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- Title
RANKS OF HOMOTOPY AND COHOMOLOGY GROUPS FOR RATIONALLY ELLIPTIC SPACES AND ALGEBRAIC VARIETIES.
- Authors
LIBGOBER, ANATOLY; SHOJI YOKURA
- Abstract
We discuss inequalities between the values of 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 stabilization threshold pp(X; e) for a simply connected rationally elliptic space X and a positive real number e, and we show that the Hilali conjecture implies that pp(X; 1) ⩽ 3.
- Subjects
ALGEBRAIC varieties; ALGEBRAIC spaces; PROJECTIVE spaces; REAL numbers; GENERATING functions; HOMOTOPY groups
- Publication
Homology, Homotopy & Applications, 2022, Vol 24, Issue 2, p93
- ISSN
1532-0073
- Publication type
Article
- DOI
10.4310/HHA.2022.v24.n2.a5