conditions.

Bochner technique on strong kähler-finsler manifolds

By using the Chern-Finsler connection and complex Finsler metric, the Bochner technique on strong Kähler-Finsler manifolds is studied. For a strong Kähler-Finsler manifold M, the authors first prove that there exists a system of local coordinate which is normalized at a point υ ∈ M ˜ = T 1 , 0 M / o ( M ) , and then the horizontal Laplace operator □ H for differential forms on PTM is defined by the horizontal part of the Chern-Finsler connection and its curvature tensor, and the horizontal Laplace operator □ H on holomorphic vector bundle over PTM is also defined. Finally, we get a Bochner vanishing theorem for differential forms on PTM. Moreover, the Bochner vanishing theorem on a holomorphic line bundle over PTM is also obtained

Twisted Poincaré lemma and twisted Čech–de Rham isomorphism in case dimension =1

For a compact Riemann surface, (n+1)-tuple x:=(x0,…,xn) of points on it, and a holomorphic vector bundle with an integrable connection on the open Riemann surface Xx deprived of (n+1) points x0,…,xn, let L be the local system of horizontal sections of the connection. In this article, we give a suitable covering of Xx to calculate the Čech cohomology and describe the isomorphism between the cohomology and the twisted de Rham cohomology, which is the cohomology of the complex with the differentials given by the connection. This isomorphism is given by the integrations over Aomoto’s regularized paths, the so-called Euler type integrals. For the family {Xx}x parametrized by x, we give a variant of the isomorphism.

Vector bundles on Sasakian manifolds

We investigate the analog of holomorphic vector bundles in the context of Sasakian manifolds.

Some results about holomorphic vector bundles over general Hopf manifolds

In the present note that grew out of my talk given at the conference in honor of Prof. Zhong Tongde, I give a survey of some recent results about holomorphic vector bundles over general Hopf manifolds.