# Balanced metrics on twisted Higgs bundles

@article{GarciaFernandez2014BalancedMO, title={Balanced metrics on twisted Higgs bundles}, author={Mario Garcia-Fernandez and Julius Ross}, journal={Mathematische Annalen}, year={2014}, volume={367}, pages={1429-1471} }

A twisted Higgs bundle on a Kähler manifold X is a pair $$(E,\phi )$$(E,ϕ) consisting of a holomorphic vector bundle E and a holomorphic bundle morphism $$\phi :M \otimes E \rightarrow E$$ϕ:M⊗E→E for some holomorphic vector bundle M. Such objects were first considered by Hitchin when X is a curve and M is the tangent bundle of X, and also by Simpson for higher dimensional base. The Hitchin–Kobayashi correspondence for such pairs states that $$(E,\phi )$$(E,ϕ) is polystable if and only if E…

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## References

SHOWING 1-10 OF 60 REFERENCES

CONSTRUCTING CO-HIGGS BUNDLES ON ℂℙ2

- Mathematics
- 2014

On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized…

Dimensional reduction and quiver bundles

- Mathematics
- 2001

The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kahler manifold admits a…

A Universal Construction for Moduli Spaces of Decorated
Vector Bundles over Curves

- Mathematics
- 2000

AbstractLet $X$ be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation $\rho\colon \mathop{\rm GL}(r)\rightarrow \mathop{\rm GL}(V)$. Then one can…

Canonical metrics on stable vector bundles

- Mathematics
- 2005

The problem of constructing moduli space of vector bundles over a projective manifold has attracted many mathematicians for decades. In mid 60’s Mumford first constructed the moduli space of vector…

On a set of polarized Kähler metrics on algebraic manifolds

- Mathematics
- 1990

A projective algebraic manifold M is a complex manifold in certain projective space CP, N > dim c M = n . The hyperplane line bundle of CP restricts to an ample line bundle L on M. This bundle L is a…

Limits of balanced metrics on vector bundles and polarised manifolds

- Mathematics
- 2011

We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter �, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X.…

Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

- Mathematics
- 1988

The fundamental group is one of the most basic topological invariants of a space. The aim of this paper is to present a method of constructing representations of fundamental groups in complex…

Moduli space of semistable pairs on a curve

- Mathematics
- 1991

Let X be a smooth projective curve over an algebraically closed field k of any characteristic. A stable pair (E, <p) on X, as defined by Hitchin [2], is a vector bundle E on X together with a…

Hitchin–Kobayashi Correspondence, Quivers, and Vortices

- Mathematics
- 2003

Abstract: A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of…

Relative Hitchin–Kobayashi Correspondences for Principal Pairs

- Mathematics
- 2002

A principal pair consists of a holomorphic principal G-bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming…