Asian Journal of Mathematics

Volume 20 (2016)

Number 5

Yang–Mills–Higgs connections on Calabi–Yau manifolds

Pages: 989 – 1000



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

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


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.


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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