Connections and Higgs fields on a principal bundle

Let M be a compact connected Kähler manifold and G a connected linear algebraic group defined over \({\mathbb{C}}\) . A Higgs field on a holomorphic principal G-bundle ε _G over M is a holomorphic section θ of \(\text{ad}(\epsilon_{G})\otimes {\Omega}^{1}_{M}\) such that θ∧ θ = 0. Let L(G) be the Levi quotient of G and (ε _G (L(G)), θ _l ) the Higgs L(G)-bundle associated with (ε _G , θ). The Higgs bundle (ε _G , θ) will be called semistable (respectively, stable) if (ε _G (L(G)), θ _l ) is semistable (respectively, stable). A semistable Higgs G-bundle (ε _G , θ) will be called pseudostable if the adjoint vector bundle ad(ε _G (L(G))) admits a filtration by subbundles, compatible with θ, such that the associated graded object is a polystable Higgs vector bundle. We construct an equivalence of categories between the category of flat G-bundles over M and the category of pseudostable Higgs G-bundles over M with vanishing characteristic classes of degree one and degree two. This equivalence is actually constructed in the more general equivariant set-up where a finite group acts on the Kähler manifold. As an application, we give various equivalent conditions for a holomorphic G-bundle over a complex torus to admit a flat holomorphic connection.     

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant Members of VBAC (Vector Bundles on Algebraic Curves). Second and Third authors partially supported by Ministerio de Educación y Ciencia and Conselho de Reitores das Universidades Portuguesas through Acción Integrada Hispano-Lusa HP2002-0017 (Spain)/E–30/03 (Portugal). First and Second authors partially supported by Ministerio de Educación y Ciencia (Spain) through Project MTM2004-07090-C03-01. Third author partially supported by the Centro de Matemática da Universidade do Porto and the project POCTI/MAT/58549/2004, financed by FCT (Portugal) through the programmes POCTI and POSI of the QCA III (2000–2006) with European Community (FEDER) and national funds. The second author visited the IHES with the partial support of the European Commission through its 6th Framework Programme “Structuring the European Research Area” and the Contract No. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action     

A Higgs bundle on a Hermitian symmetric space,II

Let M : = Γ\G/K be the quotient of an irreducible Hermitian symmetric space G/K by a torsionfree cocompact lattice G ì G{\Gamma \subset G}. Let V be a complex irreducible representation of G. We give a Hodge decomposition of the cohomology of the Γ-module V in terms of the cohomologies of automorphic vector bundles on M associated to the Lie algebra cohomologies H*(\mathfrak p⁺ ,V){H*({\mathfrak p}^{+} ,V)}.     

Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces

We obtain the full classification of coisotropic and polar isometric actions of compact Lie groups on irreducible Hermitian symmetric spaces.

     

The volume entropy of local Hermitian symmetric space of noncompact type

We calculate the volume entropy of local Hermitian symmetric spaces of noncompact type in terms of its invariant r, a, b.     

Hermitian-Poisson Metrics on Flat Bundles over Complete Hermitian Manifolds

In this paper, the author solves the Dirichlet problem for Hermitian-Poisson metric equation √?1Λ_ωG_H = λId and proves the existence of Hermitian-Poisson metrics on flat bundles over a class of complete Hermitian manifolds. When λ = 0, the HermitianPoisson metric is a Hermitian harmonic metric.     

Polar actions on Hermitian and quaternion-Kähler symmetric spaces

We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classification of polar actions on Hermitian symmetric spaces. In the second part we prove that polar actions on Wolf spaces are quaternion-coisotropic and that isometric actions on these spaces admit an orbit of special type, analogous to the existence of a complex orbit for an isometric action on a compact homogeneous simply connected Kähler manifold.     

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, ${{\rm Sp}(2,\mathbb {R})}Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, or SO(p, 2) with p ≥ 3. The symmetric spaces associated to these G’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of K?hler manifolds and Higgs bundles we study representations of the lattice Г into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G = SU(p, 2) with p ≥ 2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p, 2)/S(U(p) × U(2)), on which it acts cocompactly.     

Spheres in Hermitian Symmetric Spaces and Flag Manifolds

This paper contains a proof of the following property of compact irreducible Hermitian symmetric spaces. If H=G/K where G is a compact simply connected simple Lie group, T a maximal torus of G and F(T,H)=|E ₁,...,E _m is the fixed point set of T on H, then for each pair E _i , E _j there is a 2-dimentional sphere N _ij H such that E _i and E _j are antipodal points of N _ij.     

The stringy E‐function of the moduli space of Higgs bundles with trivial determinant

Let M be the moduli space of semistable rank 2 Higgs pairs (V, ϕ) with trivial determinant over a smooth projective curve X of genus g ≥ 2. We provide an explicit formula for the stringy E‐function of M . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

In this paper we study the space of effective -cycles in with the homology class equal to an integral multiple of the homology class of Schubert variety of type . When is a proper linear subspace of a linear space in , we know that is already complicated. We will show that for a smooth Schubert variety in a Hermitian symmetric space, any irreducible subvariety with the homology class , , is again a Schubert variety of type , unless is a non-maximal linear space. In particular, any local deformation of such a smooth Schubert variety in Hermitian symmetric space is obtained by the action of the Lie group .

     

On gap rigidity problems for compact Hermitian symmetric spaces

We prove a gap rigidity theorem for diagonal curves in maximal polyspheres of irreducible compact Hermitian symmetric spaces of tube type, which is a dual analog to a theorem obtained by Mok. Motivated by the proof we show that the Chow space of certain totally geodesic submanifolds is affine algebraic, which gives a weaker version of gap rigidity for a class of higher dimensional submanifolds.     

Hermitian Yang-Mills Metrics on Higgs Bundles over Asymptotically Cylindrical Kähler Manifolds

Let V be an asymptotically cylindrical Kahler manifold with asymptotic cross-section     

The Holomorphic 2-Number of a Hermitian Symmetric Space

This paper contains a proof a priori (i.e. independent of the classification of Hermitian symmetric spaces) of a theorem on the holomorphic 2-number of a Hermitian symmetric space. If N=G/K is a Hermitian symmetric space, where G is a compact simply connected simple Lie group, T a maximal torus of G and F(T,N) = E₁,... , E_m is the fixed point set of T in N, then for each pair E_i, E_j there is a two-dimensional sphere N_ij N such that E_i and E_j are antipodal points of N_ij.     

Several variants of the Hermitian and skew‐Hermitian splitting method for a class of complex symmetric linear systems

This paper is concerned with several variants of the Hermitian and skew‐Hermitian splitting iteration method to solve a class of complex symmetric linear systems. Theoretical analysis shows that several Hermitian and skew‐Hermitian splitting based iteration methods are unconditionally convergent. Numerical experiments from an n‐degree‐of‐freedom linear system are reported to illustrate the efficiency of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.     

Global nonautonomous Schrodinger flows on Hermitian locally symmetric spaces

In this paper, we consider the global existence of one-dimensional nonautonomous (inhomogeneous) Schrodinger flow. By exploiting geometric symmetries, we prove that, given a smooth initial map, the Cauchy problem of the nonautonomous (inhomogeneous) Schrodinger flow from S1 into a Hermitian locally symmetric space admits a unique global smooth solution, and then we address the global existence of the Cauchy problem of inhomogeneous Heisenberg spin ferromagnet system.     

Reducibility of generalized Verma modules for Hermitian symmetric pairs

In this article, we give some practical criteria to determine the reducibility of generalized Verma modules (induced from finite-dimensional modules) in the Hermitian symmetric case. Our criteria are given by the information of the corresponding highest weights of the finite-dimensional modules and relatively easy to verify. This article is inspired by earlier work of Kubo about Jantzen's criterion.     

Global nonautonomous Schr?dinger flows on Hermitian locally symmetric spaces

In this paper, we consider the global existence of one-dimensional nonautonomous (inhomogeneous) Schr?dinger flow. By exploiting geometric symmetries, we prove that, given a smooth initial map, the Cauchy problem of the nonautonomous (inhomogeneous) Schr?dinger flow from S¹ into a Hermitian locally symmetric space admits a unique global smooth solution, and then we address the global existence of the Cauchy problem of inhomogeneous Heisenberg spin ferromagnet system.     

Space of Hermitian Triples and Ashtekar–Isham Quantization

We generalize the Ashtekar–Isham construction for quantizing gauge fields to the case where the configuration variables belong to the space of Hermitian triples, not Hermitian connections.     

Schroedinger Flows on Compact Hermitian Symmetric Spaces and Related Problems

In this note,we prove that the Schroedinger flow of maps from a closed riemann surface into a compact irreducible Hermitian symmetic space admits a global weak solution.Also,we show the existence of weak solutions to the initial value problem of Heisenberg model with Lie algebra values,which is closely related to the Schroedinger flow on compact Hermitian symmetric spaces.