Geometry & Topology

Coupled equations for Kähler metrics and Yang–Mills connections

Luis Álvarez-Cónsul, Mario García-Fernández, and Oscar García-Prada

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We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kähler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kähler metric and Hermite–Yang–Mills for a connection. We provide a moment map interpretation of the equations and study obstructions for the existence of solutions, generalizing the Futaki invariant, the Mabuchi K–energy and geodesic stability. We finish by giving some examples of solutions.

Article information

Geom. Topol., Volume 17, Number 5 (2013), 2731-2812.

Received: 29 June 2012
Revised: 26 March 2013
Accepted: 25 April 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32Q20: Kähler-Einstein manifolds [See also 53Cxx] 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20]

coupled Kähler–Yang–Mills equation Kähler metric Hermitian–Yang–Mills connection generalized Futaki invariant generalized Mabuchi energy


Álvarez-Cónsul, Luis; García-Fernández, Mario; García-Prada, Oscar. Coupled equations for Kähler metrics and Yang–Mills connections. Geom. Topol. 17 (2013), no. 5, 2731--2812. doi:10.2140/gt.2013.17.2731.

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