Local comparisons of homological and homotopical mixed Hodge polynomials

For a simply connected complex algebraic variety X, by the mixed Hodge structures [mathematical expression not reproducible] of the homology group [H.sub.*](X; Q) and the homotopy groups [[pi].sub.*] (x) [cross product] Qrespectively, we have the following mixed Hodge polynomials [mathematical expression not reproducible], which are respectively called the homological mixed Hodge polynomial and the homotopical mixed Hodge polynomial. In this paper we discuss some inequalities concerning these two mixed Hodge polynomials. Key words: mixed Hodge structures; mixed Hodge polynomials; Hilali conjecture; rational homotopy theory.