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• W2040201907 abstract "Let <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> be a field, and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega> <mml:semantics> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:annotation encoding=application/x-tex>Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a universal domain over <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>. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f colon upper X right-arrow upper S> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>f:X rightarrow S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a dominant morphism defined over <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> from a smooth projective variety <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a smooth projective variety <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=less-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>leq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the general fibre of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Subscript normal upper Omega> <mml:semantics> <mml:msub> <mml:mi>f</mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>f_Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has trivial Chow group of zero-cycles. For example, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> could be the total space of a two-dimensional family of varieties whose general member is rationally connected. Suppose that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=less-than-or-equal-to 4> <mml:semantics> <mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>leq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then we prove that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a self-dual Murre decomposition, i.e., that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a self-dual Chow-Künneth decomposition which satisfies Murre’s conjectures (B) and (D). Moreover, we prove that the motivic Lefschetz conjecture holds for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and hence so does the Lefschetz standard conjecture. We also give new examples of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-folds of general type which are Kimura finite dimensional, new examples of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=4> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding=application/x-tex>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-folds of general type having a self-dual Murre decomposition, as well as new examples of varieties with finite degree three unramified cohomology." @default.
• W2040201907 created "2016-06-24" @default.
• W2040201907 creator A5089099792 @default.
• W2040201907 date "2014-01-27" @default.
• W2040201907 modified "2023-09-23" @default.
• W2040201907 title "Chow–Künneth decomposition for 3- and 4-folds fibred by varieties with trivial Chow group of zero-cycles" @default.
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