Hodge theory in the geometric sense is a field which lie in the
intersection of complex, algebraic and differential geometry and is in
particular used in mirror symmetry. The cohomology of compact Kähler
manifolds, e.g. smooth projective varities, are the main examples of
Hodge structures.

In this talk I want to give a short introduction to Hodge structures
and their variations, i.e. abstract definitions and the main examples
which are constructed from families of compact Kähler manifolds. After
that one could consider period domains and period mappings which are
an important tool in complex algebraic geometry.

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