On the \(\Gamma\)-factors of motives. II (Q5931107)

Let \(X\) be a smooth projective variety over a number field \(k\). The definition of the local Euler factor at the finite place for the motive \(H^n(X)\) involves the Galois action on the \(\ell\)-adic cohomology groups [\textit{J.-P. Serre}, Sémin. Delange-Pisot-Poitou 11 (1969/70), Exp. No. 19 (1970; Zbl 0214.48403)]. At the infinite place the local Euler factor is a product of \(\Gamma\)-factors determined by the real Hodge structure on the singular cohomology. First steps towards a uniform description of the local Euler factors were made by \textit{C. Deninger} [cf. Part I, Invent. Math. 104, 245--261 (1991; Zbl 0739.14010); ibid. 107, 135--150 (1992; Zbl 0762.14015); Proc. Symp. Pure Math. 55, Part 1, 707--743 (1994; Zbl 0816.14010)]. He gave an interpretation of local Euler factors as regularized characteristic power series on infinite-dimensional complex vector spaces with a linear flow.NEWLINENEWLINENEWLINEIn this article, the author proposes a direct dynamical description of the Archimedean \(\Gamma\)-factor. The approach, based on a result of \textit{C. Simpson} [Proc. Symp. Pure Math. 62, Part 2, 217--281 (1997; Zbl 0914.14003)], replaces the consideration of the Hodge filtration by looking at a relative de Rham complex with a deformed differential.