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
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