Algebraic families of E∞-spectra

We construct three different spectra, and hence three generalized cohomology theories, associated to an algebraic variety X: the morphic spectrum ZX—an E∞-ring spectrum—related to intersection theory when X is smooth; the multiplicative morphic spectrum MX and the holomorphic K-theory spectrum KX. The constructions use holomorphic maps from X into appropriate moduli spaces, and are functorial on X. The coefficients for the corresponding cohomology theories reflect algebraic geometric and topological invariants for the variety X. In the morphic case, the coefficients are given in terms of Friedlander–Lawsonʹs morphic cohomology for varieties (Friedlander and Lawson, 1992). The theory carries total Chern class maps and total cycle maps, extending to the stable category classical constructions in algebraic geometry.