By W. Barth, Raghavan Narasimhan
The articles during this quantity have been written to commemorate Reinhold Remmert's sixtieth birthday in June, 1990. they're surveys, intended to facilitate entry to a few of the various features of the speculation of complicated manifolds, and exhibit the interaction among advanced research and plenty of different branches of arithmetic, algebraic geometry, differential topology, representations of Lie teams, and mathematical physics being in simple terms the obvious of those branches. every one of those articles should still serve not just to explain the actual circle of principles in complicated research with which it offers but additionally as a advisor to the numerous mathematical rules on the topic of its subject.
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Additional resources for Several Complex Variables VI: Complex Manifolds (Encyclopaedia of Mathematical Sciences)
Sample text
A (weakly) pseudoconvex domain in (c2 with real analytic boundary whose automorphism group is non-compact is biholomorphic to a domain of the form {(z, w) E (II21 1z12+ Iw12P < l},p > 0 an integer. We have described above the approach to boundary regularity based on the Bergman projection and results which can be proved using related techniques. Different methods have been found. 8 to give a new and elementary proof of the Wong-Rosay theorem. Further, he and Hasanov [PHI have used it in conjunction with a C” version of the “edge of the wedge” theorem to give a direct elementary proof of Fefferman’s theorem which gives results about extensions in domains with Ck boundary (k I 00).
Let Q be a bounded strictly pseudoconvex domain in aI:” with smooth boundary. Then 0 can be properly embeddedin AN for someN. Low [Lo 21 and Forstneric [FF 31 have also shown that strictly pseudoconvex domains Q can be embedded into BN for some N. Some of these embeddings do not even extend continuously to dQ, while some are continuous but not differentiable. There have been several refinements of these theorems. It is of course natural to ask about the smallest codimension that can be allowed in these theorems.
Although the algebro-geometric aspects of the program never achieved full success, it has had a lasting influence on modern Hodge theory and algebraic geometry. Shortly after Griffiths’ work appeared, Deligne’s work on mixed Hodge theory [D4] introduced a set of new ideas and techniques into the subject. He introduced the notion of a mixed Hodge structure, which is, roughly speaking, an iterated extension of Hodge structures of increasing weight. The cohomology groups of any complex algebraic variety (not necessarily compact, not necessarily smooth) carry mixed Hodge structures.