Volume 17, issue 5 (2013)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Coupled equations for Kähler metrics and Yang–Mills connections

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

Geometry & Topology 17 (2013) 2731–2812
Abstract

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.

Keywords
coupled Kähler–Yang–Mills equation, Kähler metric, Hermitian–Yang–Mills connection, generalized Futaki invariant, generalized Mabuchi energy
Mathematical Subject Classification 2010
Primary: 32Q20, 53C07
References
Publication
Received: 29 June 2012
Revised: 26 March 2013
Accepted: 25 April 2013
Published: 14 October 2013
Proposed: Simon Donaldson
Seconded: Richard Thomas, Frances Kirwan
Authors
Luis Álvarez-Cónsul
Instituto de Ciencias Matemáticas
C/ Nicolás Cabrera 13–15
Campus Cantoblanco
28049 Madrid
Spain
Mario García-Fernández
MATHGEOM
École Polytechnique Fédéral de Lausanne
MA B1 437, Station 8
CH-1015 Lausanne
Switzerland
Oscar García-Prada
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)
C/ Nicolás Cabrera 13–15
Campus Cantoblanco
28049 Madrid
Spain