*Patrick Brosnan and Fouad El Zein*

*Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161341
- eISBN:
- 9781400851478
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161341.003.0008
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses the definition of admissible variations of mixed Hodge structure (VMHS), the results of M. Kashiwara in A study of variation of mixed Hodge structure (1986), and applications ...
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This chapter discusses the definition of admissible variations of mixed Hodge structure (VMHS), the results of M. Kashiwara in A study of variation of mixed Hodge structure (1986), and applications to the proof of algebraicity of the locus of certain Hodge cycles. It begins by recalling the relations between local systems and linear differential equations as well as the Thom–Whitney results on the topological properties of morphisms of algebraic varieties. The definition of a VMHS on a smooth variety is given, and the singularities of local systems are discussed. The chapter then studies the properties of degenerating geometric VMHS. Next it gives the definition and properties of admissible VMHS and reviews important local results of Kashiwara. Finally, the chapter recalls the definition of normal functions and explains recent results on the algebraicity of the zero set of normal functions.Less

This chapter discusses the definition of admissible variations of mixed Hodge structure (VMHS), the results of M. Kashiwara in *A study of variation of mixed Hodge structure* (1986), and applications to the proof of algebraicity of the locus of certain Hodge cycles. It begins by recalling the relations between local systems and linear differential equations as well as the Thom–Whitney results on the topological properties of morphisms of algebraic varieties. The definition of a VMHS on a smooth variety is given, and the singularities of local systems are discussed. The chapter then studies the properties of degenerating geometric VMHS. Next it gives the definition and properties of admissible VMHS and reviews important local results of Kashiwara. Finally, the chapter recalls the definition of normal functions and explains recent results on the algebraicity of the zero set of normal functions.

*Franc¸ois Charles and Christian Schnell*

*Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161341
- eISBN:
- 9781400851478
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161341.003.0011
- Subject:
- Mathematics, Geometry / Topology

This chapter surveys the theory of absolute Hodge classes. First, the chapter recalls the construction of cycle maps in de Rham cohomology, which is then used in the definition of absolute Hodge ...
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This chapter surveys the theory of absolute Hodge classes. First, the chapter recalls the construction of cycle maps in de Rham cohomology, which is then used in the definition of absolute Hodge classes. The chapter then deals with variational properties of absolute Hodge classes. After stating the variational Hodge conjecture, the chapter proves Deligne's principle B and discusses consequences of the algebraicity of Hodge bundles and of the Galois action on relative de Rham cohomology. Finally, the chapter provides some important examples of absolute Hodge classes: a discussion of the Kuga–Satake correspondence as well as a full proof of Deligne's theorem which states that Hodge classes on abelian varieties are absolute.Less

This chapter surveys the theory of absolute Hodge classes. First, the chapter recalls the construction of cycle maps in de Rham cohomology, which is then used in the definition of absolute Hodge classes. The chapter then deals with variational properties of absolute Hodge classes. After stating the variational Hodge conjecture, the chapter proves Deligne's principle B and discusses consequences of the algebraicity of Hodge bundles and of the Galois action on relative de Rham cohomology. Finally, the chapter provides some important examples of absolute Hodge classes: a discussion of the Kuga–Satake correspondence as well as a full proof of Deligne's theorem which states that Hodge classes on abelian varieties are absolute.

*Günter Harder and A. Raghuram*

- Published in print:
- 2019
- Published Online:
- September 2020
- ISBN:
- 9780691197890
- eISBN:
- 9780691197937
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691197890.003.0001
- Subject:
- Mathematics, Number Theory

This introductory chapter presents the general principle that the cohomology of arithmetic groups and the L-functions L(s, π, r) attached to irreducible “pieces” π have a strong symbiotic ...
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This introductory chapter presents the general principle that the cohomology of arithmetic groups and the L-functions L(s, π, r) attached to irreducible “pieces” π have a strong symbiotic relationship with each other. The symbiosis goes in both directions. The first is that expressions in the special values L(k, π, r) enter in the transcendental description of the cohomology. Since the cohomology is defined over ℚ one can deduce rationality (algebraicity) results for these expressions in special values. Next, these special values in turn influence the structure of the cohomology as a Hecke module; prime numbers dividing these values occur in the denominators of Eisenstein classes.Less

This introductory chapter presents the general principle that the cohomology of arithmetic groups and the *L*-functions *L*(*s*, *π*, *r*) attached to irreducible “pieces” π have a strong symbiotic relationship with each other. The symbiosis goes in both directions. The first is that expressions in the special values *L*(*k*, *π*, *r*) enter in the transcendental description of the cohomology. Since the cohomology is defined over ℚ one can deduce rationality (algebraicity) results for these expressions in special values. Next, these special values in turn influence the structure of the cohomology as a Hecke module; prime numbers dividing these values occur in the denominators of Eisenstein classes.