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366 19. Higher Dimensional Analogs of Elliptic Curves: Calabi–Yau Varieties

On U a ∩ U b , the metric and change of frames functions are related by h a =
2
|g b,a | h b or log(h a ) = log(h b ) + log(g b,a ) + log(g b,a ). For the connection form
ˇ

ω a = (d h a )h −1 = d log(h a ), we calculate its Cech coboundary
a
1 1
(δω) a,b = ω b − ω a = d(log(h b /h a )) = d(log(g a,b g a,b ))
2πi 2πi
1
= d(log(g a,b ) = dσ a,b
2πi
Hence the closed two form associated with the cocycle δσ is the first Chern class
form derived from the Hermitian metric h, namely
1

c 1 (L, h) = d d log(h a ).
2πi
§6. Characterizations of Calabi–Yau Manifolds: First Examples

(6.1) Equivalent Definitions of Calabi-Yau Manifolds. Let X be a compact n di-
mensional complex manifold. Then X is a Calabi–Yau manifold provided it satisfies
any of the following equivalent conditions:
(1) X is a K¨ ahler manifold with a vanishing first Chern class.
(2) X admits a Levi–Civita connection with SU(n) homology.
(3) X has a Ricci flat K¨ ahler metric.
(4) X is a K¨ ahler manifold with a nowhere vanishing holomorphic n-form.
(5) X is a K¨ ahler manifold with a trivial canonical line bundle ω X ,thatis, ω X is
isomorphic to O X .

(6.2) Remark. The first Chern class is represented by the trace of the curvature
2-form, that is, the Ricci tensor. Hence (3) implies (1). The Calabi conjecture and
proven by Yau is the converse implication.
(6.3) Yau’s Theorem. If X is a complex K¨ ahler manifold with K¨ ahler form ω and
vanishing first Chern class, then there exists a unique Ricci-flat metric on X whose
K¨ ahler form is in the same cohomology class as ω.

An early reference on Calabi–Yau 3-folds is Hirzebruch, Gesammelte Abhand-
lungen, T. II, no. 75, pp. 757-770, “Some examples of threefolds with trivial canoni-
cal bundle.”
The first indication of the importance of the Calabi–Yau variety concept is seen
in the many equivalent versions of the definition. Now for basic examples together
with their Euler numbers in low dimensions starting with curves of genus one.

(6.4) Example (dimension one). A Calabi–Yau in the extended sense in dimension
one can be either a smooth cubic curve in P 2 or the complete intersection of two
smooth quadrics in P 3 . The Euler number is 0. Note that the fundamental group is
infinite.