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Approximate Hitchin–Kobayashi correspondence for Higgs G-bundles
Abstract
We announce a result about the extension of the Hitchin-Kobayashi correspondence to principal Higgs bundles. A principal Higgs bundle on a compact Kähler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.
... Since the original articles of Hitchin and Simpson [21,28], Higgs bundles have played an important role in geometry and there exists a very vast literature on this topic in complex geometry and mathematical physics. In particular, we would like to emphasize that several properties of Higgs bundles can be naturally extended to a very special kind of principal bundles, usually known as Higgs G-bundles [6]. Since in general principal bundles play an important role in gauge theory, such extension could be eventually of interest also in areas of high energy physics like string theory and Yang-Mills theory. ...
... The above results appear in [14] as Theorem 1 and Proposition 1, respectively. 6 Notice that in the right hand side of the above difference, the first term does not depend on the metric (it is a topological constant), however the second term depends on h. In particular, if Φ = 0 the second term vanishes and we have that -up to an additive constant-the Yang-Mills and the Kobayashi functionals are in essence the same as far as holomorphic vector bundles is concern. ...
We study the 2k-Hitchin equations introduced by Ward \cite{Ward 2} from the geometric viewpoint of Higgs bundles. After an introduction on Higgs bundles and 2k-Hitchin equations, we review some elementary facts on complex geometry and Yang-Mills theory. Then we study some properties of holomorphic vector bundles and Higgs bundles and we review the Hermite-Yang-Mills equations and two related functionals to such equations. Using some geometric tools we show that, as far as Higgs bundles is concern, the 2k-Hitchin equations are reduced to a set of only two equations. Finally, we introduce a functional closely related to the 2k-Hitchin equations and we study some of its basic properties.
... compact Kähler manifolds. There are many interesting generalized Hitchin-Kobayashi correspondences (see [1][2][3][4][5][7][8][9][10]12,18,20,21,23,26,[28][29][30][31]33,34,36,46], etc.). It is natural to hope that geometric results dealing with closed manifolds will extend to yield interesting information for manifolds with boundary. ...
... From the above discussion, h(t) are uniformly bounded in C 1 . Using (9) (7) and (8). ...
The purpose of this paper is twofold. We first solve the Dirichlet problem for α-Hermitian–Einstein equations on I±-holomorphic bundles over bi-Hermitian manifolds. Second, by using Uhlenbeck–Yau’s continuity method, we prove that the existence of approximate α-Hermitian–Einstein structure and the α-semi-stability on I±-holomorphic bundles over compact bi-Hermitian manifolds are equivalent.
We study the [Formula: see text]-Hitchin’s equations introduced by Ward from the geometric viewpoint of Higgs bundles. After an introduction on Higgs bundles and [Formula: see text]-Hitchin’s equations, we review some elementary facts on complex geometry and Yang–Mills theory. Then, we study some properties of holomorphic vector bundles and Higgs bundles and we review the Hermite–Yang–Mills equations together with two functionals related to such equations. Using some geometric tools we show that, as far as Higgs bundles are concerned, [Formula: see text]-Hitchin’s equations are reduced to a set of two equations. Finally, we introduce a functional closely related to [Formula: see text]-Hitchin’s equations and we study some of its basic properties.
The purpose of this paper is twofold. We first solve the Dirichlet problem for -Hermitian–Einstein equations on holomorphic filtrations over compact Hermitian manifolds. Secondly, by using Uhlenbeck–Yau’s continuity method, we prove the existence of approximate -Hermitian–Einstein structure on holomorphic filtrations over closed Gauduchon manifolds.
Given a compact Kähler manifold M and a connected reductive algebraic group G over ℂ, a principal G-bundle over M admits an Einstein-Hermitian connection if and only if the principal bundle is polystable. If M is a projective manifold, a Higgs G-bundle over M admits an Einstein-Hermitian connection if and only if the Higgs bundle is polystable.
Let
X
be a compact connected Riemann surface and
G
a connected reductive complex affine algebraic group. Given a holomorphic principal
G
-bundle
over
X
, we construct a
Hermitian structure on
together with a
1
-parameter family of
automorphisms
of the principal
G
-bundle
with the following property: Let
be the connection on
corresponding to the Hermitian structure and the new holomorphic structure on
constructed using
from the original holomorphic structure. As
, the connection
converges in
Fréchet topology to the connection on
given by the Hermitian–Einstein connection on the polystable principal bundle associated to
. In particular, as
, the curvature of
converges in
Fréchet topology to the curvature of the connection on
given by the Hermitian–Einstein connection on the polystable principal bundle associated to
. The family
is constructed by generalizing the method of [6]. Given a holomorphic vector bundle
E
on
X
, in [6] a
1
-parameter family of
automorphisms of
E
is constructed such that as
, the curvature converges, in
topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.
Einstein-Hermitian vector bundles are defined by a certain curvature condition. We prove that over a compact Khler manifold a bundle satisfying this condition is semistable in the sense of Mumford-Takemoto and a direct sum of stable Einstein-Hermitian subbundles.
We study the basic properties of Higgs sheaves over compact K\"ahler
manifolds and we 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 we 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 proved some properties
of semistable Higgs sheaves.
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.
Let E_G be a principal G-bundle over a compact connected K\"ahler manifold,
where G is a connected reductive complex linear algebraic group. We show that
E_G is semistable if and only if it admits approximate Hermitian-Einstein
structures.
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\"ahler
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.
Let k be an algebraically closed field, and X a nonsingular irreducible protective algebraic variety over k . These assumptions will remain fixed throughout this paper. We will consider a family of vector bundles on X of fixed rank r and fixed Chern classes (modulo numerical equivalence). Under what condition is this family a bounded family? When X is a curve, Atiyah [1] showed that it is so if all elements of this family are indecomposable. But when I is a surface, he showed also that this condition is not sufficient. We give the definition of an H -stable vector bundle on a variety X . This definition is a generalization of Mumford’s definition on a curve. Under the condition that all elements of a family are H -stable of rank two on a surface X , we prove that the family is bounded. And we study H -stable bundles, when X is an abelian surface, the protective plane or a geometrically ruled surface.
We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact K\"ahler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, establishing semistability criteria for Higgs bundles on projective manifolds of any dimension.