Compact Hermitian Manifolds Research Articles

The universal Kobayashi-Hitchin correspondence on Hermitian manifolds

We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds. This correspondence refers to moduli spaces of holomorphic oriented pairs. Most of the classical moduli problems in complex geometry (e. g. holomorphic bundles with reductive structure groups, holomorphic pairs, holomorphic Higgs pairs, Witten triples, arbitrary quiver moduli problems) are special cases of this universal classification problem. Our Kobayashi-Hitchin correspondence relates the complex geometric concept polystable oriented holomorphic pair to the existence of a reduction solving a generalized Hermitian-Einstein equation. The proof is based on the Uhlenbeck-Yau continuity method. We also investigate metric properties of the moduli spaces in this general non-Kaehlerian framework. We discuss in more detail moduli spaces of oriented connections, Douady Quot spaces and moduli spaces of non-abelian monopoles.

Existence of K-limits of holomorphic maps

Let D be a complete hyperbolic domain in ℂn, n >1, and N a compact Hermitian manifold. We prove a criterion for the existence of the K-limit of an arbitrary holomorphic mapf: D → N at an arbitrary boundary point D under the condition that f has the corresponding radial limit at this point.

Hermitian–Einstein metrics on holomorphic vector bundles over Hermitian manifolds

In this paper, we prove the long-time existence of the Hermitian–Einstein flow on a holomorphic vector bundle over a compact Hermitian (non-Kähler) manifold, and solve the Dirichlet problem for the Hermitian–Einstein equations. We also prove the existence of Hermitian–Einstein metrics for holomorphic vector bundles on a class of complete non-compact Hermitian manifolds.

Estimation effective de la perte de positivité dans la régularisation des courants

Let ( X, ω) be a compact complex Hermitian manifold, and let T⩾ γ be a d-closed (1,1) almost positive current on X. A variant of Demailly's regularization-of-currents theorem states that T is the weak limit of a sequence of (1,1)-currents T m with analytic singularities of coefficient 1/ m, lying in the same cohomology class as T, whose Lelong numbers converge to those of T, and with a loss of positivity decaying to zero. We prove that if the (1,1)-form γ is assumed to be closed and C ∞, the regularizing currents T m can be chosen such that T m⩾γ− C m for a constant C>0 independent of m. To cite this article: D. Popovici, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Spectral theory for tensor products of Hermitian holomorphic line bundles

Let L be a quasipositive line bundle over a compact complex Hermitian manifold. It has been conjectured that the smallest positive eigenvalue of the Laplacian, acting on the k′th tensor power of L, is uniformly bounded independent of k. Various special cases of the conjecture are proved, but a counterexample is given to the general conjecture.

ON THE GROWTH OF HOLOMORPHIC PROJECTIVE FOLIATIONS

In the theory of real (non-singular) foliations, the study of the growth of the leaves has proved to be useful in the comprehension of the global dynamics as the existence of compact leaves and exceptional minimal sets. In this paper we are interested in the complex version of some of these basic results. A natural question is the following: What can be said of a codimension one (possibly singular) holomorphic foliation on a compact hermitian manifold M exhibiting subexponential growth for the leaves? One of the first examples comes when we consider the Fubini–Study metric on [Formula: see text] and dimension one foliations. In this case, under some non-degeneracy hypothesis on the singularities, we may classify the foliation as a linear logarithmic foliation. In particular, the limit set of ℱ is a union of singularities and invariant algebraic curves. Applications of this and other results we prove are given to the general problem of uniformization of the leaves of projective foliations.

The Abelian Vortex Equations on Hermitian Manifolds

We study the vortex equations for line bundles on compact Hermitian manifolds. We obtain a description of the moduli space of solutions of these equations in terms of effective divisors, which generalizes the known result in the Kahler case [B1].

Vanishing theorems andstring backgrounds

We show various vanishing theorems for the cohomology groups ofcompact Hermitian manifolds for which the Bismut connection has a(restricted) holonomy contained in SU(n) and classify all such manifoldsof dimension four. In this way we provide necessary conditions for theexistence of such structures on compact Hermitian manifolds withvanishing first Chern class of non-Kähler type. Then we apply ourresults to solutions of the string equations and show that such solutionsadmit various cohomological restrictions such as, for example, that undercertain natural assumptions the plurigenera vanish. We also find thatunder some assumptions the string equations are equivalent to thecondition that a certain vector is parallel with respect to the Bismutconnection.

On Grassmannians associated with JB*-triples

Then Γ is a complex Lie group that has precisely n+ 1 orbits E0, . . . , En in E – each Er is the (locally closed) complex submanifold of all matrices of rank r. Clearly, Er−1 is in the closure of Er for every 1 ≤ r ≤ n and En is the unique open orbit. For every pair of integers p, q ≥ 0 let Gp,q be the Grassmannian of all p-planes in Cp+q. Then Gp,q is a compact symmetric hermitian manifold of dimension pq holomorphically equivalent to Gq,p. Every Er is in a canonical way a Γ -equivariant holomorphic fibre bundle

Harmonic and holomorphic 1-forms on compact balanced Hermitian manifolds

On compact balanced Hermitian manifolds we obtain obstructions to the existence of harmonic 1-forms, ∂-harmonic (1,0)-forms and holomorphic (1,0)-forms in terms of the Ricci tensors with respect to the Riemannian curvature and the Hermitian curvature. Necessary and sufficient conditions the (1,0)-part of a harmonic 1-form to be holomorphic and vice versa, a real 1-form with a holomorphic (1,0)-part to be harmonic are found. The vanishing of the first Dolbeault cohomology groups of the twistor space of a compact irreducible hyper-Kähler manifold is shown.

HOLOMORPHIC AND KILLING VECTOR FIELDS ON COMPACT BALANCED HERMITIAN MANIFOLDS

We extend the classical theorems for nonexistence of Killing and holomorphic vector fields on compact Kähler manifolds to compact balanced Hermitian manifolds. We express the obstructions to the existence of Killing and holomorphic vector fields on compact balanced Hermitian manifolds in terms of the Ricci tensors of the Levi–Civita connection as well as in terms of the Ricci tensors of the Chern connection. We show that every affine vector field with respect to the Chern connection on a compact balanced Hermitian manifold is holomorphic.

Holomorphic Torus Actions on Compact Locally Conformal Kähler Manifolds

Given a torus action (T2, M) on a smooth manifold, the orbit map evx(t)=t·xfor each x∈Minduces a homomorphism ev*: ℤ2→H1(M;ℤ). The action is said to be Rank-kif the image of ev*has rank k(≤2) for each point of M. In particular, if ev*is a monomorphism, then the action is called homologically injective. It is known that a holomorphic complex torus action on a compact Kahler manifold is homologically injective. We study holomorphic complex torus actions on compact non-Kahler Hermitian manifolds. A Hermitian manifold is said to be a locally conformal Kahler if a lift of the metric to the universal covering space is conformal to a Kahler metric. We shall prove that a holomorphic conformal complex torus action on a compact locally conformal Kahler manifold Mis Rank-1 provided that Mhas no Kahler structure.

Hermitian harmonic maps from complete Hermitian manifolds to complete Riemannian manifolds

In this paper we study a nonlinear elliptic system of equations imposed on a map from a complete Hermitian (non-Kahler) manifold to a Riemannian manifold. This system is more appropriate to Hermitian geometry than the harmonic map system since it is compatible with the holomorphic structure of the domain manifold in the sense that holomorphic maps are Hermitian harmonic maps. It was first studied by Jost and Yau in [J-Y], and was applied to study the rigidity of compact Hermitian manifolds. We extend their existence and uniqueness results to the case where both domain and target manifolds are complete. Hopefully the results will be useful to study corresponding rigidity of complete Hermitian manifolds. Let M be a complex manifold with Hermitian metric (hαβ), and let N be a Riemannian manifold with metric (gij) and Christoffel symbols Γ i jk. A Hermitian harmonic map u : M → N satisfies the following elliptic system

Einstein metrics and stability for flat connections on compact Hermitian manifolds, and a correspondence with Higgs operators in the surface case

A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D -invariant subbundle (resp. if it is the D -invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E,D) in this sense is equivalent to the existence of a so-called harmonic metric in E . In this paper we consider flat complex vector bundles on compact Hermitian manifolds (X,g) . We propose new notions of g -(poly-)stability of such bundles, and of g -Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if g is a Kähler metric, but are different in general. Our main result is that the g -polystability in our sense is equivalent to the existence of a g -Hermitian-Einstein metric. Our notion of a g -Einstein metric in a flat bundle is motivated by a correspondence between flat bundles and Higgs bundles over compact surfaces, analogous to the correspondence in the case of Kähler manifolds [S1], [S2], [S3].

Variation de la L2 r-norme de la 1-forme de torsion sur une variété hermitienne compacte

Let ( M, g) be a complex compact hermitian manifold with dimension complex dim C = m ≥ 2, we study in this paper the critical points of the functional: Φ = 1 2 r ∫ M ¦θ¦ 2r υ g in the set of the hermitian metrics with total volume equal to one, where θ is 1 — form of torsion of the Chern connexion and r is a real number such that 1 ≤ r ≤ m. We show that the critical points of this functional are exactly a semi-kählerian metric.

On some variational problems for 2-dimensional Hermitian metrics

Let (M, J, g) be a compact complex 2-dimensional Hermitian manifold with the Kahler form Ω, and the torsion 1-form ω defined by dΩ = ω ∧ Ω. In this note we obtain the Euler-Lagrange equations for the variational functionals defined by ‖ω‖2 and ‖dω‖2, whereg runs in the space of all the Hermitian metrics onM. In the first case, the extremals are precisely the Kahler metrics [Gd]. In the second case, we also write down a formula for the second variation.

Compact Hermitian manifolds of constant holomorphic sectional curvature

Compact Hermitian manifolds of constant holomorphic sectional curvature

Volumes of certain small geodesic balls and almost-Hermitian geometry

Let D be the characteristic connection of an almost-Hermitian manifold, V D m (r) the volume of a small geodesic ball for the connection D and C C D 1 the first non-trivial term of the Taylor expansion of V D m (r). NK-manifolds are characterized in terms of C C D 1 and a family of Hermitian manifolds for which ∫ M C C D 1 dvol is a spectral invariant is given and one proves that C C D 1 and the spectrum of the complex Laplacian, together, determine the class in which a compact Hermitian manifold lines.

On the curvature of compact Hermitian manifolds

On the curvature of compact Hermitian manifolds

Families of negatively curved Hermitian manifolds

A complex analytic family of compact hermitian manifolds has negative holomorphic sectional curvature in a neighborhood of any fibre having negative holomorphic sectional curvature.