Hodge Theory and Lefschetz Linear Algebra

Abstract

The cohomology rings of smooth complex projective algebraic varieties satisfy a “package” of properties: Poincare duality, weak Lefschetz, hard Lefschetz, and the Hodge–Riemann bilinear relations. These properties impose conditions on the structure of the cohomology ring, and give rise to interesting linear operators and bilinear forms on the ring. In several contexts this “package” yields important results about geometry, representation theory, or combinatorics. We discuss the above properties purely in terms of linear algebra, so that some of the techniques and intuition from geometry can be brought to bear on more general problems. Our focus is on the techniques which appear in the proof of Soergel’s conjecture, but there have been successes in other fields as well (e.g. in the combinatorics of matroids and polytopes), and it seems there will be more to come.