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.

Presents sheaves with a clear connection to the set-theoretic foundations Strives for a maximum of rigor (concerning proofs, statements, definitions, and notation) Overcomes the canonical isomorphism paradigm; all morphisms are given/constructed explicitly Introduces a Gauß-Manin connection for families of possibly non-compact manifolds Includes supplementary material: sn.pub/extras