2 ULF PERSSON

The rest of the conditions, however, do not imply 2) in the cases

of higher dimensions.

Note that condition 6 illustrates the fact that algebraic surfaces

are characterized by their wealth of curves.

To describe the nonalgebraic ones with respect to this property we

have the following, also due to Kodaira [10 a, Thm. 4.1, 4.2, 5.1].

Theorem B: Let X be a surface. Then we have:

1. tr.deg !IR(X) = 1, iff we have a map g:X - * C, with elliptic

curves as fibers. (X is a so called elliptic surface) and

such that X has no transversal divisors with respect to the

fibration.

If this is the case then q#: ©(C) - * w(X) induces an isomor-

phism of fields.

2. tr.deg ^(X) = 0, iff there are only a finite number of curves

on X.

Given any surface X we can of course look at the Hodge spectral

sequence and the Hodge numbers h = dim H (X,n )•

We define p = h ' « dim H (X,fl ) by Serre duality

h

2

'°

=

h

0

'

2

,

= h

0A , . .. 1

= dim H (X,0) and we have

p is referred to as the geometric genus of X, and q is called the

irregularity.

We define x ~ P - Q + 1* an* ^et e denote the Euler character-

istic.

These invariants are related by Noether's formula