The MHM Project

by Claude Sabbah & Christian Schnell

Description of the project

The theory of mixed Hodge modules represents a vast generalization of classical Hodge theory. It was introduced by Morihiko Saito in two long papers in 1988 and 1990, building on the foundations created by many people during the 1970s and 1980s, especially in the study of variations of mixed Hodge structure and their degenerations, perverse sheaves, and $$\mathscr D$$-module theory. The theory is very powerful and led to striking applications right from the beginning, such as

• - a new, complex-analytic proof for the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber;
• - the construction of pure Hodge structures on the intersection cohomology of projective algebraic varieties;
• - a far-reaching generalization of Kodaira's vanishing theorem.
Over the years, new applications of mixed Hodge modules came to light, but it is fair to say that in spite of its power, the theory is not as widely known as it should be. A learning workshop was organized by Mircea Mustaţă, Claude Sabbah and Christian Schnell in Oxford, August 19-23, 2013, within the workshop program of the Clay Mathematics Institute. Its goal was to make Saito's theory more accessible by bringing together a group of people interested in this topic, in order to discuss both the fundamentals of the theory and its applications. While a handful of participants were experts in this topic (including Saito himself), the majority consisted of experts in other fields, who used some aspects of the theory and who wanted to get a better understanding of its inner workings and of the existing applications.