Chapter I
A brief survey of the structure and classification
of surfaces
By a surface X we will mean:
A compact complex analytic manifold of (complex) dimension two.
The surface will be called algebraic iff it is the classical real-
ization, i.e., the (E-points of a reduced, proper and regular scheme/(E.
We will now briefly state some general facts about surfaces; most
of the results, with the exception of some more special results, will be
stated without proofs. General references for this would be the papers
of Kodaira [10,11], and Shafarevicz seminar [18].
The following is proved in Kodaira [10a].
Theorem A (Conditions for algebraicity): Let X be an analytic sur-
face. Then the following conditions are equivalent:
1. X is algebraic.
2. X is projective.
3. tr.d *!?(X) = 2, (OT(X) the field of meromorphic functions).
4. X is Hodge, i.e., has an integral Kahler form.
5. X has a divisor D, with D 0.
6. Given any point p X, we can find two curves passing through p.
Note that 2) implies all other conditions more or less directly.
4) = 2) is the Kodaira embedding theorem (or vanishing theorem) and is
true in all dimensions.
This paper has been partially supported by the National Science Founda-
tion under grant number NSF MCS76-08181.
Received by the editor October 16, 1975.
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