A Kähler Package for Grassmannians

Abstract

One of the fundamental results of Hodge theory is that the cohomology rings of compact Kähler manifolds obey three rules: Poincaré Duality, the hard Lefschetz theorem, and the Hodge-Riemann bilinear relations. These relations, known together as the Kähler package, have appeared in many several other settings with profound implications: in polytopes and the \(\smash{g}\)-conjecture, in matroids and the log-concavity of characteristic polynomials, and perhaps in projective varieties and Grothendieck's conjecture. In this thesis, we formulate and develop an analogous Kähler package over Grassmann manifolds. Our graded ring is the function space over the Grassmannians of the vector space \(\mathbb F^n\), equipped with a natural associative convolution product, and our Lefschetz hyperplane operator is the generalized Radon transform on Grassmannians. We establish the Kähler package in full over finite fields \(\mathbb F_q\) and state partial results over the fields \(\mathbb R\) and \(\mathbb C\).