The Hodge conjecture is proved for all simple abelian varieties of prime dimension. The microweight conjecture holds for the $l$-adic representation associated to the Tate module of abelian variety over a number field. The finiteness of the Brauer group holds for an arithmetic model of a hyperkahler variety with the second Betti number greater than 3 over a number field. For all smooth complex 3-dimensional projective varieties of non-basic type the Grothendieck standard conjecture (of Lefschetz type) on algebraicity of the Hodge operator star is true.
Biography
Graduated from A.N. Kolmogorov's physico-mathematical boarding-school (1965). Graduated from Faculty of Mathematics and Mechanics of M.V. Lomonosov Moscow State University (MSU) in 1970 (department of algebra). Ph.D. thesis (MSU) was defended in 1973. D.Sci. thesis (MSU) was defended in 1985.
Main publications:
Tankeev S.G., Cycles on simple abelian varieties of prime dimension, Math. USSR-Izv.,
20:1 (1983), 157-171.
Tankeev S.G., On weights of $l$-adic representation and arithmetic of Frobenius eigenvalues, Russian Acad. Sci. Izv. Math., 63:1
(1999), 181-218.
Tankeev S.G., On the standard conjecture of Lefschetz type for complex projective 3-dimensional varieties. II ,
Izv. Math.
75:5 (2011), 1047-1062.
Tankeev S.G.,
On the Brauer group of an arithmetic model of a hyperkahler variety over a number field, Izv. Math.79:3 (2015).
S. G. Tankeev, “On the standard conjecture for a fourfold with
$1$-parameter fibration by Abelian varieties”, Izv. RAN. Ser. Mat., 88:2 (2024), 153–183; Izv. Math., 88:2 (2024), 339–368
2023
2.
S. G. Tankeev, “Hodge and Mumford–Tate groups of an Abelian Variety, Complex Multiplication, and Frobenius Elements”, Mat. Zametki, 113:4 (2023), 622–625; Math. Notes, 113:4 (2023), 601–604
2022
3.
S. G. Tankeev, “On the standard conjecture for compactifications of Néron models of 4-dimensional Abelian varieties”, Izv. RAN. Ser. Mat., 86:4 (2022), 192–232; Izv. Math., 86:4 (2022), 797–835
S. G. Tankeev, “On Algebraic Isomorphisms of the Rational Cohomology of Künnemann's Compactifications of the Néron Model of an Abelian Variety without Complex Multiplication”, Mat. Zametki, 111:4 (2022), 636–637; Math. Notes, 111:4 (2022), 652–653
2021
5.
S. G. Tankeev, “On the standard conjecture for projective compactifications of Néron models of $3$-dimensional
Abelian varieties”, Izv. RAN. Ser. Mat., 85:1 (2021), 154–186; Izv. Math., 85:1 (2021), 145–175
S. G. Tankeev, “On the Standard Conjecture for Compactifications of Néron Models of Three-Dimensional Abelian Varieties with Multiplications in an Imaginary Quadratic Field”, Mat. Zametki, 109:3 (2021), 479–480; Math. Notes, 109:3 (2021), 498–499
2020
7.
S. G. Tankeev, “On the standard conjecture for a $3$-dimensional variety fibred by curves with a non-injective Kodaira–Spencer map”, Izv. RAN. Ser. Mat., 84:5 (2020), 211–232; Izv. Math., 84:5 (2020), 1016–1035
S. G. Tankeev, “On algebraic isomorphisms of rational cohomology of a Künneman compactification of the Néron minimal model”, Sib. Èlektron. Mat. Izv., 17 (2020), 89–125
S. G. Tankeev, “On the standard conjecture for a fibre product of three elliptic surfaces with pairwise-disjoint discriminant loci”, Izv. RAN. Ser. Mat., 83:3 (2019), 213–256; Izv. Math., 83:3 (2019), 613–653
S. G. Tankeev, “On the Standard Conjecture for a 3-Dimensional Variety Fibered over a Surface”, Mat. Zametki, 105:4 (2019), 643–644; Math. Notes, 105:4 (2019), 636–637
2017
11.
S. G. Tankeev, “On an inductive approach to the standard conjecture for a fibred
complex variety with strong semistable degeneracies”, Izv. RAN. Ser. Mat., 81:6 (2017), 199–231; Izv. Math., 81:6 (2017), 1253–1285
S. G. Tankeev, “On the Brauer group of an arithmetic model of a hyperkähler variety over a number field”, Izv. RAN. Ser. Mat., 79:3 (2015), 203–224; Izv. Math., 79:3 (2015), 623–644
S. G. Tankeev, “On the standard conjecture and the existence of a Chow–Lefschetz decomposition for complex projective varieties”, Izv. RAN. Ser. Mat., 79:1 (2015), 185–216; Izv. Math., 79:1 (2015), 177–207
S. G. Tankeev, “On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models”, Izv. RAN. Ser. Mat., 78:1 (2014), 181–214; Izv. Math., 78:1 (2014), 169–200
S. G. Tankeev, “On the standard conjecture for complex 4-dimensional elliptic varieties”, Izv. RAN. Ser. Mat., 76:5 (2012), 119–142; Izv. Math., 76:5 (2012), 967–990
S. G. Tankeev, “On the standard conjecture of Lefschetz type for complex projective threefolds. II”, Izv. RAN. Ser. Mat., 75:5 (2011), 177–194; Izv. Math., 75:5 (2011), 1047–1062
S. G. Tankeev, “On the standard conjecture of Lefschetz type for complex projective threefolds”, Izv. RAN. Ser. Mat., 74:1 (2010), 175–196; Izv. Math., 74:1 (2010), 167–187
S. G. Tankeev, “On the numerical equivalence of algebraic cycles on potentially simple Abelian schemes of prime relative dimension”, Izv. RAN. Ser. Mat., 69:1 (2005), 145–164; Izv. Math., 69:1 (2005), 143–162
S. G. Tankeev, “On the standard conjecture for complex Abelian schemes over smooth projective curves”, Izv. RAN. Ser. Mat., 67:3 (2003), 183–224; Izv. Math., 67:3 (2003), 597–635
S. G. Tankeev, “On the Conjectures of Artin and Shafarevich–Tate”, Trudy Mat. Inst. Steklova, 241 (2003), 254–264; Proc. Steklov Inst. Math., 241 (2003), 238–248
2002
25.
S. G. Tankeev, “The arithmetic and geometry of a generic hypersurface section”, Izv. RAN. Ser. Mat., 66:2 (2002), 173–204; Izv. Math., 66:2 (2002), 393–424
S. G. Tankeev, “Cycles of small codimension on a simple abelian variety”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 70 (2001), 206–235; J. Math. Sci. (New York), 106:5 (2001), 3365–3382
28.
S. G. Tankeev, “On the Mumford–Tate conjecture for abelian varieties”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 33 (2001), 213–241; J. Math. Sci., 81:3 (1996), 2719–2737
S. G. Tankeev, “Cycles of small codimension on a simple $2p$- or $4p$-dimensional Abelian variety”, Izv. RAN. Ser. Mat., 63:6 (1999), 167–208; Izv. Math., 63:6 (1999), 1221–1262
S. G. Tankeev, “On weights of the $l$-adic representation and arithmetic of Frobenius eigenvalues”, Izv. RAN. Ser. Mat., 63:1 (1999), 185–224; Izv. Math., 63:1 (1999), 181–218
S. G. Tankeev, “Surfaces of type K3 over number fields and the Mumford–Tate conjecture. II”, Izv. RAN. Ser. Mat., 59:3 (1995), 179–206; Izv. Math., 59:3 (1995), 619–646
S. G. Tankeev, “Cycles on an Abelian variety without complex multiplication and $l$-adic representations”, Uspekhi Mat. Nauk, 49:1(295) (1994), 225–226; Russian Math. Surveys, 49:1 (1994), 247
1993
37.
S. G. Tankeev, “Abelian varieties and the general Hodge conjecture”, Izv. RAN. Ser. Mat., 57:4 (1993), 192–206; Russian Acad. Sci. Izv. Math., 43:1 (1994), 179–191
S. G. Tankeev, “Kuga–Satake abelian varieties and $l$-adic representations”, Izv. Akad. Nauk SSSR Ser. Mat., 55:4 (1991), 877–889; Math. USSR-Izv., 39:1 (1992), 855–867
S. G. Tankeev, “K3 surfaces over number fields and the Mumford–Tate conjecture”, Izv. Akad. Nauk SSSR Ser. Mat., 54:4 (1990), 846–861; Math. USSR-Izv., 37:1 (1991), 191–208
S. G. Tankeev, “K3 surfaces over number fields and $l$-adic representations”, Izv. Akad. Nauk SSSR Ser. Mat., 52:6 (1988), 1252–1271; Math. USSR-Izv., 33:3 (1989), 575–595
S. G. Tankeev, “Cycles on simple Abelian varieties of prime dimension over number fields”, Izv. Akad. Nauk SSSR Ser. Mat., 51:6 (1987), 1214–1227; Math. USSR-Izv., 31:3 (1988), 527–540
S. G. Tankeev, “On cycles on Abelian varieties of prime dimension over finite or number fields”, Izv. Akad. Nauk SSSR Ser. Mat., 47:2 (1983), 356–365; Math. USSR-Izv., 22:2 (1984), 329–337
S. G. Tankeev, “Cycles on simple Abelian varieties of prime dimension”, Izv. Akad. Nauk SSSR Ser. Mat., 46:1 (1982), 155–170; Math. USSR-Izv., 20:1 (1983), 157–171
S. G. Tankeev, “On algebraic cycles on simple 5-dimensional Abelian varieties”, Izv. Akad. Nauk SSSR Ser. Mat., 45:4 (1981), 793–823; Math. USSR-Izv., 19:1 (1982), 95–123
S. G. Tankeev, “On algebraic cycles on surfaces and Abelian varieties”, Izv. Akad. Nauk SSSR Ser. Mat., 45:2 (1981), 398–434; Math. USSR-Izv., 18:2 (1982), 349–380
S. G. Tankeev, “On algebraic cycles on Abelian varieties. II”, Izv. Akad. Nauk SSSR Ser. Mat., 43:2 (1979), 418–429; Math. USSR-Izv., 14:2 (1980), 383–394
S. G. Tankeev, “On homomorphisms of Abelian schemes. II”, Izv. Akad. Nauk SSSR Ser. Mat., 41:6 (1977), 1231–1251; Math. USSR-Izv., 11:6 (1977), 1175–1194
S. G. Tankeev, “Pluricanonical mappings of algebraic surfaces of general type”, Uspekhi Mat. Nauk, 30:6(186) (1975), 184
1972
51.
S. G. Tankeev, “On a global theory of moduli of algebraic surfaces of general type”, Izv. Akad. Nauk SSSR Ser. Mat., 36:6 (1972), 1220–1236; Math. USSR-Izv., 6:6 (1972), 1200–1216
S. G. Tankeev, “On $n$-dimensional canonically polarized varieties and varieties of fundamental type”, Izv. Akad. Nauk SSSR Ser. Mat., 35:1 (1971), 31–44; Math. USSR-Izv., 5:1 (1971), 29–43
S. G. Tankeev, “Моноидальные преобразования и алгебраические соответствия”, Izv. RAN. Ser. Mat., 0
Presentations in Math-Net.Ru
1.
О стандартной гипотезе Гротендика типа Лефшеца S. G. Tankeev Conference on algebra, algebraic geometry, and number theory on the
occasion of academician Igor Rostislavovich Shafarevich 100th birthday June 9, 2023 12:20
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