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Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties.
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 equipped wit...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Providence :
American Mathematical Society,
2016.
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| Colección: | Memoirs of the American Mathematical Society.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Title page; Introduction; Structure of the paper; Notation; Chapter 1. Splittings for MHS on real homotopy types; 1.1. The family of formality quasi-isomorphisms; 1.2. The reduced bar construction; 1.3. The mixed Hodge structure on Quillen's real homotopy type and on homotopy groups; Chapter 2. Non-abelian structures; 2.1. Hodge filtrations; 2.2. Twistor filtrations; 2.3. Mixed Hodge structures; 2.4. Mixed twistor structures; 2.5. Real homotopy types revisited; Chapter 3. Structures on cohomology; 3.1. Cohomology of Hodge filtrations; 3.2. Cohomology of MHS.
- 3.3. Real Deligne cohomology; 3.4. Analogies with limit Hodge structures; 3.5. Archimedean cohomology; Chapter 4. Relative Malcev homotopy types; 4.1. Review of pro-algebraic homotopy types; 4.2. Cosimplicial and DG Hopf algebras; 4.3. Equivalent formulations; 4.4. The reduced bar construction; 4.5. Equivalences of homotopy categories; 4.6. Families of homotopy types; Chapter 5. Structures on relative Malcev homotopy types; 5.1. Homotopy types; 5.2. Grouplike structures; 5.3. Splittings over \cS; 5.4. Mixed Hodge structures on homotopy groups.
- Chapter 6. MHS on relative Malcev homotopy types of compact Kähler manifolds; 6.1. Real homotopy types; 6.2. Relative Malcev homotopy types; Chapter 7. MTS on relative Malcev homotopy types of compact Kähler manifolds; 7.1. Unitary actions; Chapter 8. Variations of mixed Hodge and mixed twistor structures; 8.1. Representations in MHS/MTS; 8.2. Enriching VMTS; 8.3. Absolute Hodge and twistor homotopy types; Chapter 9. Monodromy at the Archimedean place; 9.1. Reformulation via _{∞} derivations; 9.2. Kähler identities; 9.3. Monodromy calculation; Chapter 10. Simplicial and singular varieties
- 10.1. Semisimple local systems; 10.2. The Malcev homotopy type; 10.3. Mixed Hodge structures; 10.4. Enriching twistor structures; Chapter 11. Algebraic MHS/MTS for quasi-projective varieties I; 11.1. The Hodge and twistor filtrations; 11.2. Higher direct images and residues; 11.3. Opposedness; 11.4. Singular and simplicial varieties; Chapter 12. Algebraic MHS/MTS for quasi-projective varieties II -non-trivial monodromy; 12.1. Constructing mixed Hodge structures; 12.2. Constructing mixed twistor structures; 12.3. Unitary monodromy; 12.4. Singular and simplicial varieties.
- 12.5. More general monodromy; Chapter 13. Canonical splittings; 13.1. Splittings of mixed Hodge structures; 13.2. Splittings of mixed twistor structures; 13.3. Reduced forms for Hodge and twistor homotopy types; Chapter 14. \SL₂ splittings of non-abelian MTS/MHS and strictification; 14.1. Simplicial structures; 14.2. Functors parametrising Hodge and twistor structures; 14.3. Deformations; 14.4. Quasi-projective varieties; Bibliography; Back Cover.