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

(a) If q = 0, then either X is isomorphic to P 2 so K · K = 9, or X is a ruled surface
over P 1 , i.e., a P 1 -bundle over P 1 .
(b) If q > 0, then X is a ruled surface over a curve of genus q with K ·K = 8(1−q).
Step 2. The analog of curves of genus g > 1 for surfaces are surfaces of general
type, that is, surfaces such that
(1) K · D ≥ 0 for all divisors D ≥ 0and
(2) K · K > 0.
As with curves, these surfaces can be birational embedding in projective space using
sections from (X, K m⊗ ) for large enough m.
Step 3. There are two broad classes of minimal surfaces which are the analogue
of genus g = 1 for curves, that is, elliptic curves. In both cases, K · D ≥ 0 for
all divisors D ≥ 0and K · K = 0. The first class consists of those surfaces with
12K = 0 and the second class with 12K = 0.
The second class where 12K = 0 are the properly elliptic surfaces. These sur-
faces are fibred over a curve of genus q > 1 by elliptic curves with only a finite
number of exceptional fibres.
The first class where 12K = 0 contains the surfaces with K = 0. These are
the natural generalizations of elliptic curves, that is, the class of two dimensional
Calabi–Yau manifolds.
Step 4. The classification of minimal surfaces with 12K = 0and K · D ≥ 0 for
all divisor D ≥ 0:

b 2 b 1 e(X) q p g χ(O)
K3 surfaces 22 0 24 0 1 2
Enriques surfaces 10 0 12 0 0 1
Abelian surfaces 6 4 0 2 1 0
Hyperelliptic surfaces 2 2 0 1 0 0

(10.3) Remark. Our main interest is in surfaces X with K X = 0 or equivalently
with & 2 = O X . Only K3 surfaces and abelian surfaces satisfy this condition. En-
X
riques surfaces satisfy K X = 0, but 2K X = 0, and 12K X = 0 for hyperelliptic
surfaces. There are hyperelliptic surfaces with nK X = 0 for all divisors n of 12, but
n K X = 0 for a proper divisor n of n.

(10.4) Remark. Both complex K3 surfaces and complex tori can be nonalgebraic,
but if a surface has an embedding in a projective space, then by Chow’s lemma, it is
algebraic. The term abelian surface is usually reserved for algebraic tori.

§11. Introduction to K3 Surfaces

We begin by collecting some elementary data about K3 surfaces. Firstly, we start with
a definition which has a meaning for compact complex surfaces and for algebraic
surfaces.