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<p>We show that birational hyper-Kähler varieties of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1339_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>K</mi> <msup> <mn>3</mn> <mrow> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$K3^{[n]}$</EquationSource> </InlineEquation>-type are derived equivalent, establishing the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1339_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>D</mi> </math></EquationSource> <EquationSource Format="TEX">$D$</EquationSource> </InlineEquation>-equivalence conjecture in these cases. The Fourier–Mukai kernels of our derived equivalences are constructed from projectively hyperholomorphic bundles, following ideas of Markman. Our method also proves a stronger version of the <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1339_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>D</mi> </math></EquationSource> <EquationSource Format="TEX">$D$</EquationSource> </InlineEquation>-equivalence conjecture for hyper-Kähler varieties of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1339_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>K</mi> <msup> <mn>3</mn> <mrow> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$K3^{[n]}$</EquationSource> </InlineEquation>-type with Brauer classes.</p>

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The \(D\)-equivalence conjecture for hyper-Kähler varieties via hyperholomorphic bundles

  • 期刊论文

  • 发表日期：

    2025年6月9日

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  • Davesh Maulik,
  • Junliang Shen,
  • Qizheng Yin,
  • Ruxuan Zhang

摘要

We show that birational hyper-Kähler varieties of K 3 [ n ] $K3^{[n]}$ -type are derived equivalent, establishing the D $D$ -equivalence conjecture in these cases. The Fourier–Mukai kernels of our derived equivalences are constructed from projectively hyperholomorphic bundles, following ideas of Markman. Our method also proves a stronger version of the D $D$ -equivalence conjecture for hyper-Kähler varieties of K 3 [ n ] $K3^{[n]}$ -type with Brauer classes.