Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hilali conjecture and complex algebraic varieties

Shoji Yokura

TL;DR

The paper surveys the Hilali conjecture for rationally elliptic spaces, focusing on complex algebraic varieties and deriving revised formulas and conjectures. It consolidates fundamental properties (via Halperin and Friedlander–Halperin) and presents concrete classes of rationally elliptic spaces (toric, Kähler) where the conjecture holds, including singular cases. It proposes a unifying conjecture that rationally elliptic complex algebraic varieties have Poincaré polynomials matching products of spheres and projective spaces, and it investigates the Hilali conjecture modulo products across Poincaré and mixed Hodge polynomials. The work also links formality and curvature considerations in Kähler geometry, and extends the Hilali framework to mixed Hodge structures, with several stabilization-type results for powers of spaces. Overall, it advances understanding of when homotopy ranks are controlled by homology ranks in both topological and algebro-geometric contexts, and outlines several open questions and conjectures for broader classes of varieties.

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 Poincaré polynomial and the homotopical Poincaré polynomial.

Hilali conjecture and complex algebraic varieties

TL;DR

The paper surveys the Hilali conjecture for rationally elliptic spaces, focusing on complex algebraic varieties and deriving revised formulas and conjectures. It consolidates fundamental properties (via Halperin and Friedlander–Halperin) and presents concrete classes of rationally elliptic spaces (toric, Kähler) where the conjecture holds, including singular cases. It proposes a unifying conjecture that rationally elliptic complex algebraic varieties have Poincaré polynomials matching products of spheres and projective spaces, and it investigates the Hilali conjecture modulo products across Poincaré and mixed Hodge polynomials. The work also links formality and curvature considerations in Kähler geometry, and extends the Hilali framework to mixed Hodge structures, with several stabilization-type results for powers of spaces. Overall, it advances understanding of when homotopy ranks are controlled by homology ranks in both topological and algebro-geometric contexts, and outlines several open questions and conjectures for broader classes of varieties.

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 Poincaré polynomial and the homotopical Poincaré polynomial.
Paper Structure (14 sections, 35 theorems, 71 equations)

This paper contains 14 sections, 35 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Rationally elliptic space
  3. Hilali conjecture
  4. Fundamental examples for which the Hilali conjecture holds.
  5. Rationally elliptic singular varieties for which the Hilali conjecture holds.
  6. Fundamental properties of rationally elliptic spaces
  7. S. Halperin's Theorems
  8. J. B. Friedlander--S. Halperin's Theorems
  9. Rationally elliptic smooth toric varieties
  10. Rationally elliptic Kähler manifolds
  11. Formal dimension and Hilali conjecture
  12. Hilali Conjecture modulo "products"
  13. "Hilali conjecture modulo products" for Poincaré polynomials
  14. "Hilali conjecture modulo products" for mixed Hodge polynomials

Key Result

Theorem 2.5

A simply connected Riemannian manifold with entire Grauert tube is rationally elliptic.

Theorems & Definitions (81)

  • Conjecture 2.1: Bott conjecture
  • Conjecture 2.2: Hopf conjecture Hopf(also see BG), 1953 or Chern conjecture Chern, 1966
  • Conjecture 2.3: Gromov conjecture Gromov,1981
  • Remark 2.4
  • Theorem 2.5
  • Conjecture 3.1: Hilali Conjecture, 1990
  • Example 3.2
  • Example 3.3
  • Proposition 3.4
  • proof
  • ...and 71 more