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

Yokura, Shoji

Proc. Japan Acad., Ser. A 96, No. 3, 28-31 (2020).

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.

Reviewer: Jean Claude Thomas (Angers)

MSC:

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

Keywords:

mixed Hodge structures; mixed Hodge polynomials; Hilali conjecture; rational homotopy theory

Citations:

Zbl 0219.14007; Zbl 0401.14003; Zbl 0639.14002

References:

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