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Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting
 
J. P. Pridham University of Edinburgh, United Kingdom
Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting
eBook ISBN:  978-1-4704-3448-9
Product Code:  MEMO/243/1150.E
List Price: $93.00
MAA Member Price: $83.70
AMS Member Price: $55.80
Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting
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Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting
J. P. Pridham University of Edinburgh, United Kingdom
eBook ISBN:  978-1-4704-3448-9
Product Code:  MEMO/243/1150.E
List Price: $93.00
MAA Member Price: $83.70
AMS Member Price: $55.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2432016; 178 pp
    MSC: Primary 14; Secondary 32; 55

    The author defines and constructs 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. The author also shows 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.

  • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2432016; 178 pp
MSC: Primary 14; Secondary 32; 55

The author defines and constructs 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. The author also shows 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.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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