Formality and splitting of real non-abelian mixed Hodge structures

Abstract

We define and construct mixed Hodge structures on real schematic homotopy types of complex projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x-i), giving new results even for simply connected varieties. Thus the mixed Hodge structures can be recovered from cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these all become real variations of mixed Hodge structure.

This paper has been withdrawn by the author. Withdrawn - superseded by sections 1-10 of arXiv:1104.1409; 71 pages, supersedes arXiv:math/0611686; v2 overhauled, main new results are Thm 3.10, Sections 7 & 8, & parts of Section 2

References in corpus (6)

• Deformation theory via differential graded Lie algebras
• Pro-algebraic homotopy types
• Galois actions on homotopy groups
• Non-Additive Prolegomena (to any future Arithmetic that will be able to present itself as a Geometry)
• Non-abelian real Hodge theory for proper varieties
• The Hodge theoretic fundamental group and its cohomology

Cited by in corpus (1)

• The Hodge theoretic fundamental group and its cohomology