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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.6 Calabi–Yau Manifolds

Eric Zaslow

1 Basic Definition

Calabi–Yau manifolds, named after Eugenio Calabi and Shing-Tung Yau, arise in Riemannian geometry and algebraic geometry, and play a prominent role in string theory and mirror symmetry.

In order to explain what they are, we need first to recall the notion of orientability on a real MANIFOLD [I.3 §6.9]. Such a manifold is orientable if you can choose coordinate systems at each point in such a way that any two systems x = (x1, . . . , xm) and y = (y1 , . . . , ym) that are defined on overlapping sets give rise to a positive Jacobian: det (∂ yi/∂ xj) > 0. The notion of a Calabi–Yau manifold is the natural complex analogue of this. Now the manifold is complex, and for each local coordinate ...

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