Hilali conjecture and complex algebraic varieties

TL;DR

This paper explores the Hilali conjecture in the context of rationally elliptic spaces and complex algebraic varieties, providing examples, revising formulas, and discussing related Hodge-theoretic topics.

Contribution

It offers new examples, revises existing formulas, and proposes conjectures related to the Hilali conjecture for rationally elliptic spaces and algebraic varieties.

Findings

Confirmed the Hilali conjecture for several classes of rationally elliptic spaces

Proposed revised formulas and new conjectures related to the conjecture

Discussed the role of mixed Hodge polynomials and Poincaré polynomials in this context

Abstract

A simply connected topological space is called \emph{rationally elliptic} if the rank of its total homotopy group and its total (co)homology group are both finite. A well-known Hilali conjecture claims that for a rationally elliptic space its homotopy rank \emph{does not exceed} its (co)homology rank. In this paper, after recalling some well-known fundamental properties of a rationally elliptic space and giving some important examples of rationally elliptic spaces and rationally elliptic singular complex algebraic varieties for which the Hilali conjecture holds, we give some revised formulas and some conjectures. We also discuss some topics such as mixd Hodge polynomials defined via mixed Hodge structures on cohomology group and the dual of the homotopy group, related to the ``Hilali conjecture \emph{modulo product}", which is an inequality between the usual homological Poincar\'e polynomial and the homotopical Poincar\'e polynomial.