Algebraic families of \(E_\infty\)-spectra
In the paper under review, a spectrum \(S\) is a collection of based spaces \(\{S(n)\mid n\in\mathbb{N}\}\) together with a compatible family of homeomorphisms \(s_n:S(n)\to\Omega S(n+1)\) for all \(n\in\mathbb{N}\). Recall that a spectrum determintes a generalized cohomology theory. The author shows how to associate three distinct spectra to a complex projective variety \(X\): the morphic spectrum (an \(E_\infty\)-ring spectrum), the multiplicative morphic spectra and the holomorphic \(K\)-theory spectrum. All these constructions are contravariantly functorial on \(X\) and they are based on standard infinite loop space machinery. The coefficients of cohomology theories associated to the spectra reflect algebraic geometric and topological invariants for \(X\). In the morphic case the coefficients are given in terms of Friedlander-Lawson's morphic cohomology [\textit{E. M. Friedlander} and \textit{H. B. Lawson} jun., Ann. Math., II. Ser. 136, No.~2, 361-428 (1992; Zbl 0788.14014)]. The reviewer mentions that the author tries to be as self-contained as possible in the area. For example, he introduces in Section 2 of the work, the basic moduli spaces of algebraic cycles and the mapping spaces associated to them, which are the central tool in his constructions.
scientific article; zbMATH DE number 1367564