Real non-abelian mixed Hodge structures for schematic homotopy types of quasi-projective varieties

Details Real non-abelian mixed Hodge structures for schematic homotopy types of quasi-projective varieties Real non-abelian mixed Hodge structures for schematic homotopy types of quasi-projective varieties

2011-04-07

arxiv.org/abs/1104.1409v1

Mathematics

Algebraic Geometry

48 pages

Scientific paper

The relative Malcev homotopy type of a quasi-projective variety carries a canonical non-positively weighted algebraic mixed twistor structure (MTS), provided we restrict to extensions of local systems with trivial monodromy around the components of the divisor. This can be enriched to an analytic mixed Hodge structure (MHS), which becomes algebraic if we restrict to extensions of local systems underlying VHS. We then show that every non-positively weighted MHS or MTS on homotopy types admits a canonical splitting over SL_2. For smooth varieties, this allows us to characterise the MHS or MTS in terms of the Gysin spectral sequence, together with the monodromy action at the Archimedean place. It also means that the relative Malcev homotopy groups carry canonical MTS or MHS.

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