Abstract
I give a thorough introduction to the global theory of, possibly singular, symplectic complex spaces. The spaces are assumed to be of Kahler type for the most part. My presentation focuses on the study of proper and flat deformations. The corresponding local theory of symplectic singularities is hardly touched upon. The key results are a local Torelli theorem on the second cohomology, which generalizes the local Torelli theorem for irreducible symplectic complex manifolds to potentially singular spaces, as well as an analogous generalization of the Fujiki relation. Moreover, an entire section is devoted to developing the Beauville-Bogomolov quadratic form in the generalized context.
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