Hodge Structure - Mixed Hodge Structures

Mixed Hodge Structures

It was noticed by Jean-Pierre Serre in the 1960s based on the Weil conjectures that even singular (possibly reducible) and non-complete algebraic varieties should admit 'virtual Betti numbers'. More precisely, one should be able to assign to any algebraic variety X a polynomial P_X(t), called its virtual Poincaré polynomial, with the properties

• if X is nonsingular and projective (or complete);
• if Y is a closed algebraic subset of X and U=X\Y.

The existence of such polynomials would follow from the existence of an analogue of Hodge structure in the cohomologies of a general (singular and non-complete) algebraic variety. The novel feature is that the nth cohomology of a general variety looks as if it contained pieces of different weights. This led Alexander Grothendieck to his conjectural theory of motives and motivated a search for an extension of Hodge theory, which culminated in the work of Pierre Deligne. He introduced the notion of a mixed Hodge structure, developed techniques for working with them, gave their construction (based on Hironaka's resolution of singularities) and related them to the weights on l-adic cohomology, proving the last part of the Weil conjectures.