Holomorphic Vector Bundles Research Articles

A remark on the relative Lie algebroid connections and their moduli spaces

We investigate the relative lie algebroid connections on a holomorphic vector bundle over a family of compact complex manifolds (or smooth projective varieties over [Formula: see text]). We provide a sufficient condition for the existence of a relative Lie algebroid connection on a holomorphic vector bundle over a complex analytic family of compact complex manifolds. We show that the relative Lie algebroid Chern classes of a holomorphic vector bundle admitting relative Lie algebroid connection vanish, if each of the fibers of the complex analytic family is compact and Kähler. Moreover, we consider the moduli space of relative Lie algebroid connections and we show that there exists a natural relative compactification of this moduli space.

Twistor theory of the Chen–Teo gravitational instanton**Dedicated to Nick Woodhouse on the occasion of his 75th birthday.

Abstract Toric Ricci--flat metrics in dimension four correspond to certain holomorphic vector bundles over a
twistor space. We construct these bundles explicitly, by exhibiting and characterising their patching matrices, 
for the five--parameter family of Riemannian ALF metrics constructed by Chen and Teo. The Chen--Teo family contains
a two--parameter family of asymptotically flat gravitational instantons. The patching matrices for these 
instantons take a simple rational form.

On Finite Parts of Divergent Complex Geometric Integrals and Their Dependence on a Choice of Hermitian Metric

Let X be a reduced complex space of pure dimension. We consider divergent integrals of certain forms on X that are singular along a subvariety defined by the zero set of a holomorphic section of some holomorphic vector bundle E→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E \\rightarrow X$$\\end{document}. Given a choice of Hermitian metric on E we define a finite part of the divergent integral. Our main result is an explicit formula for the dependence on the choice of metric of the finite part.

Cohesive self-dual Yang-Mills modules over four manifolds

This research is motivated by Atiyah, Hitchin, and Singers paper Self-duality in Four-Dimensional Riemannian Geometry , which introduced a relationship between self-dual Yang-Mills fields on smooth manifolds and holomorphic vector bundles on their twistor spaces. Here, self-duality is a specific structure in 4-dimensional manifolds and Yang-Mills fields are gauge fields that satisfy Yang-Mills equations in 4-dimensions and are corresponded to the holomorphic bundles on twistor spaces. In this paper, we extend the relationship from vector bundles to a generalization of the Yang-Mills fields. To achieve this purpose, we apply Atiyah, Hitchin, and Singers theorem to cohesive modules, which was originally introduced by Block in in studying coherent sheaves over complex manifolds and the relations between homomorphic torus and its dual non-commutative torus. We introduce the notion of cohesive self-dual Yang-Mills modules and show that the twistor correspondence actually induces the equivalence between the dg category of cohesive self-dual Yang Mills modules P_(A_SD ) and the dg category of holomorphic cohesive modules P_(A_Hol ) on the twistor spaces.

New characterization of plurisubharmonic functions and positivity of direct image sheaves

abstract: We discover a new characterization of plurisubharmonic functions in terms of $L^p$ extension from one point and Griffiths positivity of holomorphic vector bundles with singular Finsler metrics in terms of $L^p$ extensions. As applications, we give a stronger result or new proof of some well-known theorems on the Griffiths positivity of the holomorphic vector bundles and their direct image sheaves associated to certain holomorphic fibrations.

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} .

Berndtsson-Lempert-Szőke fields associated to proper holomorphic families of vector bundles

Drawing on work of Berndtsson and of Lempert and Szőke, we define a complex analytic structure for families of (possibly finite-dimensional) Hilbert spaces that might not fit together to form a holomorphic vector bundle but nevertheless have a reasonable definition of curvature that agrees with the curvature of the Chern connection when the family of Hilbert spaces is locally trivial. As one consequence, we obtain a new proof of a theorem of Berndtsson on the curvature of direct images of semi-positively twisted relative canonical bundles (i.e., adjoint bundles), and of its higher-rank generalization by Liu-Yang.

Subellipticity, compactness, $H^{\epsilon}$ estimates and regularity for $\bar{\partial}$ on weakly $q$-pseudoconvex/concave domains

For a weakly q -pseudoconvex (resp. q -pseudoconcave) domain \Omega in a Stein manifold X of dimension n , we give a sufficient condition for subelliptic estimates for the \bar{\partial} -Neumann problem. Moreover, we study the compactness of the \bar{\partial} -Neumann operator N on \Omega . Such compactness estimates immediately lead to smoothness of solutions, the closed range property, the L^{2} -setting and the Sobolev estimates of N on \Omega for any \bar{\partial} -closed (r, k) -form with k \geqslant q (resp. k \leqslant q ). Furthermore, we study the \bar{\partial} -problem with support conditions in \Omega for forms of type (r, k) , with values in a holomorphic vector bundle. Applications to the \bar{\partial}_{b} -problem for smooth forms on boundaries of \Omega are given.

A Kollár-type vanishing theorem for k-positive vector bundles

Abstract Given a proper holomorphic surjective morphism f : X → Y {f:X\rightarrow Y} between compact Kähler manifolds, and a Nakano semipositive holomorphic vector bundle E on X, we prove Kollár-type vanishing theorems on cohomologies with coefficients in R q ⁢ f ∗ ⁢ ( ω X ⁢ ( E ) ) ⊗ F {R^{q}f_{\ast}(\omega_{X}(E))\otimes F} , where F is a k-positive vector bundle on Y. The main inputs in the proof are the deep results on the Nakano semipositivity of the higher direct images due to Berndtsson and Mourougane–Takayama, and an L 2 {L^{2}} -Dolbeault resolution of the higher direct image sheaf R q ⁢ f ∗ ⁢ ( ω X ⁢ ( E ) ) {R^{q}f_{\ast}(\omega_{X}(E))} , which is of interest in itself.

Curvature of new Kähler metrics on the total space of Griffiths negative vector bundle and quasi-Fuchsian space

We study Kähler metrics on the total space of Griffiths negative holomorphic vector bundles over Kähler manifolds. As an application, we construct mapping class group invariant Kähler metrics on [Formula: see text], the holomorphic tangent bundle of Teichmüller space of a closed surface [Formula: see text]. Consequently,we obtain a new mapping class group invariant Kähler metric on the quasi-Fuchsian space [Formula: see text], which extends the Weil–Petersson metric on the Teichmüller space [Formula: see text]. We also calculate its curvature and prove non-positivity for the curvature along the tautological directions.

Principal bundles with holomorphic connections over a Kähler Calabi-Yau manifold

We prove that any holomorphic vector bundle admitting a holomorphic connection, over a compact Kähler Calabi-Yau manifold, also admits a flat holomorphic connection. This addresses a particular case of a question asked by Atiyah and generalizes a result previously obtained in [6] for simply connected compact Kähler Calabi-Yau manifolds. We give some applications of it in the framework of Cartan geometries and foliated Cartan geometries on Kähler Calabi-Yau manifolds.

J-equation on holomorphic vector bundles

We introduce the J-equation on holomorphic vector bundles over compact Kähler manifolds and investigate some fundamental properties as well as examples of solutions. In particular, we provide an algebraic condition called (asymptotic) J-stability in terms of subbundles on compact Kähler surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, we discuss an application for the vector bundle version of the deformed Hermitian–Yang–Mills equation in the small volume regime.

Vector bundles on real abelian varieties

This paper is about real holomorphic vector bundles on real abelian varieties. The main result of the paper gives several conditions that are necessary and sufficient for the existence of a holomorphic connection on a real holomorphic vector bundle over a real abelian variety. Also proved is an analogue, for real abelian varieties, of a result of Simpson, which gives a criterion for a holomorphic vector bundle to arise by successive extensions of stable vector bundles with vanishing Chern classes.

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.

The Atiyah class of generalized holomorphic vector bundles

We introduce the notion of Atiyah class of a generalized holomorphic vector bundle, which captures the obstruction to the existence of generalized holomorphic connections on the bundle. As in the classical holomorphic case, this Atiyah class can be defined in three different ways: using Čech cohomology, using the first-jet short exact sequence, or adopting the Lie pair point of view.

Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds

In this paper, we study the semi-stable twisted holomorphic vector bundles over compact Gauduchon manifolds. By using Uhlenbeck–Yau's continuity method, we show that the existence of approximate Hermitian–Einstein structure and the semi-stability of twisted holomorphic vector bundles are equivalent over compact Gauduchon manifolds. As its application, we show that the Bogomolov type inequality is also valid for a semi-stable twisted holomorphic vector bundle.

Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces

Let G be a Lie group and g its Lie algebra. The aim here is to develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in g⊗g and one of general not necessarily non-degenerate quasi Hamiltonian structures relative to a not necessarily non-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of G as symmetry group involving the 2-tensor in the tensor square of the Lie algebra g measure how that bracket fails to satisfy the Jacobi identity. The present approach involves a novel concept of momentum mapping and yields, in the non-degenerate case, a bijective correspondence, in terms of explicit algebraic expressions, between non-degenerate quasi Poisson structures and non-degenerate quasi Hamiltonian structures. The new theory applies to various not necessarily non-singular moduli spaces and yields thereupon, via reduction with respect to an appropriately defined momentum mapping, not necessarily non-degenerate ordinary Poisson structures arising from the data including the symmetric 2-tensor in g⊗g. Among these moduli spaces are representation spaces, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and moduli spaces of semistable holomorphic vector bundles as well as Higgs bundle moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the symplectic part of a Kähler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. Along the way, a side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.

Expected local topology of random complex submanifolds

Let n ≥ 2 n\geq 2 and r ∈ { 1 , ⋯ , n − 1 } r\in \{1, \cdots , n-1\} be integers, M M be a compact smooth Kähler manifold of complex dimension n n , E E be a holomorphic vector bundle with complex rank r r and equipped with a Hermitian metric h E h_E , and L L be an ample holomorphic line bundle over M M equipped with a metric h h with positive curvature form. For any d ∈ N d\in \mathbb N large enough, we equip the space of holomorphic sections H 0 ( M , E ⊗ L d ) H^0(M,E\otimes L^d) with the natural Gaussian measure associated to h E h_E , h h and its curvature form. Let U ⊂ M U\subset M be an open subset with smooth boundary. We prove that the average of the ( n − r ) (n-r) -th Betti number of the vanishing locus in U U of a random section s s of H 0 ( M , E ⊗ L d ) H^0(M,E\otimes L^d) is asymptotic to ( n − 1 r − 1 ) d n ∫ U c 1 ( L ) n {n-1 \choose r-1} d^n\int _U c_1(L)^n for large d d . On the other hand, the average of the other Betti numbers is o ( d n ) o(d^n) . The first asymptotic recovers the classical deterministic global algebraic computation. Moreover, such a discrepancy in the order of growth of these averages is new and contrasts with all known other smooth Gaussian models, in particular the real algebraic one. We prove a similar result for the affine complex Bargmann-Fock model.