Steven B. Bradlow, James F. Glazebrook, Franz W. Kamber
Reduction of the Hermitian-Einstein Equation on Kählerian Fiber Bundles
Preprint series: ESI preprints

MSC:

58C25 Differentiable maps

58A30 Vector distributions (subbundles of the tangent bundles)

53C12 Foliations (differential geometric aspects), See Also {57R30, 57R32}

53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions, See also {58G30}

53C55 Hermitian and Kahlerian manifolds, See also {32Cxx}

83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems)

Abstract: The technique of dimensional reduction of an integrable system
usually requires symmetry arising from a group action.
In this paper we study a situation in which a dimensional reduction
can be achieved despite the absence of any such global symmetry.
We consider certain holmorphic vector bundles over a Kahler manifold
which is itself the total space of a fiber bundle over a Kahler
manifold. We establish an equivalence between invariant solutions to the
Hermitian--Einstein equations on such bundles, and general solutions
to a coupled system of equations defined on holomorphic bundles over the
base Kahler manifold. The latter equations are the Coupled Vortex Equations.
Our results thus generalize the dimensional reduction results of
Garc\'ia--Prada, which apply when the fiber bundle is a product
and the fiber is the complex projective line.
Keywords: Coupled vortex equation, Kaehlerian fiber bundles, projective bundles, homogeneous bundles, Hermitian-Einstein equation, stability, foliations, Hitchin-Kobayashi correspondence