Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties

TL;DR

This paper investigates inequalities between homotopical and cohomological Poincaré polynomials for rationally elliptic spaces and algebraic varieties, introducing invariants and exploring implications for mixed Hodge structures and the Hilali conjecture.

Contribution

It establishes new inequalities for Poincaré polynomials of rationally elliptic spaces and varieties, introduces the stabilization threshold invariant, and relates these to the Hilali conjecture.

Findings

Homotopical and cohomological Poincaré polynomials satisfy specific inequalities.

The homotopical mixed Hodge polynomial of a rationally elliptic toric manifold decomposes into sums of polynomials of projective spaces.

The stabilization threshold is bounded above by 3 if the Hilali conjecture holds.

Abstract

We discuss inequalities between the values of \emph{homotopical and cohomological Poincar\'e polynomials} of the self-products of rationally elliptic spaces. For rationally elliptic quasi-projective varieties, we prove inequalities between the values of generating functions for the ranks of the graded pieces of the weight and Hodge filtrations of the canonical mixed Hodge structures on homotopy and cohomology groups. Several examples of such mixed Hodge polynomials and related inequalities for rationally elliptic quasi-projective algebraic varieties are presented. One of the consequences is that the homotopical (resp. cohomological) mixed Hodge polynomial of a rationally elliptic toric manifold is a sum (resp. a product) of polynomials of projective spaces. We introduce an invariant called \emph{stabilization threshold} $\frak{pp} (X;\varepsilon)$ for a simply connected rationally elliptic space $X$ and a positive real number $\varepsilon$, and we show that the Hilali conjecture implies that $\frak{pp} (X;1) \le 3$.