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Approximate Hermitian-Yang-Mills structures on semistable principal Higgs bundles

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Abstract

We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.

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... This is a Higgs bundle version of the Donaldson-Uhlenbeck-Yau theorem [8,27]. In [5,6,7], it shows that a Higgs bundle is semistable if only if it is admits approximate Hermitian-Einstein structure, i.e., for every positive ε, there is a Hermitian metric h ε on E such that ...
... In fact, the existence of approximate Hermitian-Einstein structure on semistable Higgs vector bundle was proved by Cardona [6] when dim(X) = 1, and by Li-Zhang [15] for arbitrary dimension, Bruzzo-Otero [5] used Li-Zhang's result to prove the semistable principal Higgs bundles case. The existence of stable (semistable) Higgs bundles is depends on the geometry of the underlying manifold. ...
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... 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. ...
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... Later, Simpson [29] proved an analogue of the Donaldson-Uhlenbeck-Yau theorem for the Higgs bundle over higer dimensional Kähler manifolds, influenced by the work of Hitchin. In the compact case, the Higgs version of Donaldson-Uhlenbeck-Yau has been extensively studied during the last two decades, see references [1,2,5,6,12,13,16,17,23]. ...
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Reprint of the 1969 original
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  • K Nomizu