By Alfred Barnard Basset

This quantity is made out of electronic photographs from the Cornell collage Library ancient arithmetic Monographs assortment.

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**Example text**

If E is an elliptic curve, we denote by A(E) the group of its biholomorphic automorphisms. After fixing an origin 0 E E this group can be described in the following way; E, acting on itself by translations, forms a normal subgroup of A ( E ) and the quotient A ( E ) / E can be identified with the group of automorphisms leaving 0 fixed. So this quotient is the cyclic group 71~ of order 2, 4 or 6. If B is any smooth, compact, connected curve, then the holomorphic fibre bundles with typical fibre E and base B are classified by the cohomology set H i ( B , AB) and there is an exact sequence of cohomology sets H I ( B , ~ B ) --+ H I ( B , AB) --+ HI(B,~,~) , where ,As, resp.

E. a compact surface which is the total space of an analytic fibre bundle with fibre IP 1 and structural group PGL(2, C) over a smooth, connected curve B. Examples are provided by the projective bundles IP(E) of algebraic 2-vector bundles E over a smooth, connected, compact 40 2. Facts on compact complex surfaces curve. In fact there are no other examples since any ruled surface is isomorphic to IP(E), where E is an algebraic 2-vector bundle over B. 1). The surfaces IP(O~, @ 0 ~ (n)) are denoted by E,~ and are called the Hirzebruch surfaces.

27 (1) The minimal compact surfaces X with a ( X ) = 1, kod(X) = - o o are exactly the Hopf surfaces. (2) A compact surface X with a(X) = 0 is a Hopf surface if and only if bl(X) = 1 , b2(X) = 0 and there is a curve on X . Thus, besides some Hopf surfaces with a(X) = 1 all minimal surfaces in class VII (class (2) here) have a(X) = 0. For surfaces X in class VII with a ( X ) = 0 and b2(X) = 0, there are, apart from Hopf surfaces, other three kind of examples given by Inoue [In]. We shall present only one type of Inoue surfaces (see also [B-H]).