Invariant Cycles on Abelian Schemes
- PDF / 172,311 Bytes
- 7 Pages / 594 x 792 pts Page_size
- 91 Downloads / 188 Views
Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
INVARIANT CYCLES ON ABELIAN SCHEMES O. V. Makarova A. G. and N. G. Stoletov Vladimir State University 87, Gor’kogo St., Vladimir 600000, Russia [email protected]
UDC 512.7
We prove the Grothendieck conjecture on invariant cycles for an Abelian scheme over a smooth connected complex algebraic curve and the Tate conjecture for codimension r algebraic cycles on an Abelian variety over a number field. Bibliography: 18 titles.
1
Introduction
The Grothendieck conjecture on invariant cycles (cf. [1] and [2, Conjecture B.181]) asserts that if π : X → C is a smooth projective morphism of smooth connected quasiprojective complex varieties and α ∈ H 0 (C, R2r π∗ Q) (where r is a natural number) so that for some point s0 ∈ C the restriction α|Xs0 ∈ H 2r (Xs0 , Q) is an algebraic cohomology class, then for any s ∈ C the class α|Xs ∈ H 2r (Xs , Q) is algebraic. Let VQ be a rational Hodge structure of weight r. The Hodge decomposition VQ ⊗Q C = ⊕p+q=r VCp,q determines a morphism of real algebraic groups h : ResC/R (Gm) → GL(VQ ⊗Q R), where ResC/R (Gm) is a real algebraic group obtained from the multiplicative group Gm over C by restriction of the field of scalars to R [2]. An element z ∈ CC × = ResC/R (Gm)(R) acts on VCp,q by the rule [3, formula (2.1.5.1)] z · v = z p z q v. Consider the group U 1 = {z ∈ C× = ResC/R (Gm)(R) | |z| = 1}. Definition 1.1. The smallest Q-subgroup of GL(VQ ) whose group of R-points contains h(U 1 ), is called the Hodge group of the Hodge structure VQ and is denoted by Hg(VQ ). Definition 1.2 ([2, Definition B.51]). The smallest Q-subgroup of GL(VQ ) whose group of R-points contains h(C× ), is called the Mumford–Tate group of the Hodge structure VQ and is denoted by MT(VQ ) If VQ is a polarizable rational Hodge structure, then Hg(VQ ) and MT(VQ ) are reductive Q-groups naturally acting in VQ [2]. It is known (cf. [2]) that H 2r (X, Q)Hg(H
2r (X,Q))
= H 2r (X, Q) ∩ H r,r (X, C),
Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 63-68. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0069
69
where H r,r (X, C) is the component of type (r, r) of the Hodge decomposition H 2r (X, Q) ⊗Q C = ⊕p+q=2r H p,q (X, C). The Hodge conjecture [4] asserts that the Q-space H 2r (X, Q) ∩ H r,r (X, C) is spanned by cohomology classes of algebraic cycles of codimension r on X. Let J be an absolutely simple g-dimensional Abelian variety over a number field k ⊂ C, [k : Q] < ∞. Then for any prime number l and a natural l-adic representation ρl : Gal(k/k) → GL(H´e1t (J ⊗k k, Ql )) the Mumford–Tate conjecture asserts that there exists a canonical isomorphism of Lie algebras [5] def
Lieρl (Gal(k/k)) → LieMT(J ⊗k C)(Ql ) = LieMT(H 1 (J ⊗k C, Q))(Ql ). The Tate conjecture asserts that the Ql -space H´e2rt (J ⊗k k, Ql (r))Lieρl (Gal(k/k)) is spanned by cohomology classes of algebraic cycles of codimension r on the variety J ⊗k k [6]. This conjecture is known to be equivalent to the finiteness of the group of Galois invariants
Data Loading...
