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Calabi–Yau Manifolds

Dominic Joyce

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Abstract

Abstract In this chapter we study compact Riemannian manifolds with holonomy SU(m). A Calabi–Yau manifold is a compact Kähler manifold (M, J, g) of dimension m 2, with Hol(g) = SU(m). Calabi–Yau manifolds are Ricci-flat. They are called Calabi– Yau manifolds because one can use Yau’s solution of the Calabi conjecture to provethe existence of K ähler metrics g with holonomy SU(m) on suitable compact complex manifolds (M, J).

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What this paper is about

Abstract In this chapter we study compact Riemannian manifolds with holonomy SU(m). A Calabi–Yau manifold is a compact Kähler manifold (M, J, g) of dimension m 2, with Hol(g) = SU(m). Calabi–Yau manifolds are Ricci-flat. They are called Calabi– Yau manifolds because one can use Yau’s solution of the Calabi conjecture to provethe existence of K ähler metrics g with holonomy SU(m) on suitable compact complex manifolds (M, J).

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Available abstract

Abstract In this chapter we study compact Riemannian manifolds with holonomy SU(m). A Calabi–Yau manifold is a compact Kähler manifold (M, J, g) of dimension m 2, with Hol(g) = SU(m). Calabi–Yau manifolds are Ricci-flat. They are called Calabi– Yau manifolds because one can use Yau’s solution of the Calabi conjecture to provethe existence of K ähler metrics g with holonomy SU(m) on suitable compact complex manifolds (M, J).

Key concepts: Holonomy, Calabi–Yau manifold, Hyperkähler manifold, Ricci-flat manifold, Manifold (fluid mechanics), Pure mathematics, Conjecture, Mathematics

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