In
mathematics, and especially
differential geometry and
algebraic geometry, a stable principal bundle is a generalisation of the notion of a
stable vector bundle In mathematics, a stable vector bundle is a ( holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable ...
to the setting of
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
s. The concept of stability for principal bundles was introduced by
Annamalai Ramanathan for the purpose of defining the moduli space of G-principal bundles over a
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
, a generalisation of earlier work by
David Mumford
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
and others on the moduli spaces of vector bundles.
[Ramanathan, A., 1996, August. Moduli for principal bundles over algebraic curves: I. In Proceedings of the Indian Academy of Sciences-Mathematical Sciences (Vol. 106, No. 3, pp. 301-328). Springer India.][Ramanathan, A., 1996, November. Moduli for principal bundles over algebraic curves: II. In Proceedings of the Indian Academy of Sciences-Mathematical Sciences (Vol. 106, No. 4, pp. 421-449). Springer India.]
Many statements about the stability of vector bundles can be translated into the language of stable principal bundles. For example, the analogue of the
Kobayashi–Hitchin correspondence In differential geometry, algebraic geometry, and gauge theory, the Kobayashi–Hitchin correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The co ...
for principal bundles, that a holomorphic principal bundle over a compact
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
admits a
Hermite–Einstein connection if and only if it is polystable, was shown to be true in the case of projective manifolds by Subramanian and Ramanathan, and for arbitrary compact Kähler manifolds by Anchouche and
Biswas.
[Subramanian, S. and Ramanathan, A., 1988. Einstein-Hermitian connections on principal bundles and stability.][Anchouche, B. and Biswas, I., 2001. Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold. American Journal of Mathematics, 123(2), pp.207-228.]
Definition
The essential definition of stability for principal bundles was made by Ramanathan, but applies only to the case of Riemann surfaces.
In this section we state the definition as appearing in the work of Anchouche and Biswas which is valid over any Kähler manifold, and indeed makes sense more generally for
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
.
This reduces to Ramanathan's definition in the case the manifold is a Riemann surface.
Let
be a
connected
Connected may refer to:
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* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
reductive algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Ma ...
over the complex numbers
. Let
be a compact Kähler manifold of complex dimension
. Suppose
is a holomorphic principal
-bundle over
. Holomorphic here means that the transition functions for
vary holomorphically, which makes sense as the structure group is a
complex Lie group
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mat ...
. The principal bundle
is called stable (resp. semi-stable) if for every
reduction of structure group
In differential geometry, a ''G''-structure on an ''n''- manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''.
The notion of ''G''-structures includes var ...
for
a maximal
parabolic subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgro ...
where
is some open subset with the codimension
, we have
:
Here
is the relative
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of the fibre bundle
otherwise known as the
vertical bundle
Vertical is a geometric term of location which may refer to:
* Vertical direction, the direction aligned with the direction of the force of gravity, up or down
* Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...
of
. Recall that the degree of a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
(or
coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
)
is defined to be
:
where
is the first
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
of
. In the above setting the degree is computed for a bundle defined over
inside
, but since the codimension of the complement of
is bigger than two, the value of the integral will agree with that over all of
.
Notice that in the case where
, that is where
is a
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
, by assumption on the codimension of
we must have that
, so it is enough to consider reductions of structure group over the entirety of
,
.
Relation to stability of vector bundles
Given a principal
-bundle for a complex Lie group
there are several natural vector bundles one may associate to it.
Firstly if
, the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
, then the standard representation of
on
allows one to construct the
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
. This is a
holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
over
, and the above definition of stability of the principal bundle is equivalent to slope stability of
. The essential point is that a maximal parabolic subgroup
corresponds to a choice of flag
, where
is invariant under the subgroup
. Since the structure group of
has been reduced to
, and
preserves the vector subspace
, one may take the associated bundle
, which is a sub-bundle of
over the subset
on which the reduction of structure group is defined, and therefore a subsheaf of
over all of
. It can then be computed that
:
where
denotes the
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of the vector bundles.
When the structure group is not
there is still a natural associated vector bundle to
, the
adjoint bundle , with fibre given by the
Lie algebra of
. The principal bundle
is semistable if and only if the adjoint bundle
is slope semistable, and furthermore if
is stable, then
is slope polystable.
Again the key point here is that for a parabolic subgroup
, one obtains a parabolic subalgebra
and can take the associated subbundle. In this case more care must be taken because the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
of
on
is not always
faithful or
irreducible, the latter condition hinting at why stability of the principal bundle only leads to ''polystability'' of the adjoint bundle (because a representation that splits as a direct sum would lead to the associated bundle splitting as a direct sum).
Generalisations
Just as one can generalise a vector bundle to the notion of a
Higgs bundle
In mathematics, a Higgs bundle is a pair (E,\varphi) consisting of a holomorphic vector bundle ''E'' and a Higgs field \varphi, a holomorphic 1-form taking values in the bundle of endomorphisms of ''E'' such that \varphi \wedge \varphi=0. Such pai ...
, it is possible to formulate a definition of a principal
-Higgs bundle. The above definition of stability for principal bundles generalises to these objects by requiring the reductions of structure group are compatible with the Higgs field of the principal Higgs bundle. It was shown by Anchouche and Biswas that the analogue of the
nonabelian Hodge correspondence
In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after Kevin Corlette and Carlos Simpson) is a correspondence between Higgs bundles and representations of the fundame ...
for Higgs vector bundles is true for principal
-Higgs bundles in the case where the base manifold
is a
complex projective variety.
References
{{Reflist
Fiber bundles
Algebraic geometry
Differential geometry