The focus of this book is on the further development of the
classical achievements in analysis of several complex variables,
the analytic continuation and the analytic structure of sets, to
settings in which the q-pseudoconvexity in the sense of Rothstein
and the q-convexity in the sense of Grauert play a crucial role.
After giving a brief survey of notions of generalized convexity and
their most important results, the authors present recent statements
on analytic continuation related to them. Rothstein (1955) first
introduced q-pseudoconvexity using generalized Hartogs figures.
Slodkowski (1986) defined q-pseudoconvex sets by means of the
existence of exhaustion functions which are q-plurisubharmonic in
the sense of Hunt and Murray (1978). Examples of q-pseudoconvex
sets appear as complements of analytic sets. Here, the relation of
the analytic structure of graphs of continuous surfaces whose
complements are q-pseudoconvex is investigated. As an outcome, the
authors generalize results by Hartogs (1909), Shcherbina (1993),
and Chirka (2001) on the existence of foliations of pseudoconcave
continuous real hypersurfaces by smooth complex ones. A similar
generalization is obtained by a completely different approach using
L(2)-methods in the setting of q-convex spaces. The notion of
q-convexity was developed by Rothstein (1955) and Grauert (1959)
and extended to q-convex spaces by Andreotti and Grauert (1962).
Andreotti-Grauert's finiteness theorem was applied by Andreotti and
Norguet (1966-1971) to extend Grauert's solution of the Levi
problem to q-convex spaces. A consequence is that the sets of
(q-1)-cycles of q-convex domains with smooth boundaries in
projective algebraic manifolds, which are equipped with complex
structures as open subsets of Chow varieties, are in fact
holomorphically convex. Complements of analytic curves are studied,
and the relation of q-convexity and cycle spaces is explained.
Finally, results for q-convex domains in projective spaces are
shown and the q-convexity in analytic families is investigated.
General
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