Asian Journal of Mathematics

Volume 20 (2016)

Number 5

Yang–Mills–Higgs connections on Calabi–Yau manifolds

Pages: 989 – 1000

DOI: http://dx.doi.org/10.4310/AJM.2016.v20.n5.a8

Authors

Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Bombay, India)

Ugo Bruzzo (Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis, SC, Brazil; Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Italy; and Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy)

Beatriz Graña Otero (Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá, Colombia bgrana@javeriana.edu.co)

Alessio Lo Giudice (Departamento de Matemática, Cidade Universitária, Campinas, SP, Brazil)

Abstract

Let $X$ be a compact connected Kähler–Einstein manifold with $c_1 (TX) \geq 0$. If there is a semistable Higgs vector bundle $(E, \theta)$ on $X$ with $\theta \neq 0$, then we show that $c_1 (TX) = 0$; any $X$ satisfying this condition is called a Calabi–Yau manifold, and it admits a Ricci-flat Kähler form. Let $(E, \theta)$ be a polystable Higgs vector bundle on a compact Ricci-flat Kähler manifold $X$. Let $h$ be an Hermitian structure on $E$ satisfying the Yang–Mills–Higgs equation for $(E, \theta)$. We prove that $h$ also satisfies the Yang–Mills–Higgs equation for $(E, \theta)$. A similar result is proved for Hermitian structures on principal Higgs bundles on $X$ satisfying the Yang–Mills–Higgs equation.

Keywords

Calabi–Yau manifold, approximate Hermitian–Yang–Mills structures, Hermitian–Yang–Mills metrics, polystability, Higgs field

2010 Mathematics Subject Classification

14F05, 14J32, 32L05, 53C07, 58E15

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Published 22 February 2017