Lecture Notes in Mathematics

By: Tim Kirschner

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Paperback | 25 September 2015

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298 Pages

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22.86 x 15.24 x 1.27

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The text consists in three parts. The first part develops a theory of period mappings of "Hodge-de Rham type" for families of complex manifolds, the essential argument being that the manifolds in question need neither be compact nor Kahler. For compact Kahler manifolds we fall back on Griffiths' classical theory of period mappings. The second part investigates the degeneration behavior of the relative Frolicher spectral sequence associated to a submersive morphism of complex manifolds, the main point again being that the fibers of the morphism in question need neither be compact nor Kahler. In turn, the third part focuses on applications of the previous parts to the study of irreducible symplectic complex spaces. The latter notion generalizes the idea of an irreducible symplectic manifold, dubbed a hyperkahler manifold in differential geometry, to possibly singular spaces. The three parts of the work offer valuable resources, both individually and as a combined whole.

Industry Reviews

"The book under review aims to extend a number of methods and results from algebraic geometry (schemes and algebraic varieties) to the theory of complex analytic spaces. ... The book is very clearly written, with almost all prerequisites collected in two appendices. In this way it is interesting not only for the original results it contains, but also as an introduction to this area lying at the intersection of algebraic and complex geometry." (Andrei D. Halanay, Mathematical Reviews, December, 2016)

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