Hermitian Holomorphic Vector Bundles Research Articles

Characterizations of Griffiths positivity, pluriharmonicity and flatness

Deng-Ning-Wang-Zhou showed that a Hermitian holomorphic vector bundle is Griffiths semi-positive if it satisfies the optimal L2-extension condition. As a generalization, we present a quantitative characterization of Griffiths positivity in terms of certain L2-extension conditions. We also show that a R-valued measurable function is pluriharmonic if and only if it satisfies the equality part of the optimal Lp-extension condition. This answers a conjecture of Inayama affirmatively. Moreover, the flatness of a possibly singular Hermitian metric is also equivalent to the equality part of the optimal Lp-extension condition.

Compact convergence, deformation of the $L^{2}$-$\overline{\partial}$-complex and canonical $K$-homology classes

Let (X,\gamma) be a compact, irreducible Hermitian complex space of complex dimension m and with \dim(\mathrm{sing}(X))=0 . Let (F,\tau)\rightarrow X be a Hermitian holomorphic vector bundle over X , and let us denote by \overline{\eth}_{F,m,\mathrm{abs}} the rolled-up operator of the maximal L^{2} - \overline{\partial} -complex of F -valued (m,\bullet) -forms. Let \pi:M\rightarrow X be a resolution of singularities, g a metric on M , E:=\pi^{*}F and \rho:=\pi^{*}\tau . In this paper, under quite general assumptions on \tau , we prove the following equality of analytic K -homology classes [\overline{\eth}_{F,m,\mathrm{abs}}]=\pi_{*}[\overline{\eth}_{E,m}] , with \overline{\eth}_{E,m} the rolled-up operator of the L^{2} - \overline{\partial} -complex of E -valued (m,\bullet) -forms on M . Our proof is based on functional analytic techniques developed in Kuwae and Shioya (2003) and provides an explicit homotopy between the even unbounded Fredholm modules induced by \overline{\eth}_{F,m,\mathrm{abs}} and \overline{\eth}_{E,m} .

Recursive formula for covariant derivatives and geometric classification of quotient modules

Let H be a vector-valued holomorphic reproducing kernel function space and H0n be the subspace of functions vanishing to order n on an analytic hyper-surface. We determine unitary equivalence of the quotient space H⊖H0n by covariant derivatives of the curvature on the vector bundle associated to the reproducing kernel of H. The proof is a combination of some geometric and combinatorial results. In particular, we show that on a holomorphic Hermitian vector bundle, matrix representations for covariant derivatives of its curvature satisfy a recursive formula, which is of independent interest.

A geometric approach to the compressed shift operator on the Hardy space over the bidisk

This paper studies the compressed shift operator Sz on the Hardy space over the bidisk via the geometric approach. We calculate the spectrum and essential spectrum of Sz on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for Sz⁎ to be a Cowen-Douglas operator. Then we extend the concept of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and show that Sz⁎ is a generalized Cowen-Douglas operator. Moreover, we establish the connection between the reducibility of the Hermitian holomorphic vector bundle induced by kernel spaces and the reducibility of the generalized Cowen-Douglas operator. By using the geometric approach, we study the reducing subspaces of Sz on certain polynomial quotient modules.

Geometric similarity invariants of Cowen-Douglas operators

In 1978, M. J. Cowen and R.G. Douglas introduced a class of operators B_n(\Omega) (known as Cowen-Douglas class of operators) and associated a Hermitian holomorphic vector bundle to such an operator. They gave a complete set of unitary invariants in terms of the curvature and its covariant derivatives. At the same time they asked whether one can use geometric ideas to find a complete set of similarity invariants of Cowen-Douglas operators. We give a partial answer to this question. In this paper, we show that the curvature and the second fundamental form completely determine the similarity orbit of a norm dense class of Cowen-Douglas operators. As an application we show that uncountably many (non-similar) strongly irreducible operators in B_n(\mathbb{D}) can be constructed from a given operator in B_1(\mathbb{D}) . We also characterize a class of strongly irreducible weakly homogeneous operators in B_n(\mathbb{D}) .

Pointwise universal Gysin formulae and applications towards Griffiths’ conjecture

Let $X$ be a complex manifold, $(E,h)\to X$ be a rank $r$ holomorphic hermitian vector bundle, and $\rho$ be a sequence of dimensions $0 = \rho_0 < \rho_1 < \cdots < \rho_m = r$. Let $Q_{\rho,j}$, $j=1,\dots,m$, be the tautological line bundles over the (possibly incomplete) flag bundle $\mathbb{F}_{\rho}(E) \to X$ associated to $\rho$, endowed with the natural metrics induced by that of $E$, with Chern curvatures $\Xi_{\rho,j}$. We show that the universal Gysin formula \textsl{\`{a} la} Darondeau--Pragacz for the push-forward of a homogeneous polynomial in the Chern classes of the $Q_{\rho,j}$'s also hold pointwise at the level of the Chern forms $\Xi_{\rho,j}$ in this hermitianized situation. As an application, we show the positivity of several polynomials in the Chern forms of a Griffiths (semi)positive vector bundle not previously known, thus giving some new evidences towards a conjecture by Griffiths, which in turn can be seen as a pointwise hermitianized version of the Fulton--Lazarsfeld Theorem on numerically positive polynomials for ample vector bundles.

Globally conformally Kähler Einstein metrics on certain holomorphic bundles

The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally Kähler–Einstein metrics on certain Hermitian holomorphic vector bundles and their subbundles over complete Kähler–Einstein manifolds. In special cases, we give the explicit expressions of these metrics. These examples show that there are a compact Kähler manifold M and its subvariety N whose codimension is greater than 1 such that there is a complete conformally Kähler–Einstein metric on M–N.

Curvature positivity of invariant direct images of Hermitian vector bundles

We prove that the invariant part, with respect to a compact group action satisfying certain condition, of the direct image of a Nakano positive Hermitian holomorphic vector bundle over a bounded pseudoconvex domain is Nakano positive. We also consider the action of the noncompact group \({\mathbb {R}}^m\) and get the same result for a family of tube domains, which leads to a new method to the matrix-valued Prekopa’s theorem originally proved by Raufi.

A note on Griffiths' conjecture about the positivity of Chern–Weil forms

Let (E,h) be a Griffiths semipositive Hermitian holomorphic vector bundle of rank 3 over a complex manifold. In this paper, we prove the positivity of the characteristic differential form c1(E,h)∧c2(E,h)−c3(E,h), thus providing a new evidence towards a conjecture by Griffiths about the positivity of the Schur polynomials in the Chern forms of Griffiths semipositive vector bundles. As a consequence, we establish a new chain of inequalities between Chern forms. Moreover, we point out how to obtain the positivity of the second Chern form c2(E,h) in any rank, starting from the well-known positivity of such form if (E,h) is just Griffiths positive of rank 2. The final part of the paper gives an overview on the state of the art of Griffiths' conjecture, collecting several remarks and open questions.

Homogeneous Hermitian holomorphic vector bundles and operators in the Cowen-Douglas class over the poly-disc

In this article, we obtain two sets of results. The first set of results are for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc.A classification of irreducible hermitian holomorphic vector bundles over D2, homogeneous with respect to Möb×Möb, is obtained assuming that the associated representations are multiplicity-free. Among these the ones that give rise to an operator in the Cowen-Douglas class of D2 of rank 1,2 or 3 are determined.Any hermitian holomorphic vector bundle of rank 2 over Dn, homogeneous with respect to the n-fold direct product of the group Möb is shown to be a tensor product of n hermitian holomorphic vector bundles over D. Among them, n−1 are shown to be the line bundles and one is shown to be a rank 2 bundle. Also, each of the bundles are homogeneous with respect to Möb.The classification of irreducible homogeneous hermitian holomorphic vector bundles over D2 of rank 3 (as well as the corresponding Cowen-Douglas class of operators) is extended to the case of Dn, n>2.It is shown that there is no irreducible n - tuple of operators in the Cowen-Douglas class B2(Dn) that is homogeneous with respect to Aut(Dn), n>1. Also, pairs of operators in B3(D2) homogeneous with respect to Aut(D2) are produced, while it is shown that no n - tuple of operators in B3(Dn) is homogeneous with respect to Aut(Dn), n>2.

The metrics of Hermitian holomorphic vector bundles and the similarity of Cowen-Douglas operators

The metrics of Hermitian holomorphic vector bundles and the similarity of Cowen-Douglas operators

Characterizations of curvature positivity of Riemannian vector bundles and convexity or pseudoconvexity of bounded domains in [formula omitted] or [formula omitted] in terms of L2-estimate of d or [formula omitted] equation

The first main result is a characterization of Nakano positivity of Riemannian vector bundles over bounded domains in terms of solvability of the d-equation with certain good L2 estimate condition. As an application, we give an alternative proof of the matrix-valued Prekopa's theorem that is originally proved by Raufi. The second main result contains a characterization of convexity of smoothly bounded domains in Rn in terms L2 estimate condition for the d-equation, and a characterization of pseudoconvexity of smoothly bounded domains in Cn in terms L2 estimate condition for the ∂¯-equation. Our methods are inspired by the recent works of Deng-Ning-Wang-Zhou on characterization of Nakano positivity of Hermitian holomorphic vector bundles and positivity of direct image sheaves associated to holomorphic fibrations.

On unitary invariants of quotient Hilbert modules along smooth complex analytic sets

Let Ω⊂Cm be an open, connected and bounded set and A(Ω) be a function algebra of holomorphic functions on Ω. Suppose that M is a reproducing kernel Hilbert module over A(Ω). In this article, we first obtain a model for the quotient Hilbert modules obtained from submodules of functions in M vanishing to order k along a smooth irreducible complex analytic set Z⊂Ω of codimension at least 2, assuming that the Hilbert module M is in the Cowen-Douglas class over Ω. This model is used to show that such a quotient module happens to be in the Cowen-Douglas class over Z∩Ω which then enables us to determine unitary equivalence classes of the aforementioned quotient modules in terms of the geometric invariants of hermitian holomorphic vector bundles. As an application, we obtain that unitary equivalence classes of a large family of these Hilbert modules are completely determined by those of certain quotient modules of the above kind.

Factorization of generalized holomorphic curve and homogeneity of operators

The present paper concerns the homogeneity and similarity of operators in Cowen-Douglas class $$B_n(\Omega )$$ . Let E be the Hermitian holomorphic vector bundle induced by $$T\in B_n(\mathbb {D})$$ , and $$E_{\alpha }$$ be the Hermitian holomorphic vector bundle induced by $$\phi _{\alpha }(T)$$ , where $$\phi _{\alpha }$$ is a M $$\ddot{o}$$ bius transformation of the unit disk $$\mathbb {D}$$ . Assume that the holomorphic Hermitian vector bundle $$E_{\alpha }$$ is congruent to $$E\otimes \mathcal {L}_{\alpha }$$ for some line bundle $$\mathcal {L}_{\alpha }$$ over $$\mathbb {D}$$ , for each $$\alpha \in \mathbb {D}$$ . Then it is shown that $$\mathcal {L}_{\alpha }$$ must be the trivial bundle and T is homogeneous. Furthermore, we investigate the similarity of operators with Fredholm index n associate with Hermitian holomorphic bundles. This characterization is given in terms of the factorization of generalized holomorphic curve induced by the corresponding holomorphic bundles.

Analytic Torsion for Surfaces with Cusps I: Compact Perturbation Theorem and Anomaly Formula

Let $\overline{M}$ be a compact Riemann surface and let $g^{TM}$ be a metric over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite set of points. We suppose that $g^{TM}$ is equal to the Poincar\'e metric over a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the twisted canonical line bundle $\omega_M(D)$ with the induced Hermitian norm $\|\cdot\|_M$ over $\overline{M} \setminus D_M$. Let $(\xi, h^{\xi})$ be a holomorphic Hermitian vector bundle over $\overline{M}$. In this article we define the analytic torsion $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ associated with $(M, g^{TM})$ and $(\xi \otimes \omega_M(D)^n, h^{\xi} \otimes \|\cdot\|_M^{2n})$ for $n \leq 0$. We prove that $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ is related to the analytic torsion of non-cusped surfaces. Then we prove the anomaly formula for the associated Quillen norm. The results of this paper will be used in the sequel to study the regularity of the Quillen norm and its asymptotics in a degenerating family of Riemann surfaces with cusps and to prove the curvature theorem. We also prove that our definition of the analytic torsion for hyperbolic surfaces is compatible with the one obtained through Selberg trace formula by Takhtajan-Zograf.

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.

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 Lie algebra on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. Our first main result is the construction of an explicit differential operator intertwining the bundle with the direct sum of its factors. Next, we study Hilbert spaces of sections of these bundles. We use this to get, in particular, a full description and a similarity theorem for homogeneous n-tuples of operators in the Cowen-Douglas class of the Euclidean unit ball in Cn.

Curvature Formulas of Holomorphic Curves on $$C^*$$ C ∗ -Algebras and Cowen–Douglas Operators

For \(\Omega \subseteq \mathbb {C}\) a connected open set, and \({\mathcal {U}}\) a unital \(C^*\)-algebra, let \({\mathcal {I}} ({\mathcal {U}})\) and \({\mathcal {P}}({\mathcal {U}})\) denote the sets of all idempotents and projections in \({\mathcal {U}}\) respectively. A set \({\mathcal {P}}({\mathcal {U}})\) is called the Grassmann manifold of \(\mathcal {U}\) and \({\mathcal {I}} ({\mathcal {U}})\) is called the extended Grassmann manifold. If \(P:\Omega \rightarrow {\mathcal {P}}({\mathcal {U}})\) is a real-analytic \({\mathcal {U}}\)-valued map which satisfies \(\overline{\partial } PP=0\), then P is called a holomorphic curve on \({\mathcal {P}}({\mathcal {U}})\). In this note, we will define the formulas of curvature and it’s covariant derivatives for holomorphic curves on \(C^*\)-algebras. It can be regarded as a generalization of curvature and it’s covariant derivatives from complex geometry. By using the curvature formula, we give the unitarily and similarity classification theorems for the holomorphic curves and extended holomorphic curves on \(C^*\)-algebras respectively. Furthermore, we also give a description of the trace of the covariant derivatives of curvature for any Hermitian holomorphic vector bundles. Applications include the similarity of holomorphic Hermitian vector bundles and Cowen–Douglas operators.

A local theory for operator tuples in the Cowen–Douglas class

We present a local theory for a commuting m-tuple S=(S1,S2,⋯,Sm) of Hilbert space operators lying in the Cowen–Douglas class. By representing S on a Hilbert module M consisting of vector-valued holomorphic functions over Cm, we identify and study the localization of S on an analytic hyper-surface in Cm. We completely determine unitary equivalence of the localization and relate it to geometric invariants of the Hermitian holomorphic vector bundle associated to S. It turns out that the localization coincides with an important class of quotient Hilbert modules, and our result concludes its classification problem in full generality.

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.