A. Van de Ven

Hilbert modular surfaces and the classification of algebraic surfaces

Introduction Around 1900 Hilbert, Hecke ([7]), Blumenthal ([2]) and others started the study of certain 2-dimensional complex spaces, which are closely related to the classification of special types of 2-dimensional abelian varieties. These complex spaces can easily be described. In fact, let n be a natural number, n > 1, which is square free and let K = Q q/n). Then, if o K is the ring of algebraic integers of K, the group SLz(oK) operates in a natural way on ~ • ~ and on ~ • ~ where ~ is the upper and ~ the lower half plane of C. The quotients of See • ~ and Jg x J g by the action of SL2(ov,) are the 2-dimensional complex spaces mentioned above. If the field K has a unit of negative norm, then the two actions on ~ • ~ and ~ • J g are isomorphic. This is true if n is a prime congruent 1 rood 4, the only case we shall consider in this paper. Therefore, from now on we assume that K= Q (l,/p), where p is a prime congruent 1 mod 4. The complex space Jg x .)~/SLz(oK) can be compactified by means of a finite number of points, called the cusps, to a compact 2-dimensional complex space. After resolving the cusps and also the quotient singularities on ~ x ~ / S L 2 (OK) , both in a canonical, explicit way, a nonsingular compact complex surface Y(R) is obtained, which in fact is an algebraic surface. The field of meromorphic (i. e. rational) functions on Y(p) is isomorphic to the field of meromorphic functions of • ~ / S L 2 (OK). On the other hand, although no complete classification of algebraic surfaces is known, there exists a rough classification in several classes (for most of which the surfaces contained in that class can be classified completely, at least in principle). In big outline this classification was already known to the italian school, but its precise formulation (this time also covering the non-algebraic case) and many of the proofs involved are due to Kodaira. Now the question considered in this paper is the following: where are the surfaces Y(p) to be placed in the rough classification of algebraic surfaces?

On the Normal Bundles of Smooth Rational Space Curves

in this note we consider smooth rational curves C of degree n in threedimensional projective space IP 3 (over a closed field of characteristic 0). To avoid trivial exceptions we shall always assume that n ~ 4 (this does not hold however for certain auxiliary curves we shall consider). Let N = N c be the normal bundle of C in IP 3. Since degel(IP3)=4, and d e g c l ( l P 0 = 2 , we have that d e g c l ( N ) = 4 n 2 . By a well-known theorem of Grothendieck the bundle N is a direct sum of two line bundles. Hence N ~ O c ( 2 n l a ) G O c ( 2 n 1 +a) for some non-negative a=a(C), which is uniquely determined by C. The question we would like to answer is an obvious one: which values of a occur? We shall show (Theorem 4 below) that a value a occurs if and only if 0_ =0, therefore Hi(C, N)=O. It follows [K, p. 150] that C represents a smooth point on the Chow variety Ch(3, 1, n) of effective cycles of dimension 1 and degree n in IP 3. Since the set of all smooth rational curves with a fixed degree is obviously connected, we see that the smooth Cs represent a smooth, irreducible, 4n-dimensional (Zariski-)open subset S of Ch(3, 1, n). In a for thcoming paper [ E V ] we shall prove the following

Homotopy groups of pullbacks of varieties

In [2, § 9] there is a general result of Fulton and Lazarsfeld relating the homotopy groups of a subvariety of in a certain range of dimensions with those of its pullback under a holomorphic map in the corresponding range of dimensions. It is asked in [2, § 10] whether here is a corresponding result with replaced by a general rational homogeneous manifold, Y , and with the range of dimensions alluded to above shifted by the ampleness of the holomorphic tangent bundle of Y in the sense of [4]. In this paper we use the techniques of [4, 5, 6, 7] to answer this question in the affirmative.

Chern Classes and Complex Manifolds

As usual, we shall denote by Z the ring of integers, and by R and C the fields of real and complex numbers. If K is a field Kh will stand for the K-vectorspace of ordered n-tuples of elements of K. GL(n, K) for the general lineair group, operating on Kn, and PGL(n, K) for the projective lineair group, operating on the projective space of Kn+1