Abelian schemes

This chapter reproduces the theory of degeneration data for polarized abelian varieties, following D. Mumford's 1972 monograph, An analytic construction of degenerating abelian varieties over complete rings, as well as the first three chapters of the Degeneration of abelian varieties (1990), by G. Faltings and C.-L. Chai. Although there is essentially nothing new in this chapter, some modifications have been introduced to make the statements compatible with a certain understanding of the proofs. Moreover, since Mumford and Faltings and Chai have supplied full details only in the completely degenerate case, this chapter balances the literature by avoiding the special case and treating all cases equally.

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Geometric Invariant Theory ◽

10.1007/978-3-662-00095-3_7 ◽

1965 ◽

pp. 115-127

Author(s):

David Mumford

Keyword(s):

Abelian Schemes

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Invariant Cycles on Abelian Schemes

Journal of Mathematical Sciences ◽

10.1007/s10958-020-04998-5 ◽

2020 ◽

Vol 250 (1) ◽

pp. 69-75

Author(s):

O. V. Makarova

Keyword(s):

Abelian Schemes

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The realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne–Beilinson cohomology

International Journal of Number Theory ◽

10.1142/s1793042117501378 ◽

2017 ◽

Vol 13 (09) ◽

pp. 2471-2485 ◽

Cited By ~ 1

Author(s):

Danny Scarponi

Keyword(s):

Chow Groups ◽

Degree Zero ◽

Deligne Cohomology ◽

Abelian Scheme ◽

Logarithmic Singularities ◽

Abelian Schemes

In 2014, Kings and Rössler showed that the realization of the degree zero part of the abelian polylogarithm in analytic Deligne cohomology can be described in terms of a class of currents which was previously defined by Maillot and Rössler and strongly related to the Bismut–Köhler higher torsion form of the Poincaré bundle. In this paper we show that, if the base of the abelian scheme is proper, Kings and Rössler’s result can be refined to hold already in Deligne–Beilinson cohomology. More precisely, by means of Burgos’ theory of arithmetic Chow groups, we prove that the class of currents defined by Maillot and Rössler has a representative with logarithmic singularities at the boundary and therefore defines an element in Deligne–Beilinson cohomology. This element coincides with the realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne–Beilinson cohomology.

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