Holomorphic Bundles Research Articles

Holomorphic Bundles Trivializable by Proper Surjective Holomorphic Map

AbstractGiven a compact complex manifold $M$, we investigate the holomorphic vector bundles $E$ on $M$ such that $\varphi ^* E$ is holomorphically trivial for some surjective holomorphic map $\varphi $, to $M$, from some compact complex manifold. We prove that these are exactly those holomorphic vector bundles that admit a flat holomorphic connection with finite monodromy homomorphism. A similar result is proved for holomorphic principal $G$-bundles, where $G$ is a connected reductive complex affine algebraic group.

The Haydys monopole equation

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a K\"ahler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold -- an $A$-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we have shown a vanishing theorem in [12].

Holomorphic string algebroids

We introduce the category of holomorphic string algebroids, whose objects are Courant extensions of Atiyah Lie algebroids of holomorphic principal bundles and whose morphisms correspond to inner morphisms of the underlying holomorphic Courant algebroids. This category provides natural candidates for Atiyah Lie algebroids of holomorphic principal bundles for the (complexified) string group and their morphisms. Our main results are a classification of string algebroids in terms of Čech cohomology and the construction of a locally complete family of deformations of string algebroids via a differential graded Lie algebra.

Gauge theory and $\mathrm G_2$-geometry on Calabi–Yau links

The 7-dimensional link K of a weighted homogeneous hypersurface on the round 9-sphere in \mathbb{C}^5 has a nontrivial null Sasakian structure which is contact Calabi–Yau, in many cases. It admits a canonical co-calibrated \mathrm G_2 -structure \varphi induced by the Calabi–Yau 3-orbifold basic geometry. We distinguish these pairs (K,\varphi) by the Crowley–Nordström \mathbb{Z}_{48} -valued \nu invariant, for which we prove odd parity and provide an algorithmic formula. We describe moreover a natural Yang–Mills theory on such spaces, with many important features of the torsion-free case, such as a Chern–Simons formalism and topological energy bounds. In fact, compatible \mathrm G_2 -instantons on holomorphic Sasakian bundles over K are exactly the transversely Hermitian Yang–Mills connections. As a proof of principle, we obtain \mathrm G_2 -instantons over the Fermat quintic link from stable bundles over the smooth projective Fermat quintic, thus relating in a concrete example the Donaldson–Thomas theory of the quintic threefold with a conjectural \mathrm G_2 -instanton count.

Correction to: The regular quantizations of certain holomorphic bundles

The wrong funding number has been given in the acknowledgements section. It should be read: Nos. ZZ201818 and LZD014.

Stability and holomorphic connections on vector bundles over LVMB manifolds

We characterize all LVMB manifolds X such that the holomorphic tangent bundle TX is spanned at the generic point by a family of global holomorphic vector fields, each of them having non-empty zero locus. We deduce that holomorphic connections on semi-stable holomorphic vector bundles over LVMB manifolds with this previous property are always flat.

Bergman kernel on Riemann surfaces and Kähler metric on symmetric products

Let [Formula: see text] be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer [Formula: see text], we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle [Formula: see text], where [Formula: see text] is the holomorphic cotangent bundle of [Formula: see text]. Our first main result estimates the corresponding Bergman metric on [Formula: see text] in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of [Formula: see text] into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of [Formula: see text]. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of [Formula: see text]. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of [Formula: see text] and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on [Formula: see text].

The regular quantizations of certain holomorphic bundles

In this paper, we study the regular quantizations of Kahler manifolds by using the first two coefficients of Bergman function expansions. Firstly, we obtain sufficient and necessary conditions for certain Hermitian holomorphic vector bundles and their ball subbundles to be regular quantizations. Secondly, we obtain that some projective bundles over the Fano manifolds M admit regular quantizations if and only if M are biholomorphically isomorphism to the complex projective spaces. Finally, we obtain the balanced metrics on certain Hermitian holomorphic vector bundles and their ball subbundles over the Riemann sphere.

Fiberwise bimeromorphic maps of conic bundles

Given a holomorphic conic bundle without sections, we show that the orders of finite groups acting by its fiberwise bimeromorphic transformations are bounded. This provides an analog of a similar result obtained by Bandman and Zarhin for quasi-projective conic bundles.

A Brief Survey of Higgs Bundles

Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.

Canonical metrics on holomorphic bundles over compact bi-Hermitian manifolds

The purpose of this paper is twofold. We first solve the Dirichlet problem for α-Hermitian–Einstein equations on I±-holomorphic bundles over bi-Hermitian manifolds. Second, by using Uhlenbeck–Yau’s continuity method, we prove that the existence of approximate α-Hermitian–Einstein structure and the α-semi-stability on I±-holomorphic bundles over compact bi-Hermitian manifolds are equivalent.

The geometry of maximal components of the PSp(4, ℝ) character variety

In this paper we describe the space of maximal components of the character variety of surface group representations into PSp(4,R) and Sp(4,R). For every rank 2 real Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups PSp(4,R) and Sp(4,R), we give a mapping class group invariant parameterization of each maximal component as an explicit holomorphic fiber bundle over Teichm\"uller space. Special attention is put on the connected components which are singular, we give a precise local description of the singularities and their geometric interpretation. We also describe the quotient of the maximal components for PSp(4,R) and Sp(4,R) by the action of the mapping class group as a holomorphic submersion over the moduli space of curves. These results are proven in two steps, first we use Higgs bundles to give a non-mapping class group equivariant parameterization, then we prove an analogue of Labourie's conjecture for maximal PSp(4,R) representations.

Branched Projective Structures on a Riemann Surface and Logarithmic Connections

We study the set \mathcal{P}_S consisting of all branched holomorphic projective structures on a compact Riemann surface X of genus g \geq 1 and with a fixed branching divisor S := \sum_{i=1}^d n_i\cdot x_i , where x_i \in X . Under the hypothesis that n_i,=1 , for all i , with d a positive even integer such that d \neq 2g-2 , we show that \mathcal{P}_S coincides with a subset of the set of all logarithmic connections with singular locus S , satisfying certain geometric conditions, on the rank two holomorphic jet bundle J^1(Q) , where Q is a fixed holomorphic line bundle on X such that Q^{\otimes 2}= TX \otimes \mathcal{O}_X(S) . The space of all logarithmic connections of the above type is an affine space over the vector space H^0(X, K^{\otimes 2}_X \otimes\mathcal{O}_X(S)) of dimension 3g-3+d . We conclude that \mathcal{P}_S is a subset of this affine space that has codimenison d at a generic point.

Holomorphic Vanishing Theorems on Finsler Holomorphic Vector Bundles and Complex Finsler Manifolds

AbstractIn this paper, we investigate the holomorphic sections of holomorphic Finsler bundles over both compact and non-compact complete complex manifolds. We also inquire into the holomorphic vector fields on compact and non-compact complete complex Finsler manifolds. We get vanishing theorems in each case according to different certain curvature conditions. This work can be considered as generalizations of the classical results on Kähler manifolds and hermitian bundles.

Cycles Cohomology and Geometrical Correspondences of Derived Categories to Field Equations

The integral geometry methods are the techniques could be the more naturally applied to study of the characterization of the moduli stacks and solution classes (represented cohomologically) obtained under the study of the kernels of the differential operators of the corresponding field theory equations to the space-time. Then through a functorial process a classification of differential operators is obtained through of the co-cycles spaces that are generalized Verma modules to the space-time, characterizing the solutions of the field equations. This extension can be given by a global Langlands correspondence between the Hecke sheaves category on an adequate moduli stack and the holomorphic bundles category with a special connection (Deligne connection). Using the classification theorem given by geometrical Langlands correspondences are given various examples on the information that the geometrical invariants and dualities give through moduli problems and Lie groups acting.

Nilpotency in instanton homology, and the framed instanton homology of a surface times a circle

In the description of the instanton Floer homology of a surface times a circle due to Muñoz, we compute the nilpotency degree of the endomorphism u2−64. We then compute the framed instanton homology of a surface times a circle with non-trivial bundle, which is closely related to the kernel of u2−64. We discuss these results in the context of the moduli space of stable rank two holomorphic bundles with fixed odd determinant over a Riemann surface.

Existence criteria for special locally conformally Kähler metrics

We investigate the relation between holomorphic torus actions on complex manifolds of locally conformally Kahler (LCK) type and the existence of special LCK metrics. We show that if the group of biholomorphisms of such a manifold (M, J) contains a compact torus which is not totally real, then there exists a Vaisman metric on the manifold, generalising a result of Kamishima–Ornea. Also, we obtain a new obstruction to the existence of LCK structures on a given complex manifold in terms of its automorphism group. As an application, we obtain a classification of manifolds of LCK type among all the manifolds having the structure of a holomorphic principal torus bundle. Moreover, we show that if the group of biholomorphisms contains a compact torus whose dimension is half the real dimension of M, then (M, J) admits an LCK metric with positive potential. Finally, we obtain new non-existence results for LCK metrics on certain products of complex manifolds.

Multiplicity-freeness of Unitary Representations in Sections of Holomorphic Hilbert Bundles

Abstract We prove several results asserting that the action of a Banach–Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity-free. These results require the existence of compatible anti-holomorphic bundle maps and certain multiplicity-freeness assumptions for stabilizer groups. For the group action on the base, the notion of an $(S,\sigma )$-weakly visible action (generalizing T. Koboyashi’s visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for group actions on homogeneous bundles for larger groups. We illustrate these general results by several examples related to operator groups and von Neumann algebras.

5. The Universal Connection for Principal Bundles over Homogeneous Spaces and Twistor Space of Coadjoint Orbits

Given a holomorphic principal bundle $Q\longrightarrow X$, the universal space of holomorphic connections is a torsor$C_1(Q)$ for $\ad Q\otimes T^*X$ such that the pullback of $Q$ to $C_1(Q)$ has a tautological holomorphic connection. When$X= G/P$, where $P$ is a parabolic subgroup of a complex simple group $G$, and $Q$ is the frame bundle of an ample linebundle, we show that $C_1(Q)$ may be identified with $G/L$, where $L \subset P$ is a Levi factor. We use this identificationto construct the twistor space associated to a natural hyper-K\ahler metric on $T^*(G/P)$, recovering Biquard's description ofthis twistor space, but employing only finite-dimensional, Lie-theoretic means.

Construction of constant scalar curvature Kähler cone metrics

Over a compact K\"ahler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a K\"ahler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature K\"ahler metrics with conic singularities: existence result under small deformations of K\"ahler classes, existence result over a Fano manifold, existence result over certain ruled manifolds. In this last case, we consider the projectivisation of a parabolic stable holomorphic bundle. This leads us to prove that the existing Hermitian-Einstein metric on this bundle enjoys a regularity property along the divisor on the base.