Local comparisons of homological and homotopical mixed Hodge polynomials. (English) Zbl 1442.32037

According to P. Deligne [Publ. Math., Inst. Hautes Étud. Sci. 40, 5–57 (1971; Zbl 0219.14007)], J. W. Morgan [Publ. Math., Inst. Hautes Étud. Sci. 48, 137–204 (1978; Zbl 0401.14003)] and V. Navarro Aznar [Invent. Math. 90, 11–76 (1987; Zbl 0639.14002)], the rational cohomology and the homotopy Lie algebra of every simply connected algebraic variety \(X\) over \(C\) are endowed with functorial mixed Hodge structures. This means that there exists a finite increasing filtration \(W\) on the \(Q\)-vector spaces \(H_*(X,Q)\) and \(\pi_k(X)\otimes Q\) called the weight filtration and a finite decreasing filtration \(F\) of the \(C\)-vector space \(H_*(X,C)\) and \(\pi_k(X)\otimes C\), called the Hodge filtration, satisfying extra conditions.
In this paper the author assumes that \(\dim k \pi_k(X)\otimes Q<+\infty\) and considers the following mixed Hodge polynomials \[QH_X(t,u,v):=\sum_{k,p,q}\dim\left(Gr^p_{F_*}Gr^{W_\bullet}_{p+q}H_k(X;C)\right)t^ku^{-p}v^{-q}\] and \[QH^\pi_X(t,u,v):=\sum_{k,p,q}\dim\left(Gr^p_{\tilde F_*}Gr^{\tilde W_\bullet}_{p+q}(\pi_k(X)\otimes C)\right)t^ku^{-p}v^{-q}.\]
In this short note the author announces some inequalities between these two mixed Hodge polynomials. The author announces also a joint paper with Libgober including detailed proofs of the results as well as calculations and further information about mixed Hodge polynomials of elliptic spaces.


32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
55P62 Rational homotopy theory
55Q40 Homotopy groups of spheres
55N99 Homology and cohomology theories in algebraic topology
Full Text: DOI arXiv Euclid


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