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<p>We show that the upper motives of projective homogeneous varieties under absolutely simple algebraic groups of all types other than <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1285_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(^6\!{{\textsf {\textit{D}}}}_4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mrow /> <mn>6</mn> </mmultiscripts> <mspace width="-0.166667em" /> <msub> <mi mathvariant="italic">D</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> satisfy Poincaré duality. New are the cases of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1285_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(^2\!{{\textsf {\textit{A}}}}_n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mrow /> <mn>2</mn> </mmultiscripts> <mspace width="-0.166667em" /> <msub> <mi mathvariant="italic">A</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1285_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(^3\!{{\textsf {\textit{D}}}}_4.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mrow /> <mn>3</mn> </mmultiscripts> <mspace width="-0.166667em" /> <msub> <mi mathvariant="italic">D</mi> <mn>4</mn> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Poincaré duality for upper motives of absolutely simple algebraic groups

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • Nikita A. Karpenko

摘要

We show that the upper motives of projective homogeneous varieties under absolutely simple algebraic groups of all types other than \(^6\!{{\textsf {\textit{D}}}}_4\) 6 D 4 satisfy Poincaré duality. New are the cases of \(^2\!{{\textsf {\textit{A}}}}_n\) 2 A n and \(^3\!{{\textsf {\textit{D}}}}_4.\) 3 D 4 .