An analogue of Hartshorne and Serre problems for 1-convex surfaces

Let C be an elliptic curve and let L∈Pic(C). If c₁(L)<0, a well known result of Grauert tells us that L is rigid. On the other hand, Arnold provided a criterion for the rigidity of L when c₁(L)=0. However, a concrete example of such a bundle is hard to come by. In this paper, we construct explicitly such an L which turns out to be the line bundle associated to some toroidal group , viewed as topologically trivial -bundle over C. This example turns out to be the counterexample to the following analogue of a problem of Serre for 1-convex surfaces:Let X be a compactifiable surface such that - for all p?0 and q?1. Is X always 1-convex?Also a cohomological characterization of toroidal groups of finite type is established, as well as an analogue of a problem of Hartshorne for 1-convex surfaces will be discussed.