Hermitian Holomorphic Vector Bundles Research Articles

𝐿²-metrics, projective flatness and families of polarized abelian varieties

We compute the curvature of the L 2 L^{2} -metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kähler manifolds. As an application, we show that the L 2 L^{2} -metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kähler manifold, this leads to the poly-stability of the direct image with respect to any Kähler form on the parameter space.

A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions

We study a decomposition of a holomorphic vector bundle with connection which need not be endowed with any metrics, which is a generalization of an orthogonal decomposition of a Hermitian holomorphic vector bundle. We first derive several results on the induced connections, the second fundamental forms of subbundles and curvature forms of the connections. We next apply these results to a complex affine immersion. Especially, we give elementary self-contained proofs of the fundamental theorems for a complex affine immersion to a complex affine space.

Lp-curvature and the Cauchy-Riemann equation near an isolated singular point

Let X be a complex n-dimensional reduced analytic space with isolated singular point x0, and with a strongly plurisubharmonic function ρ : X → [0, ∞) such that ρ(x0) = 0. A smooth Kähler form on X \ {x0} is then defined by i∂∂ρ. The associated metric is assumed to have -curvature, to admit the Sobolev inequality and to have suitable volume growth near x0. Let E → X \ {x0} be a Hermitian-holomorphic vector bundle, and ξ a smooth (0, 1)-form with coefficients in E. The main result of this article states that if ξ and the curvature of E are both then the equation has a smooth solution on a punctured neighbourhood of x0. Applications of this theorem to problems of holomorphic extension, and in particular a result of Kohn-Rossi type for sections over a CR-hypersurface, are discussed in the final section.

ANALYTIC CONTINUATION OF VECTOR BUNDLES WITH Lp –CURVATURE

This article addresses the problem of removable singularities for a Hermitian-holomorphic vector bundle ℰ, defined on the complement of an analytic set A of complex codimension at least two in a complex n-dimensional manifold X. In particular it is shown here that there exists a unique holomorphic bundle [Formula: see text] on X, such that [Formula: see text], when the curvature of ℰ belongs to Ln (X\A). This result is in fact sharp, as counterexamples exist for the extensibility of ℰ with curvature in Lp, p < n. Extension across general closed subsets of finite (2n - 4)-dimensional Hausdorff measure then follows directly from a slicing theorem of Bando and Siu.

Extension of holomorphic vector bundles across a totally real submanifold

Let ϵ be a Hermitian-holomorphic vector bundle with compatible Chern connection ▽, defined on the complement o f R n inside a polydisc Δ ⊇ C n . The main result of this notes provides a necessary and sufficient condition for unique extension of ϵ to Δ, namely that the curvature F ▽, when contracted with a non-vanishing holomorphic vector field ξ, is ∂-exact. Applications to removable singularities for anti-self-dual connections across a real time line in C 2 and solutions of the static monopole equation across a point in R 3 are also given as corollaries.

Extrinsic hermitian geometry of functional determinants for vector subbundles and the Drinfeld-Sokolov ghost system

In this paper, a novel method is presented for the study of the dependence of the functional determinant of the Laplace operator associated to a subbundle $F$ of a hermitian holomorphic vector bundle $E$ over a Riemann surface $\Sigma$ on the hermitian structure $(h,H)$ of $E$. The generalized Weyl anomaly of the effective action is computed and found to be expressible in terms of a suitable generalization of the Liouville and Donaldson actions. The general techniques worked out are then applied to the study of a specific model, the Drinfeld--Sokolov (DS) ghost system arising in $W$--gravity. The expression of generalized Weyl anomaly of the DS ghost effective action is found. It is shown that, by a specific choice of the fiber metric $H_h$ depending on the base metric $h$, the effective action reduces into that of a conformal field theory. Its central charge is computed and found to agree with that obtained by the methods of hamiltonian reduction and conformal field theory. The DS holomorphic gauge group and the DS moduli space are defined and their dimensions are computed.

Supercoherent states, super-Kähler geometry and geometric quantization

Generalized coherent states provide a means of connecting square integrable representations of a semi-simple Lie group with the symplectic geometry of some of its homogeneous spaces. In the first part of the present work this point of view is extended to the supersymmetric context, through the study of the OSp(2/2) coherent states. These are explicitly constructed starting from the known abstract typical and atypical representations of osp(2/2). Their underlying geometries turn out to be those of supersymplectic OSp(2/2) homogeneous spaces. Moment maps identifying the latter with coadjoint orbits of OSp(2/2) are exhibited via Berezin's symbols. When considered within Rothstein's general paradigm, these results lead to a natural general definition of a super K\"ahler supermanifold, the supergeometry of which is determined in terms of the usual geometry of holomorphic Hermitian vector bundles over K\"ahler manifolds. In particular, the supergeometry of the above orbits is interpreted in terms of the geometry of Einstein-Hermitian vector bundles. In the second part, an extension of the full geometric quantization procedure is applied to the same coadjoint orbits. Thanks to the super K\"ahler character of the latter, this procedure leads to explicit super unitary irreducible representations of OSp(2/2) in super Hilbert spaces of $L^2$ superholomorphic sections of prequantum bundles of the Kostant type. This work lays the foundations of a program aimed at classifying Lie supergroups' coadjoint orbits and their associated irreducible representations, ultimately leading to harmonic superanalysis. For this purpose a set of consistent conventions is exhibited.

HOMOGENEOUS VECTOR BUNDLES AND COWEN-DOUGLAS OPERATORS

In this paper we obtain an algebraic classification of all homogeneous Hermitian holomorphic vector bundles of arbitrary rank over a bounded symmetric domain. This classification result is used in order to classify, up to unitary equivalence, all irreducible homogeneous bounded linear operators on a separable infinite-dimensional Hilbert space that belong to the Cowen-Douglas class B2 (∆), where ∆ is the open unit disk.

Bott-Chern currents, excess normal bundles and the Chern character

LetY andY′ be two complex submanifolds of a complex manifoldX, and assume thatU=Y∩Y′ is also a submanifold ofX. Let N be the excess normal bundle toU inX. We attach certain Bott-Chern currents to holomorphic Hermitian vector bundles onY, Y′ and to resolutions of the corresponding sheaves by holomorphic Hermitian complexes overX. We show that these currens verify natural functorial properties. Our results extend earlier results of Bismut-Gillet-Soule to the case where N is nonzero.

PROLONGEMENT D’UN FIBRE HOLOMORPHE HERMITIEN A COURBURE Lp SUR UNE COURBE OUVERTE

Let X be a Riemann surface and S a finite set of marked points on X. If [Formula: see text] is a hermitian holomorphic vector bundle with Lp-curvature for some p>1, we study the asymptotic behaviour of the Chern connection around the marked points; by solving directly a [Formula: see text]-problem in a weighted Sobolev space, we extend the holomorphic structure of [Formula: see text] over S to get a parabolic bundle. We deduce a proof of the classification of these hermitian metrics.

Complex geometrical interpretation of BRST algebra in topological Yang-Mills theory

Abstract We show that the BRST and anti-BRST algebra can be obtained by constructing a complex vector bundle over (M × X), where M is the real four-manifold and X an additional complex manifold. Witten's BRST algebra is derived from a hermitian holomorphic vector bundle, and the complex manifold X can be identified with the instanton moduli space.

Superconnection currents and complex immersions

Let i: M′ → M be an immersion of complex manifolds, and let (ζ, ν) be a complex of holomorphic Hermitian vector bundles on M which provides a projective resolution of the sheaf of sections of a holomorphic vector bundle η on M′. We study the convergence as u → + ∞ of a class of superconnection currents ωu introduced by Quillen, and we calculate the limit. Microlocal estimates of the speed of convergence are also given.

Koszul complexes, harmonic oscillators, and the Todd class

In this paper, we construct secondary characteristic classes associated with a short exact sequence of holomorphic Hermitian vector bundles. These secondary invariants are generalized analytic torsion forms associated with a family of elliptic operators. They are calculated in terms of Bott-Chern forms and of a certain complicated characteristic class. In a joint calculation with C. Soulé, we relate this characteristic class to the arithmetic Todd genus of Gillet and Soulé.

Analytic torsion and holomorphic determinant bundles I. Bott-Chern forms and analytic torsion

We attach secondary invariants to any acyclic complex of holomorphic Hermitian vector bundles on a complex manifold. These were first introduced by Bott and Chern [Bot C]. Our new definition uses Quillen's superconnections. We also give an axiomatic characterization of these classes. These results will be used in [BGS2] and [BGS3] to study the determinant of the cohomology of a holomorphic vector bundle.

Anomalies as curvature in complex geometry

We study two-dimensional bosonic and fermionic field theories in the language of infinite-dimensional complex geometry. We show that all local (“perturbative”) anomalies canbe interpreted as curvature of a Fock bundle over the space of complex structures. We discuss the example of the gauge anomaly of a chiral fermion in detail. A method to compute the curvature of a homogeneous hermitian holomorphic vector bundle is developed in the appendix.

Hermitian Vector Bundles and Value Distribution for Schubert Cycles

R. Bott and S. S. Chern used the theory of characteristic differential forms of a holomorphic hermitian vector bundle to study the distribution of zeroes of a holomorphic section. In this paper their methods are extended to study how often a holomorphic mapping into a Grassmann manifold hits Schubert cycles of fixed type.

Hermitian vector bundles and value distribution for Schubert cycles

R. Bott and S. S. Chern used the theory of characteristic differential forms of a holomorphic hermitian vector bundle to study the distribution of zeroes of a holomorphic section. In this paper their methods are extended to study how often a holomorphic mapping into a Grassmann manifold hits Schubert cycles of fixed type.

RC-positivity, rational connectedness and Yau’s conjecture

In this paper, we introduce a concept of RC-positivity for Hermitian holomorphic vector bundles and prove that, if $E$ is an RC-positive vector bundle over a compact complex manifold $X$, then for any vector bundle $A$, there exists a positive integer $c_A=c(A,E)$ such that $$H^0(X,\mathrm{Sym}^{\otimes \ell}E^*\otimes A^{\otimes k})=0$$ for $\ell\geq c_A(k+1)$ and $k\geq 0$. Moreover, we obtain that, on a compact K\"ahler manifold $X$, if $\Lambda^p T_X$ is RC-positive for every $1\leq p\leq \dim X$, then $X$ is projective and rationally connected. As applications, we show that if a compact K\"ahler manifold $(X,\omega)$ has positive holomorphic sectional curvature, then $\Lambda^p T_X$ is RC-positive and $H_{\bar\partial}^{p,0}(X)=0$ for every $1\leq p\leq \dim X$, and in particular, we establish that $X$ is a projective and rationally connected manifold, which confirms a conjecture of Yau([57, Problem 47]).