Geodesic
Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties
Homology, homotopy, and applications, Tome 24 (2022) no. 2, pp. 93-113.

Voir la notice de l'article provenant de la source International Press of Boston

We discuss inequalities between the values of homotopical and cohomological Poincaré 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 stabilization threshold $\mathfrak{pp}(X; \varepsilon)$ for a simply connected rationally elliptic space $X$ and a positive real number $\varepsilon$, and we show that the Hilali conjecture implies that $\mathfrak{pp}(X; 1) \leqslant 3$.
DOI : 10.4310/HHA.2022.v24.n2.a5
Classification : 32S35, 55N99, 55P62, 55Q40
Keywords: mixed Hodge structure, mixed Hodge polynomial, Hilali conjecture, rational homotopy theory 
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     title = {Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties},
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     url = {http://geodesic.mathdoc.fr/articles/10.4310/HHA.2022.v24.n2.a5/}
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Anatoly Libgober; Shoji Yokura. Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties. Homology, homotopy, and applications, Tome 24 (2022) no. 2, pp. 93-113. doi : 10.4310/HHA.2022.v24.n2.a5. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2022.v24.n2.a5/

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