Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles. I: generalities and the one-dimensional case

We review the notions of (weak) Hermitian–Yang–Mills structure and approximate Hermitian–Yang–Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact K?hler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be written in terms of the mean curvature of the Hitchin–Simpson connection. We also study some properties of the solutions of the evolution equation associated with that functional. Next, we study the problem of the existence of approximate Hermitian–Yang–Mills structures and its relation with the algebro-geometric notion of semistability and we show that for a compact Riemann surface, the notion of approximate Hermitian–Yang–Mills structure is in fact the differential- geometric counterpart of the notion of semistability. Finally, we review the notion of admissible Hermitian structure on a torsion-free Higgs sheaf and define the Donaldson functional for such an object.     

Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles II: Higgs sheaves and admissible structures

We study the basic properties of Higgs sheaves over compact Kähler manifolds and establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we prove some properties of semistable Higgs sheaves.     

HERMITIAN-EINSTEIN METRICS FOR HIGGS BUNDLES OVER COMPLETE HERMITIAN MANIFOLDS

In this paper, we solve the Dirichlet problem for the Hermitian-Einstein equations on Higgs bundles over compact Hermitian manifolds. Then we prove the existence of the Hermitian-Einstein metrics on Higgs bundles over a class of complete Hermitian manifolds.     

Some curvature properties of complex surfaces

Summary In this paper, we are investigating curvature properties of complex two-dimensional Hermitian manifolds, particularly in the compact case. To do this, we start with the remark that the fundamental form of such a manifold is integrable, and we use the analogy with the locally conformal KÄhler manifolds, which follows from this remark. Among the obtained results, we have the following: a compact Hermitian surface for which either the Riemannian curvature tensor satisfies the KÄhler symmetries or the Hermitian curvature tensor satisfies the Riemannian Bianchi identity is KÄhler; a compact Hermitian surface of constant sectional curvature is a flat KÄhler surface; a compact Hermitian surface M with nonnegative nonidentical zero holomorphie Hermitian bisectional curvature has vanishing plurigenera, c₁(M) 0, and no exceptional curves; a compact Hermitian surface with distinguished metric, and positive integral Riemannian scalar curvature has vanishing plurigenera, etc.     

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.     

The log-term of the Bergman kernel of the disc bundle over a homogeneous Hodge manifold

We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kähler–Einstein manifold of classical type. Our results extends that in (Engli? and Zhang, Math Z 264(4):901–912, 2010) for the case of Hermitian symmetric spaces of compact type.     

Small Eigenvalues of the Laplacians on Vector Bundles over Towers of Covers

Under the conditions that a compact Riemannian manifold is of sufficiently pinched negative sectional curvature and that a smooth Hermitian vector bundle over the manifold is also of sufficiently small curvature, we prove some pinching results on the asymptotic behavior of the numbers of small eigenvalues of the Laplacians on the induced Hermitian vector bundles over a tower of covers of the manifold. In the process we also obtain interesting results on the non-existence of square integrable 'almost harmonic' vector bundle-valued forms omitting the middle degree(s) on the universal cover. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonorientable Manifolds, Complex Structures, and Holomorphic Vector Bundles

A generalization of the notion of almost complex structure is defined on a nonorientable smooth manifold M of even dimension. It is defined by giving an isomorphism J from the tangent bundle TM to the tensor product of the tangent bundle with the orientation bundle such that JJ=–Id _TM . The composition JJ is realized as an automorphism of TM using the fact that the orientation bundle is of order two. A notion of integrability of this almost complex structure is defined; also the Kähler condition has been extended. The usual notion of a complex vector bundle is generalized to the nonorientable context. It is a real vector bundle of even rank such that the almost complex structure of a fiber is given up to the sign. Such bundles have generalized Chern classes. These classes take value in the cohomology of the tensor power of the local system defined by the orientation bundle. The notion of a holomorphic vector bundle is extended to the context under consideration. Stable vector bundles and Einstein–Hermitian connections are also generalized. It is shown that a generalized holomorphic vector bundle on a compact nonorientable Kähler manifold admits an Einstein–Hermitian connection if and only if it is polystable.     

Reduction theorems for principal and classical connections

We prove general reduction theorems for gauge natural operators transforming principal connections and classical linear connections on the base manifold into sections of an arbitrary gauge natural bundle. Then we apply our results to the principal prolongation of connections. Finally we describe all such gauge natural operators for some special cases of a Lie group G.     

Compactness theorems for coupled Yang-Mills fields

In this paper, we consider coupled Yang-Mills fields on vector bundle E over compact Riemannian manifold M. Under appropriate conditions on the curvature and the Higgs field, two compactness theorems are proved.     

Vanishing cohomology theorems and stability of complex analytic foliations

In this paper we deal with a complex analytic foliation of a compact complex manifold endowed with a bundle-like metric and give a transversally holomorphic rigidity theorem (Theorem 9.1) for these foliations, depending on curvature conditions. We give some examples for which we study holomorphic rigidity. The classical vanishing theorems of Nakano, Griffiths and Le Potier are the main tools we use to prove our results.     

复Finsler流形上的Hodge-Laplace算子

本文定义了强拟凸复Finsler流形上的Hodge-Laplace算子,并给出其水平部分的局部坐标表示．     

Notes on the Sasaki metric

We survey on the geometry of the tangent bundle of a Riemannian manifold, endowed with the classical metric established by S. Sasaki 60 years ago. Following the results of Sasaki, we try to write and deduce them by different means. Questions of vector fields, mainly those arising from the base, are related as invariants of the classical metric, contact and Hermitian structures. Attention is given to the natural notion of extension or complete lift of a vector field, from the base to the tangent manifold. Few results are original, but finally new equations of the mirror map are considered.     

Geodesics of Sasakian Metrics on Tensor Bundles

On the basis of the phase completion the notion of vertical and horizontal lifts of vector fields is defined in the tensor bundles over a Riemannian manifold. Such a tensor bundle is made into a manifold with a Riemannian structure of special type by endowing it with Sasakian metric. The components of the Levi-Civita and other metric connections with respect to Sasakian metrics on tensor bundles with respect to the adapted frame are presented. This having been done, it is shown that it is possible to study geodesics of Sasakian metrics dealing with geodesics of the base manifolds. Dedicated to the memory of Vladimir Vishnevskii (1929-2007)     

Deformations of Compact Complex Manifolds with Ample Canonical Bundles*

In this paper, the author discusses the deformations of compact complex manifolds with ample canonical bundles. It is known that a complex manifold has unobstructed deformations when it has a trivial canonical bundle or an ample anti-canonical bundle.When the complex manifold has an ample canonical bundle, the author can prove that this manifold also has unobstructed deformations under an extra condition.     

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szegö kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kähler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szegö kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Engli? (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.     

Asymptotically Holomorphic Families of Symplectic Submanifolds

We construct a wide range of symplectic submanifolds in a compact symplectic manifold as the zero sets of asymptotically holomorphic sections of vector bundles obtained by tensoring an arbitrary vector bundle by large powers of the complex line bundle whose first Chern class is the symplectic form. We also show that, asymptotically, all sequences of submanifolds constructed from a given vector bundle are isotopic. Furthermore, we prove a result analogous to the Lefschetz hyperplane theorem for the constructed submanifolds. Submitted: December 1996, final version: October 1997     

Stable Higgs bundles with trivial Chern classes. Several examples

We propose a new interpretation of Higgs bundles as vector bundles together with actions of the symmetric tangent tensor algebra. Via this interpretation, we construct new nontrivial examples of stable Higgs bundles with vanishing Chern classes. To the memory of Professor V.A. Iskovskikh     

Applications harmoniques, applications pluriharmoniques et existence de 2-formes parallèles non nulles

In this paper we study harmonic maps from a compact riemannian manifold equiped with a non trivial parallel 2-form, to a K?hler manifold of strongly negative curvature tensor or a riemannian manifold of strictly negative complex sectional curvature. In a first part we set up some rigidity results of Siu type. Then we obtain an upper bound for the rank of such maps in terms of the rank of the 2-form and deduce some vanishing theorems. Received: March 4, 1996     

Traceless component of the conformal curvature tensor in Kähler manifold

We investigate the traceless component of the conformal curvature tensor defined by (2.1) in Kähler manifolds of dimension ? 4, and show that the traceless component is invariant under concircular change. In particular, we determine Kähler manifolds with vanishing traceless component and improve some theorems (for example, [4, pp. 313–317]) concerning the conformal curvature tensor and the spectrum of the Laplacian acting on p (0 ? p ? 2)-forms on the manifold by using the traceless component.