2006/11/22 by J. P. Pridham, Pridham, J. P.
Mathematics · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.math/0611686
We show that if X is any proper complex variety, there is a weight decomposition on the real schematic homotopy type, in the form of an algebraic Gm-action. This extends to a real Hodge structure, in the form of a discrete C^*-action, such that C^* x X -> Xsch is real analytic. If the fundamental group is algebraically good, and the higher homotopy groups have finite rank, this gives bigraded decompositions on the complexified homotopy groups. For smooth proper varieties, the Hodge structure can be recovered from the cohomology ring with coefficients in the universal semisimple local system.