Holomorphic Vector Bundles Research Articles

∂¯-tangential invariants of certain vector bundles over complex foliations

The Dolbeault cohomology plays an important role in the study of some ∂ ¯ -invariants of complex and holomorphic vector bundles as ∂ ¯ -Chern classes and Atiyah classes. In this paper we generalize similar invariants and their properties in the tangential Dolbeault cohomology. More exactly, we introduce and we study tangential Atiyah classes for F -holomorphic vector bundles and ∂ ¯ -tangential Chern classes for tangentially smooth vector bundles over a manifold M endowed with a complex foliation F . Also, ∂ ¯ -tangential secondary invariants are studied following similar constructions for Lie algebroids. The notions are introduced by a global formalism that is used in the tangential theory of foliated spaces.

Vector-valued modular forms and the modular orbifold of elliptic curves

This paper presents the theory of holomorphic vector-valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector-valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of [Formula: see text] play in the holomorphic theory of vector-valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line [Formula: see text].

Sobolev regularity of the Bergman projection on certain pseudoconvex domains

In this paper we study the Sobolev regularity of the Bergman projection B and the ∂¯-Neumann operator N on a certain pseudoconvex domain. We show that if Ω is a domain with Lipschitz boundary, which is relatively compact in an n-dimensional compact Kähler manifold and satisfies some “logδ-pseudoconvexity” condition, the operators B, N and ∂¯∗N are regular in the Sobolev spaces Wr,sk(Ω,E) for forms with values in a holomorphic vector bundle E and for any k<η/2, 0<η<1, 0≤r≤n, 0≤s≤n−1.

Yang–Mills instanton sheaves

The SL(2,C) Yang–Mills instanton solutions constructed recently by the biquaternion method were shown to satisfy the complex version of the ADHM equations and the monad construction. Moreover, we discover that, in addition to the holomorphic vector bundles on CP3 similar to the case of SU(2) ADHM construction, the SL(2,C) instanton solutions can be used to explicitly construct instanton sheaves on CP3. Presumably, the existence of these instanton sheaves is related to the singularities of the SL(2,C) instantons on S4 which do not exist for SU(2) instantons.

Positivity of character subbundles and minimum principle for noncompact group actions

Based on the results of Berndtsson on the positivity of holomorphic vector bundles associated to holomorphic fibration, we prove that, under the setting of the compact Lie group actions, the character subbundles of such bundles are holomorphic vector bundles and positive in the sense of Nakano. The second main purpose of the present paper is to establish Berndtsson’s integral form of Kiselman’s minimum principle in the setting of the noncompact Lie group actions.

A Donaldson type functional on a holomorphic Finsler vector bundle

In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by introducing a Donaldson type functional on the space $F^+(E)$ of strongly pseudo-convex complex Finsler metrics on $E$ -- a holomorphic vector bundle over a closed K\"ahler manifold $M$. This Donaldson type functional is a generalization in the complex Finsler geometry setting of the original Donaldson functional and has Finsler-Einstein metrics on $E$ as its only critical points, at which this functional attains the absolute minimum.

On connections on principal bundles

A new construction of a universal connection was given in Biswas, Hurtubise and Stasheff (2012). The main aim here is to explain this construction. A theorem of Atiyah and Weil says that a holomorphic vector bundle E over a compact Riemann surface admits a holomorphic connection if and only if the degree of every direct summand of E is zero. In Azad and Biswas (2002), this criterion was generalized to principal bundles on compact Riemann surfaces. This criterion for principal bundles is also explained.

Dressed elliptic genus of heterotic compactifications with torsion and general bundles

We define and compute the dressed elliptic genus of N = 2 heterotic compactifications with torsion that are principal two-torus bundles over a K3 surface. We consider the most general gauge bundle compatible with supersymmetry, a stable holomorphic vector bundle over the base together with an Abelian bundle over the total space, generalizing the computation previously done by the authors in the absence of the latter. Starting from a (0,2) gauged linear sigma-model with torsion we use supersymmetric localization to obtain the result. We provide also a mathematical definition of the dressed elliptic genus as a modified Euler characteristic and prove that both expressions agree for hypersurfaces in weighted projective spaces. Finally we show that it admits a natural decomposition in terms of N = 4 superconformal characters, that may be useful to investigate moonshine phenomena for this wide class of N = 2 vacua, that includes K3*T2 compactifications as special cases.

Birationality of moduli spaces of twisted $$\mathrm {U}(p,q)$$ U ( p , q ) -Higgs bundles

A $\mathrm{U}(p,q)$-Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter $\alpha$. In this paper we study wall crossing for the moduli spaces of $\alpha$-polystable twisted $\mathrm{U}(p,q)$-Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin-Kobayashi correspondence play an essential role.

Balanced metrics on twisted Higgs bundles

A twisted Higgs bundle on a Kähler manifold X is a pair $$(E,\phi )$$ consisting of a holomorphic vector bundle E and a holomorphic bundle morphism $$\phi :M \otimes E \rightarrow E$$ for some holomorphic vector bundle M. Such objects were first considered by Hitchin when X is a curve and M is the tangent bundle of X, and also by Simpson for higher dimensional base. The Hitchin–Kobayashi correspondence for such pairs states that $$(E,\phi )$$ is polystable if and only if E admits a hermitian metric solving the Hitchin equation. This correspondence is a powerful tool to decide whether there exists a solution of the equation, but it provides little information as to the actual solution. In this paper we study a quantization of this problem that is expressed in terms of finite dimensional data and balanced metrics that give approximate solutions to the Hitchin equation. Motivation for this study comes from work of Donagi–Wijnholt (JHEP 05:068, 2013) concerning balanced metrics for the Vafa–Witten equations.

A note on actions of some monoids

Smooth actions of the multiplicative monoid (R,⋅) of real numbers on manifolds lead to an alternative, and for some reasons simpler, definitions of a vector bundle, a double vector bundle and related structures like a graded bundle (Grabowski and Rotkiewicz (2011) [10]). For these reasons it is natural to study smooth actions of certain monoids closely related with the monoid (R,⋅). Namely, we discuss geometric structures naturally related with: smooth and holomorphic actions of the monoid of multiplicative complex numbers, smooth actions of the monoid of second jets of punctured maps (R,0)→(R,0), smooth actions of the monoid of real 2 by 2 matrices and smooth actions of the multiplicative reals on a supermanifold. In particular cases we recover the notions of a holomorphic vector bundle, a complex vector bundle and a non-negatively graded manifold.

Positivity of metrics on conic neighborhoods of 1-convex submanifolds

Let [Formula: see text] be a holomorphic submersion from a complex manifold [Formula: see text] onto a 1-convex manifold [Formula: see text] with exceptional set [Formula: see text] and [Formula: see text] a holomorphic section. Let [Formula: see text] be a plurisubharmonic exhaustion function which is strictly plurisubharmonic on [Formula: see text] with [Formula: see text] For every holomorphic vector bundle [Formula: see text] there exists a neighborhood [Formula: see text] of [Formula: see text] for [Formula: see text] conic along [Formula: see text] such that [Formula: see text] can be endowed with Nakano strictly positive Hermitian metric. Let [Formula: see text] [Formula: see text] be a given holomorphic function. There exist finitely many bounded holomorphic vector fields defined on a Stein neighborhood [Formula: see text] of [Formula: see text] conic along [Formula: see text] with zeroes of arbitrary high order on [Formula: see text] and such that they generate [Formula: see text] Moreover, there exists a smaller neighborhood [Formula: see text] such that their flows remain in [Formula: see text] for sufficiently small times thus generating a local dominating spray.

Chern forms of holomorphic Finsler vector bundles and some applications

In this paper, we present two kinds of total Chern forms [Formula: see text] and [Formula: see text] as well as a total Segre form [Formula: see text] of a holomorphic Finsler vector bundle [Formula: see text] expressed by the Finsler metric [Formula: see text], which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in Finsler Geometry, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133–144] to some extent. As some applications, we show that the signed Segre forms [Formula: see text] are positive [Formula: see text]-forms on [Formula: see text] when [Formula: see text] is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler–Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou’s one [Finsler geometry on complex vector bundles, in A Sampler of Riemann–Finsler Geometry, MSRI Publications, Vol. 50 (Cambridge University Press, 2004), pp. 83–105] and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.

The Yang-Mills flow and the Atiyah-Bott formula on compact Kähler manifolds

We study the Yang-Mills flow on a holomorphic vector bundle $E$ over a compact K\"ahler manifold $X$. Along a solution of the flow, we show that the curvature endomorphism $i\Lambda F(A_t)$ approaches in $L^2$ an endomorphism with constant eigenvalues given by the slopes of the quotients from the Harder-Narasimhan filtration of $E$. This proves a sharp lower bound for the Hermitian-Yang-Mills functional and thus the Yang-Mills functional, generalizing to arbitrary dimension a formula of Atiyah and Bott first proven on Riemann surfaces. Furthermore, we show any reflexive extension to all of $X$ of the limiting bundle $E_\infty$ is isomorphic to ${\rm Gr}^{{\rm hns}}(E)^{**}$, verifying a conjecture of Bando and Siu.

A Riemann–Roch–Grothendieck theorem for flat fibrations with complex fibers

We consider a proper flat fibration with real base and complex fibers. First we construct odd characteristic classes for such fibrations by a method that generalizes constructions of Bismut–Lott [5]. Then we consider the direct image of a fiberwise holomorphic vector bundle, which is a flat vector bundle on the base. We give a Riemann–Roch–Grothendieck theorem calculating the odd real characteristic classes of this flat vector bundle.

Segre forms and Kobayashi–Lübke inequality

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–Lubke inequality.

Expected volume and Euler characteristic of random submanifolds

In a closed manifold of positive dimension $n$, we estimate the expected volume and Euler characteristic for random submanifolds of codimension $r\in \{1,...,n\}$ in two different settings. On one hand, we consider a closed Riemannian manifold and some positive $\lambda$. Then we take $r$ independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues less than $\lambda$ and consider the random submanifold defined as the common zero set of these $r$ functions. We compute asymptotics for the mean volume and Euler characteristic of this random submanifold as $\lambda$ goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle $\mathcal{L}$ and a rank $r$ holomorphic vector bundle $\mathcal{E}$ that are also defined over the reals. Then we get asymptotics for the expected volume and Euler characteristic of the real vanishing locus of a random real holomorphic section of $\mathcal{E}\otimes\mathcal{L}^d$ as $d$ goes to infinity. The same techniques apply to both settings.

Negative vector bundle and complex Finsler structures

Given a holomorphic vector bundle [Formula: see text] with a strongly pseudoconvex Finsler metric [Formula: see text], an intrinsic approach to complex Finsler geometry as given by Kobayashi is proposed in order to study the negativity of [Formula: see text].

Kohn decomposition for forms on coverings of complex manifolds constrained along fibres

The classical result of J.J. Kohn asserts that over a relatively compact subdomain $D$ with $C^\infty$ boundary of a Hermitian manifold whose Levi form has at least $n-q$ positive eigenvalues or at least $q+1$ negative eigenvalues at each boundary point, there are natural isomorphisms between the $(p,q)$ Dolbeault cohomology groups defined by means of $C^\infty$ up to the boundary differential forms on $D$ and the (finite-dimensional) spaces of harmonic $(p,q)$-forms on $D$ determined by the corresponding complex Laplace operator. In the present paper, using Kohn's technique, we give a similar description of the $(p,q)$ Dolbeault cohomology groups of spaces of differential forms taking values in certain (possibly infinite-dimensional) holomorphic Banach vector bundles on $D$. We apply this result to compute the $(p,q)$ Dolbeault cohomology groups of some regular coverings of $D$ defined by means of $C^\infty$ forms constrained along fibres of the coverings.

Homogeneous Hermitian holomorphic vector bundles and the Cowen–Douglas class over bounded symmetric domains

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite-dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We write down an equivariant constant coefficient differential operator that intertwines the bundle with the direct sum of its irreducible factors. As an application, we show that in the case of the closed unit ball in Cn all homogeneous n-tuples of Cowen–Douglas operators are similar to direct sums of certain basic n-tuples.