Abelian surfaces in toric 4-folds

There are embeddings of complex abelian surfaces in P but it was shown by Van de Ven in [17] that no abelian d-fold can be embedded in P if d ≥ 3. Hulek [9], Lange [13], Birkenhake [4] and Bauer and Szemberg [3] have all investigated the possibility of replacing P with a product of projective spaces. Furthermore, Lange [14] studies the rank 2 bundle on P × P that arises from the abelian surfaces in P × P by Serre’s construction. The analogous bundle associated with the abelian surfaces in P is, of course, the HorrocksMumford bundle. In this paper we shall work over the complex numbers and consider embeddings of abelian surfaces in slightly more general ambient spaces of dimension 4, namely smooth toric varieties. The most tractable and probably the most interesting cases are when the ambient variety has small Picard number. We shall therefore consider the following question. Suppose X is a smooth complete toric variety of dimension 4 and ρ(X) ≤ 2. Does there exist an abelian surface A ⊆ X and, if so, can we describe the embedding? In the first section we shall give some numerical conditions that such an embedding must satisfy and show that for many X of this type there can be no totally nondegenerate (see Definition 1.1, below) abelian surfaces in X . In Section 2, which is based on unpublished joint work with Professor T. Oda, we show how to describe morphisms into smooth toric varieties in a particularly simple way. The results of this section overlap with work of Cox [6], Guest [8] and Kajiwara [11] but it is useful to us to have them in the form given here. We apply this description in Section 3, in which we exhibit a new 2-dimensional family of abelian surfaces embedded in a particular toric 4-fold X of Picard number 2. The normal bundles of the surfaces described in Section 3 give rise to rank 2 bundles on X which should be interesting to study. However, we do not attempt this here, but in Section 4 we make a few comments on this and other related matters. Much of this work was carried out in 1997 while the author was visiting the Research Institute for Mathematical Sciences in Kyoto, supported by the JSPS International Project Research “Infinite Analysis and Geometry”.