If X is a smooth complex projective variety, then each singular cohomology group of X (with complex coefficients) has a certain canonical direct-sum decomposition, called the Hodge decomposition. Moreover, the summands of the Hodge decomposition may be identified explicitly in terms of the differential forms on X.

The goal of Deligne’s mixed Hodge theory is to generalize the Hodge decomposition to the case when X is an arbitrary scheme of finite type over the complex numbers. Deligne’s main result, developed in his series of papers Théorie de Hodge I–III, asserts that each cohomology group of any such X naturally carries a certain (rather elaborate) structure, called a mixed Hodge structure, which recovers the Hodge decomposition when X is smooth and projective.

The goal of this course is to explain the ideas and techniques of Deligne’s theory. Topics will include the basics of mixed Hodge structures, logarithmic differentials, and the method of cohomological descent. Some additional topics, such as limit mixed Hodge structures and cohomological invariants of singularities, may be discussed.

Prerequisites: Sheaves, sheaf cohomology. Smooth morphisms, proper morphisms (otherwise, little algebraic geometry is needed). The basics of spectral sequences (degeneration, filtration on the limit terms, the Leray spectral sequence). Derived functors. The basics of derived categories may be helpful.