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Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting

About this Title

J. P. Pridham, School of Mathematics and Maxwell Institute of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 243, Number 1150
ISBNs: 978-1-4704-1981-3 (print); 978-1-4704-3448-9 (online)
DOI: https://doi.org/10.1090/memo/1150
Published electronically: April 12, 2016
Keywords: Non-abelian Hodge theory
MSC: Primary 14C30; Secondary 14F35, 32S35, 55P62

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Table of Contents

Chapters

  • Introduction
  • 1. Splittings for MHS on real homotopy types
  • 2. Non-abelian structures
  • 3. Structures on cohomology
  • 4. Relative Malcev homotopy types
  • 5. Structures on relative Malcev homotopy types
  • 6. MHS on relative Malcev homotopy types of compact Kähler manifolds
  • 7. MTS on relative Malcev homotopy types of compact Kähler manifolds
  • 8. Variations of mixed Hodge and mixed twistor structures
  • 9. Monodromy at the Archimedean place
  • 10. Simplicial and singular varieties
  • 11. Algebraic MHS/MTS for quasi-projective varieties I
  • 12. Algebraic MHS/MTS for quasi-projective varieties II — non-trivial monodromy
  • 13. Canonical splittings
  • 14. $\mathrm {SL}_2$ splittings of non-abelian MTS/MHS and strictification

Abstract

We define and construct mixed Hodge structures on real schematic homotopy types of complex quasi-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 $\mathbb {R}[x]$ equipped with the Hodge filtration given by powers of $(x-i)$, giving new results even for simply connected varieties. The mixed Hodge structures can thus be recovered from the Gysin spectral sequence of cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these structures all become real variations of mixed Hodge structure.

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