ON DEGENERATIONS OF ALGEBRAIC SURFACES

3

K2 + e

X — "—7^— t where K is the canonical divisor defined by

O(K)

= n2.

In case X is Kahler (e.g., if X is algebraic, as implied by

Theorem A) we have of course the Hodge decomposition and isomorphisms

Hp(x,nq) *Hq(x,np).

This is not true in general for analytic surfaces, but we have the

following result, by Kodaira [11 a, Thm. 3].

Theorem C: If X is a surface, then the Hodge spectral sequence of X

degenerates at the E1 term. That is, we have

^ — i

hn = ^ hpq, n = 0,1,2,3,4.

p+q=n

Furthermore, we have:

a) if h is even, then h = 2q, hence h = h ,

b) if h is odd, then h = 2q - 1, i.e.,

dim H (X,0) = dum H (X,n ) + 1.

It is conjectured that case a) implies Kabler. (See [14.2] for the

result that b even implies Kahler in the case of elliptic surfaces).

Case b) very much occurs, hence in particular there are non-Kahler

surfaces.

The Hodge invariants furnish some conditions for algebraicity and

ellipticity of a surface X. Indeed, according to Kodaira we have:

Theorem D: Let X be a surface. Then

a) p 2 or h * ^ 3 implies X is algebraic or elliptic,

b) p = 0 and h jt 1 implies X is algebraic.

See [11 a, Thm. 6.7] .

Given any surface X, we can at each point p € B, define the blow up