III.6 Calabi-Yau Manifolds by Eric Zaslow: Basic Definitions, Complex Manifolds and Hermitian Structure, Holonomy, and Calabi-Yau Manifolds in Riemannian Geometry: Page 163.

Calabi-Yau manifolds play a prominent role in string theory and mirror symmetry. 

A manifold is called orientable if you can choose coordinate systems, defined on overlapping sets, at each point so that the Jacobian between the two sets of points is positive, det(yi/xj) > 0.  The Calabi-Yau manifold is the natural extension of this to complex numbers.  In this case for each coordinate system z there is a holomorphic function f(z) that is nonvanishing and satisfies the compatibility condition.  That condition is if there is another corresponding coordinate system w(z) then the corresponding function g is related to f by the equation: f = g det(wa/zb) .  A Calabi-Yau manifold may be thought of as a complex manifold with complex orientation.

The notion  of a holomorphic function on a complex manifold does not depend on the coordinates used to express the function.   The local geometry of a complex manifold looks like an open set in Cn and the tangent space at a point looks like Cn.

One may utilize Hermitian inner products represented by Hermitian matrices,gab’, with respect to a basis, ea, on a complex vector space.

On a Riemannian manifold you can move a vector of constant length and pointing in the same direction along a path and return to the same point.  Curvature reflects the fact that the end vector depends upon the path you take.  When traversing a closed loop the end vector can also be expressed by a matrix operator or holonomy matrix that sends the starting vector to the ending vector.  The group generated by these vectors is called the holonomy group of the manifold.  The holonomy group lies in the orthogonal group of length preserving matrices, O(m).  If the manifold is oriented then the holonomy group must lie in SO(m), as exhibited by transporting an oriented vector basis around the loop.

Complex manifolds of dimension n are also real manifolds of dimension 2n.  Real manifolds obtained in this way have additional structure.  One can multiply complex coordinate directions by i so there must be an operator on the real tangent space that squares to -1.   This operator has eigenvalues ±1.  One can think of these as holomorphic and anti-holomorphic directions.  The Hermitian property states that these directions are orthogonal.  If the directions stay orthogonal after transport around a closed loop then the manifold is called Kähler.

If Kähler then the holonomy group is a subgroup of U(n) which itself is a subgroup of SO(2m).  So, complex manifolds always have real orientations.  A characterization of the Kähler property is if gab’are components of the Hermitian metric in some coordinate patch then there exists a function φ such that gab’ = 2φ/zab.   

Given the nonmetric definition of a Calabi-Yau manifold a compatible Kähler structure leads to a metric definition of the Calabi-Yau manifold.  That the Kähler structure leads to a holonomy that lies in SU(n) Ì U(n) the natural analogue of the case of real orientation.

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