Non-abelian real Hodge theory for proper varieties

Details Non-abelian real Hodge theory for proper varieties Non-abelian real Hodge theory for proper varieties

2006-11-22

arxiv.org/abs/math/0611686v5

Mathematics

Algebraic Geometry

32 pages; v2 references updated; v3 corrections in Sect. 1.4 & 4.1, new title; v4 final section reworked. Superseded by 0902.0

Scientific paper

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 G_m-action. This extends to a real Hodge structure, in the form of a discrete C^*-action, such that C^* x X -> X^{sch} 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.

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