The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds: For instance we prove that stability and semi-stability are Zariski open properties in families when the Gauduchon degree map is a topological invariant, or when the parameter manifold is compact. Second we show that, for a generically stable family of bundles over a Kähler manifold, the Petersson-Weil form extends as a closed positive current on the whole parameter space of the family. This extension theorem uses classical tools from Yang-Mills theory developed by Donaldson (e.g. the Donaldson functional and the heat equation for Hermitian metrics on a holomorphic bundle). We apply these results to study families of bundles over a Kählerian manifold $Y$ parameterized by a non-Kählerian surface $X$, proving that such families must satisfy very restrictive conditions. These results play an important role in our program to prove existence of curves on class VII surfaces.
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tangent bundles are rational homogeneous spaces.
We generalize a construction of Hitchin to prove that, given any compact Kähler manifold $M$ with positive holomorphic sectional curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a Kähler metric with positive holomorphic sectional curvature.
The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces.
We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.
We construct an irreducible holomorphic connection with SL(2,R)-monodromy on the trivial holomorphic vector bundle of rank two over a compact Riemann surface. This answers a question of Calsamiglia, Deroin, Heu and Loray in \cite{CDHL}.
We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.
Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We…
We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, ampleness is again obtained. The methods used come from both complex differential geometry and complex algebraic geometry.
We prove the classical Nakano vanishing theorem with Hörmander $L^2$-estimates on a compact Kähler manifold using Siu's so called $\partial\dbar$-Bochner-Kodaira method, thereby avoiding the Kähler identities completely. We then introduce singular hermitian metrics on holomorphic vector bundles, and proceed to prove a vanishing theorem of Demailly-Nadel type for these in the special case where the base manifold is a Riemann surface.
Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,H)$ on $X\times M$ (the action of $G$ on $X$ is trivial), such that the corresponding Quillen determinant line bundle $({\mathcal Q}, h_Q)$, which is a $G$--equivariant holomorphic Hermitian line bundle on $M$, is isomorphic to the given $G$--equivariant holomorphic Hermitian lin…
We prove the Hodge symmetry type result on the Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in the direct sum of holomorphic line bundles. Consequently, we show the vanishing and non-vanishing of Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in holomorphic line bundles.
In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.
A vector bundle $E$ over a projective variety $M$ is called finite if it satisfies a nontrivial polynomial equation with nonnegative integral coefficients. Introducing finite bundles, Nori proved that $E$ is finite if and only if the pullback of $E$ to some finite étale covering of $M$ is trivializable \cite{No1}. The definition of finite bundles extends naturally to holomorphic vector bundles over compact complex manifolds. We prove that a holomorphic vector bundle over a compact complex manifold $M$ is finite if and only if the pullback of $E$ to some finite étale covering of $M$ is holomorp…
In the present paper, we study the properties of singular Nakano positivity of singular hermitian metrics on holomorphic vector bundles, and establish an optimal $L^2$ extension theorem for holomorphic vector bundles with singular hermitian metrics on weakly pseudoconvex Kähler manifolds. As applications, we give a necessary condition for the holding of the equality in optimal $L^2$ extension theorem, and present singular hermitian holomorphic vector bundle versions of some $L^2$ extension theorems with optimal estimate.
We calculate curvature tensors of metrics on the total spaces of holomorphic fibrations. Our main tool is a theory of Chern connections and curvature forms for possibly degenerate Hermitian forms on holomorphic vector bundles. We prove a version of the Codazzi-Griffiths equations for curvatures of sub- and quotient bundles in that setting and apply them to the study of honest Hermitian metrics on fibrations. We apply this to calculate the curvature of a metric on Grassmannian bundles and prove they have positive holomorphic sectional curvature if the base does.
In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type was introduced from the moment map framework. In this paper we establish a Kobayashi-Hitchin correspondence, that is, the equivalence of the existence of an Einstein-Hermitian metric and $ψ$-polystability of a generalized holomorphic vector bundle over a compact generalized Kahler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles and we obtain $ψ$-stable Poisson mo…
In this Note we establish a relation between sections in globally generated holomorphic vector bundles on Kähler manifolds, isotropic with respect to a non-degenerate quadratic form, and totally geodesic foliations on Euclidean open domains. We find a geometric condition for a totally geodesic foliation to originate in a holomorphic vector bundle. For codimension-two foliations, this description recovers of P. Baird and J. C. Wood. The universal objects that play a key role are the orthogonal Grassmannians.
In this paper, we introduce the notions of $α$-Hermitian-Einstein metric and $α$-stability for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include generalized holomorphic bundles on generalized Kähler manifolds. We also show that $α$-stability of a vector bundle, in this sense, can depend on the parameter $α$.
Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of $k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere $\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a class of holomorphic sections of line bundles on $J^k(t_0, \mathbb{P}^1)$.
These notes are based on a series of five lectures given at the 2009 Villa de Leyva Summer School on Geometric and Topological Methods for Quantum Field Theory. The purpose of the lectures was to give an introduction to differential-geometric methods in the study of holomorphic vector bundles on a compact connected Riemann surface.
We study critical points of holomorphic sections of $\ocal(m)$ on $\CP^n$. For quadrics, we give a complete discription of their critical points. When $n=1$, we prove a spherical Gauss-Lucas theorem. For general situation, we prove that a general section has all its critical points isolated and non-degenerate.
We study anti-holomorphic involutions of the moduli space of principal $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both $X$ and $G$. We analyze the fixed point locus in the moduli space and their relation with representations of the orbifold fundamental group of $X$ equipped with the anti-holomorphic involution. We also study the relation with branes. This generalizes work by Biswas--Garc\'ıa-Prada--Hurtubise and Baraglia--Schaposnik.
The loop space $L\mathbb{P}_1$ of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to $\mathbb{P}_1$ is an infinite dimensional complex manifold. The loop group $LPGL(2,\mathbb{C})$ acts on $L\mathbb{P}_1$ . We prove that the group of $LPGL(2,\mathbb{C})$ invariant holomorphic line bundles on $L\mathbb{P}_1$ is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of these bundles is finite dimensional, and compute the dimension for a generic bundle.
We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is described in a systematic way.
Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, ω)$. We prove that if $E$ admits a $ω$-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L. Biliotti and A. Ghigi implies the existence and uniqueness of $ω$-balanced metrics of certain direct sums of irreducible homogeneous vector bundles over rational homogeneous varieties. We finally apply our result to show the rigidity of $ω$-balanced Kaehler maps into Grassmannians.
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.
For each holomorphic vector bundle we construct a holomorphic bundle 2-gerbe that geometrically represents its second Beilinson-Chern class. Applied to the cotangent bundle, this may be regarded as a higher analogue of the canonical line bundle in complex geometry. Moreover, we exhibit the precise relationship between holomorphic and smooth gerbes. For example, we introduce an Atiyah class for gerbes and prove a Koszul-Malgrange type theorem.
We study a tower of normal coverings over a compact Kähler manifold with holomorphic line bundles. When the line bundle is sufficiently positive, we obtain an effective estimate, which implies the Bergman stability. As a consequence, we deduce the equidistribution for zero currents of random holomorphic sections. Furthermore, we obtain a variance estimate for those random zero currents, which yields the almost sure convergence under some geometric condition.
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.
A Mall bundle on a Hopf manifold H is a holomorphic vector bundle whose pullback to the universal cover of H is trivial. We define resonant and non-resonant Mall bundles, generalizing the notion of the resonance in ODE, and prove that a non-resonant Mall bundle always admits a flat holomorphic connection. We use this observation to prove a version of Poincare-Dulac linearization theorem, showing that any non-resonant invertible holomorphic contraction of a complex space is linear in appropriate holomorphic coordinates. We define the notion of resonance in Hopf manifolds, and show that all non-…
We prove an analogue of the Kobayashi-Hitchin correspondence oncompact connected 3-folds that is fibered on orbifold Riemann surfaces and satisfy an integrability condition, which contains compact connected Sasakian 3-folds. We define mini-holomorphic bundles on such 3-folds and the algebraic Dirac-type singularities on mini-holomorphic bundles, and prove that there exists a special Hermitian metric (admissible BHE-metric) on a Dirac-type singular mini-holomorphic bundle if the bundle satisfies a slope stability.
By using asymptotic Morse inequalities we give a lower bound for the space of holomorphic sections of high tensor powers in a positive line bundle over a q-concave domain. The curvature of the positive bundle induces a hermitian metric on the manifold. The bound is given explicitely in terms of the volume of the domain in this metric and a certain integral on the boundary involving the defining function and its Levi form. As application we study the perturbattion of the complex structure of a q-concave manifold.
Motivated by gauge theory under special holonomy, we present techniques to produce holomorphic bundles over certain noncompact $3-$folds, called building blocks, satisfying a stability condition `at infinity'. Such bundles are known to parametrise solutions of the Yang-Mills equation over the $\rm G_2-$manifolds obtained from asymptotically cylindrical Calabi-Yau $3-$folds studied by Kovalev and by Corti-Haskins-Nordström-Pacini et al. The most important tool is a generalisation of Hoppe's stability criterion to holomorphic bundles over smooth projective varieties $X$ with $\operatorname{Pic}{…
We investigate the symplectic geometric and differential geometric aspects of the moduli space of connections on a compact Riemann surface $X$. Fix a theta characteristic $K^{1/2}_X$ on $X$; it defines a theta divisor on the moduli space ${\mathcal M}$ of stable vector bundles on $X$ of rank $r$ degree zero. Given a vector bundle $E \in {\mathcal M}$ lying outside the theta divisor, we construct a natural holomorphic connection on $E$ that depends holomorphically on $E$. Using this holomorphic connection, we construct a canonical holomorphic isomorphism between the following two: \begin{enumer…
We study the holomorphic vector bundles E over the twistor space Tw(M) of a compact simply connected hyperkähler manifold $M$. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection π:Tw(M)\rightarrow CP^1. It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on Tw(M) of prime rank, we prove that its restriction to the generic fibre of πis stable. On the other hand,…
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 of these metrics. These examples show that there is a compact Kähler manifold $M$ and its subvariety $N$ whose codimension is greater than 1 such that there is a complete conformally Kä…
Let $A$ be a diagonal linear operator on $\C^n$, with all eigenvalues satisfying $0<|α_i|<1$, and $M = (\C^n\backslash 0)/<A>$ the corresponding Hopf manifold. We show that any stable holomorphic bundle on $M$ can be lifted to a $G$-equivariant coherent sheaf on $\C^n$, where $G=(\C^*)^l$ is a Lie group acting on $\C^n$ and containing $A$. This is used to show that all stable bundles on $M$ are filtrable, that is, admit a filtration by a sequence $F_i$ of coherent sheaves, with all subquotients $F_i/F_{i-1}$ of rank 1.
Let $X$ be a compact normal complex space of dimension $n$, and $L$ be a holomorphic line bundle on $X$. Suppose $Σ=(Σ_1,\ldots,Σ_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $τ=(τ_1,\ldots,τ_\ell)$ is an $\ell$-tuple of positive real numbers, and consider the space $H^0_0 (X, L^p)$ of global holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $τ_{j}p$ along $Σ_{j}$, $1\leq j\leq\ell$. We find necessary and sufficient conditions which ensure that $\dim H^0_0(X,L^p)\sim p^n$, analogous to Ji-Shiffman's criterion for big line bundles. W…
The purpose of this paper is first to give an asymptotic formula for the holomorphic analytic torsion forms of a fibration associated with increasing powers of a given line bundle. Secondly, we generalize this formula, thanks to the theory of Toeplitz operators, in the case where the powers of the line bundle is replaced by the direct image of powers of a line bundle on a bigger manifold. In both cases we have to make fiberwise positivity assumption on the line bundle. This results are the family versions of the results of Bimsut and Vasserot on the asymptotic of the holomorphic torsion.
We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern numbers of such a holomorphic vector bundle are nonnegative and can be bounded below and above respectively by two special Chern numbers. As applications, we obtain a family of new results on compact connected complex manifolds which are homogeneous or can be holomorphically imm…
Let $K\backslash G$ be an irreducible Hermitian symmetric space of noncompact type and $Γ\,\subset\, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kähler manifold and $ρ\, :\, π_1(X, x_0)\,\longrightarrow\, Γ$ a homomorphism such that the adjoint action of $ρ(π_1(X, x_0))$ on $\text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to $ρ$ a harmonic map $X\, \longrightarrow\, K\backslash G/Γ$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti--holomorphic.
We study the behavior of the Kähler-Ricci flow on compact manifolds developing finite-time singularities, in particular, when the flow contracts exceptional divisors or collapses the Fano fibers of a holomorphic fiber bundle. We present a technique using holomorphic vector fields to prove estimates related to the work of Song-Weinkove and Fu-Zhang.
We give a comparative description of the Poisson structures on the moduli spaces of flat connections on real surfaces and holomorphic Poisson structures on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic leaves of the latter are classified by restrictions of the bundles to certain divisors. This can be regarded as fixing a "complex analogue of the holonomy" of a connection along a "complex analogue of the boundary" in analogy with the real case.
We give a complete proof of a propagation theorem of multiplicity-free property from fibers to spaces of global sections for holomorphic vector bundles. The propagation theorem is formalised in three ways, aiming for producing various multiplicity-free theorems in representation theory for both finite and infinite dimensional cases in a systematic and synthetic manner. The key geometric condition in our theorem is an orbit-preserving anti-holomorphic diffeomorphism on the base space, which brings us to the concept of visible actions on complex manifolds.
On a generalized complex manifold there is an associated definition of a generalized holomorphic bundle, introduced by Gualtieri. This notion in the case of an ordinary complex structure yields an object which we call a co-Higgs bundle and we consider the B-field action of a closed form of type (1,1), both local and global. The effect makes contact with both Nahm's equations and holomorphic gerbes.
Let $M$ be a compact connected Fujiki manifold, $G$ a semisimple affine algebraic group over $\mathbb C$ with one simple factor and $P$ a fixed proper parabolic subgroup of $G$. For a holomorphic principal $G$--bundle $E_G$ over $M$, let ${\mathcal E}_P$ be the holomorphic principal $P$-bundle $E_G\rightarrow E_G/P$ given by the quotient map. We prove that the following three statements are equivalent: (1) ${\rm ad}(E_G)$ is numerically flat, (2) the holomorphic line bundle $\bigwedge^{\rm top} {\rm ad}({\mathcal E}_P)^*$ is nef, and (3) for every reduced irreducible compact complex analytic s…
