Direct Limits of Jaffard Domains and S-Domains oy

Abstract. It is proved under mild assumptions that the class of Jaffard domains and the class of S-domains are each stable under direct limit. New examples of Jaffard domains obtained thereby include the factorial domain of Fujita, and Nagata rings in arbitrarily many indeterminates over a Jaffard domain. New examples of S-domains are the polynomial rings in arbitrarily many indeterminates over any domain. Also, any locally finite-dimensional directed union of universally catenarian going-down domains is itself a universally catenarian going-down domain. However, many related types of rings (such as [stably] strong S-domains or [universally] catenarian domains) are not preserved by direct limit. Numerous examples illustrate the need for various hypotheses, the failure of various converses, etc., as well as the sharpness of bounds that we give for the dimension and the valuative dimension of a direct limit

On holomorphic forms on compact complex threefolds

Abstract. We study the structure of holomorphic 1. forms on compact complex threefolds of positive algebraic dimension. We obtain a rather detailed description of integrable 1. forms. We use this result to extend Castelnuovo -De Franchis lemma (as well as Catanese's generalization) to non-Kähler threefolds. Mathematics Subject Classification (1991). 32J17, 32L30, 32J10. Keywords. Complex manifolds, holomorphic 1. forms, algebraic dimension, foliations, Castelnuovo-De Franchis lemma. A very useful tool in the study of Kähler manifolds is the classical Castelnuovo -De Franchis lemma: it says that if ω 1 and ω 2 are two linearly independent holomorphic 1-forms on a connected compact Kähler manifold M , which satisfy the collinearity relation ω 1 ∧ ω 2 ≡ 0, then there exists a holomorphic map π : M → C onto an algebraic curve C of genus greater or equal than 2 such that ω 1 , ω 2 are pull-back by π of two holomorphic 1-forms on C. This simple lemma has several nontrivial consequences on the topology of Kähler manifolds [Cat], their Chern numbers [Bog] [Miy], and many others things. A nice generalization has recently been found by Catanese [Cat] (and, independently, other mathematicians; see [Cat] for the references). We state only a particular case, sufficient for our purposes: if ω 1 , ω 2 and ω 3 are holomorphic 1-forms on a connected compact Kähler manifold M , such that ω 1 ∧ ω 2 ∧ ω 3 ≡ 0 and ω 1 ∧ ω 2 , ω 2 ∧ ω 3 , ω 3 ∧ ω 1 are linearly independent, then there exists a holomorphic map π : M → S onto a normal algebraic surface S of Albanese general type such that ω 1 , ω 2 , ω 3 are pull-back by π of three holomorphic 1-forms on S. We shall recall in section 1 the definition of "variety of Albanese general type", for the moment we only say that it is one of the possible higherdimensional generalizations of "curve of genus greater or equal than 2". These results are based on the closedness of holomorphic forms on compact Kähler manifolds; in fact, the Kähler assumption is exploited only to ensure that closedness. There are, however, many examples of compact complex non-Kähler manifolds which support non-closed holomorphic 1-forms: the most classical ones are compact quotients of certain Lie groups [Ue1, §17]. It is not clear to us if Castelnuovo -De Franchis -Catanese statement is still true outside the Kähler world. For instance, it is false on algebraic varieties in positive characteristic, an

Platonic Surfaces

Abstract. If SO is a Riemann surface with a complete metric of finite area and constant curvature −1, let SC denote the conformal compactification of SO. We show that, under the assumption that the cusps of SO are large, there is a close relationship between the hyperbolic metrics on SO and SC. We use this relationship to show that liminfk→ ∞ λ1(Pk) ≥ 5/36, where the Platonic surface Pk is the conformal compactification of the modular surface Sk

c ○ 1998 Birkhäuser Verlag, Basel Fano contact manifolds and nilpotent orbits

Abstract. A contact structure on a complex manifold M is a corank 1 subbundle F of TM such that the bilinear form on F with values in the quotient line bundle L = TM/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample. If g is a simple Lie algebra, the unique closed orbit in P(g) (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry. In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M ��� P(H 0 (M,L) ∗ ) associated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski an