On the Chow motive of an abelian scheme with non-trivial endomorphisms

This chapter introduces the notion of a Rost motive, which is a summand of the motive of a Rost variety 𝑋. It highlights the theorem that, assuming that Rost motives exist and H90(n − 1) holds, then 𝐻𝑛+1 ét(𝑘, ℤ(𝑛)) injects into 𝐻𝑛+1 ét(𝑘(𝑋), ℤ(𝑛)). While there may be many Rost varieties associated to a given symbol, there is essentially only one Rost motive. The Rost motive captures the part of the cohomology of a Rost variety 𝑋. Since a Rost motive is a special kind of symmetric Chow motive, the chapter begins by recalling what this means. It then introduces the notion of 𝔛-duality. This duality plays an important role in the axioms defining Rost motives, as well as a role in the construction of the Rost motive in the next chapter. Finally, this chapter assumes that Rost motives exist and proves a key theorem.

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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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ON THE RESIDUE OF EISENSTEIN CLASSES OF SIEGEL VARIETIES

Glasgow Mathematical Journal ◽

10.1017/s0017089517000271 ◽

2017 ◽

Vol 60 (3) ◽

pp. 539-553

Author(s):

FRANCESCO LEMMA

Keyword(s):

Motivic Cohomology ◽

Abelian Scheme ◽

Arbitrary Genus ◽

Direct Corollary

AbstractEisenstein classes of Siegel varieties are motivic cohomology classes defined as pull-backs by torsion sections of the polylogarithm prosheaf on the universal abelian scheme. By reduction to the Hilbert–Blumenthal case, we prove that the Betti realization of these classes on Siegel varieties of arbitrary genus have non-trivial residue on zero-dimensional strata of the Baily–Borel–Satake compactification. A direct corollary is the non-vanishing of a higher regulator map.

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Relations between the Chow motive and the noncommutative motive of a smooth projective variety

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2015.05.002 ◽

2015 ◽

Vol 219 (11) ◽

pp. 5068-5077 ◽

Cited By ~ 4

Author(s):

Marcello Bernardara ◽

Gonçalo Tabuada

Keyword(s):

Projective Variety ◽

Smooth Projective Variety ◽

Chow Motive

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On the motive of O’Grady’s ten-dimensional hyper-Kähler varieties

Communications in Contemporary Mathematics ◽

10.1142/s0219199720500340 ◽

2020 ◽

pp. 2050034

Author(s):

Salvatore Floccari ◽

Lie Fu ◽

Ziyu Zhang

Keyword(s):

Recent Result ◽

Moduli Spaces ◽

Betti Number ◽

Defect Group ◽

Crepant Resolution ◽

Tate Conjecture ◽

Chow Motive ◽

Crepant Resolutions ◽

The Difference ◽

Cubic Fourfold

We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bülles to the O’Grady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of (semi-)stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-Kähler varieties of O’Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the André motive of projective hyper-Kähler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford–Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-Kähler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-Kähler varieties, all Hodge and Tate classes are motivated, the motivated Mumford–Tate conjecture 7.3 holds, and the André motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.

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