Algebraic cycles and the Hodge structure of a Kuga fiber variety

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo stretchy="false">~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\tilde A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote a smooth compactification of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-fold fiber product of the universal family <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Superscript 1 Baseline right-arrow upper M"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{A^1} \to M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of elliptic curves with level <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> structure. The purpose of this paper is to completely describe the algebraic cycles in and the Hodge structure of the Betti cohomology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript asterisk Baseline left-parenthesis upper A overTilde comma double-struck upper Q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo stretchy="false">~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^{\ast } }(\tilde A,\mathbb {Q})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo stretchy="false">~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\tilde A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for by doing so we are able (a) to verify both the usual and generalized Hodge conjectures for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo stretchy="false">~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\tilde A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; (b) to describe both the kernel and the image of the Abel-Jacobi map from algebraic cycles algebraically equivalent to zero (modulo rational equivalence) into the Griffiths intermediate Jacobian; and (c) to verify Tate’s conjecture concerning the algebraic cycles in the étale cohomology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript et Superscript asterisk Baseline left-parenthesis upper A overTilde circled-times double-struck upper Q overbar comma double-struck upper Q Subscript l Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>et</mml:mtext> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo stretchy="false">~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mi>l</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H_{{\text {et}}}^{\ast } (\tilde A \otimes \bar {\mathbb {Q}},{\mathbb {Q}_l})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The methods used lead also to a complete description of the Hodge structure of the Betti cohomology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript asterisk Baseline left-parenthesis upper E Superscript k Baseline comma double-struck upper Q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^{\ast } }({E^k},\mathbb {Q})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-fold product of an elliptic curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> without complex multiplication, and a verification of the generalized Hodge conjecture for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Superscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{E^k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

Locations

• Transactions of the American Mathematical Society - View - PDF