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Vortex equation and reflexive sheaves. (English) Zbl 1270.53063
Given a stable holomorphic pair $$(E,\phi)$$, where $$E$$ is a holomorphic vector bundle on a compact Kähler manifold $$X$$ and $$\phi$$ is a holomorphic section of $$E$$, the vector bundle $$E$$ admits a Hermitian metric solving the vortex equation. This is generalized to pairs $$(\mathcal{E},\phi)$$, where $$\mathcal{E}$$ is a reflexive sheaf on $$X$$, see also [S. B. Bradlow, Commun. Math. Phys. 135, No. 1, 1–17 (1990; Zbl 0717.53075); G. Tian and B. Yang, J. Reine Angew. Math. 553, 17–41 (2002; Zbl 1022.53025)].
##### MSC:
 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 53B35 Local differential geometry of Hermitian and Kählerian structures
##### Keywords:
stable holomorphic pair; Hermitian metric; vortex equation
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