Evangelos Mouroukos,
An introduction to Deligne's Hodge Theory
The classical Hodge theorem asserts that the cohomology groups
of a compact Kaehler manifold (e.g. a non-singular complex projective
variety) have a remarkable property: the decomposition of a differential
form according to (p,q)-type descends to cohomology. For an arbitrary
complex algebraic variety (not necessarily non-singular or projective)
Deligne showed that the cohomology groups enjoy a much richer structure,
encoded in the data of what he called a mixed Hodge structure. This
theory has had numerous applications to the topology of algebraic
varieties. My plan is to give an elementary introduction to Deligne's
theory, some computations of mixed Hodge structures and some suggestions
of how to use them in practice.