Algebraic Families of E 1 -spectra

We construct three di erent spectra, and hence three generalized cohomology theories, associated to an algebraic variety X: the morphic spectrum ZX an E1-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 coe cients for the corresponding cohomology theories re ect algebraic geometric and topological invariants for the variety X. In the morphic case, the coe cients are given in terms of FriedlanderLawson's morphic cohomology for varieties [FL92]. The theory carries total Chern class maps and total cycle maps, extending to the stable category classical constructions in algebraic geometry.