Invariant Cycles on Abelian Schemes

2020 ◽  
Vol 250 (1) ◽  
pp. 69-75
Author(s):  
O. V. Makarova
Keyword(s):  
Abelian Schemes

ON HOMOMORPHISMS OF ABELIAN SCHEMES

1976 ◽  
Vol 10 (4) ◽  
pp. 731-747
Author(s):  
Sergei G Tankeev
Keyword(s):  
Abelian Schemes

Theory of Degeneration for Polarized Abelian Schemes

2013 ◽  
Author(s):  
Kai-Wen Lan
Keyword(s):  
Abelian Varieties ◽  
Degenerate Case ◽  
Analytic Construction ◽  
Abelian Schemes ◽  
Special Case

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.

Abelian schemes

1965 ◽  
pp. 115-127
Author(s):  
David Mumford
Keyword(s):  
Abelian Schemes

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

2017 ◽  
Vol 13 (09) ◽  
pp. 2471-2485 ◽  
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.

scholarly journals Torsion hypersurfaces on abelian schemes and Betti coordinates

2017 ◽  
Vol 371 (3-4) ◽  
pp. 1013-1045 ◽  
Author(s):  
Pietro Corvaja ◽  
David Masser ◽  
Umberto Zannier
Keyword(s):  
Abelian Schemes

Structures of Semi-Abelian Schemes

2013 ◽  
Author(s):  
Kai-Wen Lan
Keyword(s):  
Abelian Varieties ◽  
Fundamental Property ◽  
Fundamental Importance ◽  
Multiplicative Type ◽  
Abelian Schemes

This chapter explains well-known notions important for the study of semi-abelian schemes. It first studies groups of multiplicative type and the torsors under them. A fundamental property of groups of multiplicative type is that they are rigid in the sense that they cannot be deformed. The chapter then turns to biextensions, cubical structures, semi-abelian schemes, Raynaud extensions, and certain dual objects for the last two notions extending the notion of dual abelian varieties. Such notions are, as this chapter shows, of fundamental importance in the study of the degeneration of abelian varieties. The main objective here is to understand the statement and the proof of the theory of degeneration data.

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