Local comparisons of homological and homotopical mixed Hodge polynomials (Q1988510)

According to \textit{P. Deligne} [Publ. Math., Inst. Hautes Étud. Sci. 40, 5--57 (1971; Zbl 0219.14007)], \textit{J. W. Morgan} [Publ. Math., Inst. Hautes Étud. Sci. 48, 137--204 (1978; Zbl 0401.14003)] and \textit{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.