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Extending Griffiths' classical theory of period mappings for compact Kähler manifolds, this book develops and applies a theory of period mappings of "Hodge-de Rham type" for families of open complex manifolds. The text consists of three parts. The first part develops the theory. The second part investigates the degeneration behavior of the relative Frölicher spectral sequence associated to a submersive morphism of complex manifolds. The third part applies the preceding material to the study of irreducible symplectic complex spaces. The latter notion generalizes the idea of an irreducible symplectic manifold, dubbed an irreducible hyperkähler manifold in differential geometry, to possibly singular spaces. The three parts of the work are of independent interest, but intertwine nicely.
"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)