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§11. Introduction to K3 Surfaces 379

(11.1) Definition. A K3 surface is either a compact complex surface or an algebraic
surface over an algebraically closed field k with
(1) K X = 0 or equivalently & 2 and O X are isomorphic as O X -sheaves.
X/k
1
(2) b 1 (X) = 0 or in the algebraic case X is regular, that is, q = dim k H (X, O X ) =
0.
(11.2) Numerical Invariants from Cohomology. The Betti numbers are b 0 = b 4 =
1, b 1 = b 3 = 0, b 2 = 22 in the sense of singular cohomology for k = C or in ´ etale
cohomology in general. Therefore,
24 = e(X) = 12χ(O X ) = 12(h 0,0 − h 0,1 + h 0,2 )
so that the other Hodge numbers are h 2,0 = h 0,2 = 1and h 1,1 = 22. In the Hodge
p,q q p
to de Rham spectral sequence E 1 = E ∞ where as usual E = H (X,& )
1 X/k
p+q
converges to H (X/k). It is also the case that a K3 surfaces is simply connected.
DR
(11.3) Tangent and Cotangent Sheaves. The tangent sheaf T X/k = (& 1 )ˇ is by
X/k
definition the dual of the cotangent sheaf & 1 . Since (O) X and & 2 are isomorphic
X/k X/k
as O X -sheaves, we deduce that T X/k and & X/k are isomorphic.
The main assertion about the tangent sheaf on a K3 surface is that there are no
nonzero tangent vector fields, or equivalently
0
H (X, T X/k ) = 0.
There are two references for this result.
0
(a) Rudakov and Shafarevitch, n 6, Akad. Sc. SSSR 40 (1976), pp. 1264–1307.
(b) Nygaard, Annals 110 (1979), pp. 515–528.
0
Since H (X,& X/k ) = 0 by the above discussion, Serre duality gives
2 1
H (X,& ) = 0,
X/k
2
0
an isomorphism between H (X, O X ) and H (X,& 2 ) = 0 which is just k,and an
X/k
2
0
isomorphism between H (X,& 2 ) = 0and H (X, O X ) which is just k also. The
X/k
possible nonzero groups are:
2
1
2
2
2
H (X, O X ) = k H (X,& ) = 0 H (X,& ) = k
X X
1
2
1
1
1
H (X, O X ) = 0 H (X,& ) = k 20 H (X,& ) = 0.
X X
2
0
0
1
0
H (X, O X ) = k H (X,& ) = 0 H (X,& ) = k
X X
2
1
Here q = dimH (X, O X ) is the irregularity of X,and p g = dimH (X, O X ) is the
geometric genus of the surface X,and q = 0and p q = 1.