q-cohomologically complete and q-pseudoconvex domains

This dissertation is devoted to the study of the different viewpoints from which an open subset D of a Stein manifold M can be considered, as the geometric concepts of q-completeness and q- pseudoconvexity and the analytic ideas of vanishing of cohomology groups after a certain level and inextendibility of cohomology clas­ses or holomorphic functions. The idea is to generalize to any integer q the well known equivalence between O-complateness and O-cohonological completeness ( see theorem 2.81). A step in this direction, namely that if D is q-complete then it is also q-cohonologically complete, was done in 1962 by Andreotti and Grauert, but the converse implication is still an open problem. Using a rather indirect tool involving certain cohomology classes called "test classes" we can manage to prove that if D is cohomologically q-complete and has C2 boundary then it is q-complete, and this is probably the most interesting result appearing in this thesis ( see theorem 3.3.1) This method however can also be applied to answer certain natural questions about inaxtendibility of cohomology classes, analogous to inextendibility of hoiomorphic functions for domains of holomorphy, and the answer turns out to be not surprising if D has C2 boundary (see theorem 4.1.8) but less intuitive in the general case and counterexamples illustrating this behaviour are discussed in chapter 3. section 4. In particular we describe a particularly interesting applica­tion of the test classes that gives a lower bound on the number of analytic functions needed to define an analytic subvariety just touching D at a point x belonging to its boundary provided the behavi­our of ꝽD near x 13 known (see theorem 4.2.3). All these results can be deduced without knowing the explicit expression for these cohomology classes, but such an expression in terms of Dolbeault cohomology and Cech cohomology is given in the last chapters It can be observed that the test classes are related to the Bochner-Martinelli kernel