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<p>In this paper, we study the relative Gromov–Witten classes (with negative contact orders) of a relative pair <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2361_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(({\underline{X}}/Z)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <munder> <mi>X</mi> <mo>̲</mo> </munder> <mo stretchy="false">/</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of blowups of <i>X</i> along <i>S</i>. We show that when the (absolute) Gromov–Witten classes of both <i>X</i> and <i>S</i> are tautological, then so are the relative Gromov–Witten classes (with negative contact orders) of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2361_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(({\underline{X}}/Z)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <munder> <mi>X</mi> <mo>̲</mo> </munder> <mo stretchy="false">/</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. As applications, we study the Gromov–Witten classes of blowups and blowdowns.</p>

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On the Tautological Property of Relative Gromov–Witten Classes and Applications

  • 期刊论文

  • 发表日期：

    2025年2月11日

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  • Bohui Chen,
  • Cheng-Yong Du

摘要

In this paper, we study the relative Gromov–Witten classes (with negative contact orders) of a relative pair \(({\underline{X}}/Z)\) ( X ̲ / Z ) of blowups of X along S. We show that when the (absolute) Gromov–Witten classes of both X and S are tautological, then so are the relative Gromov–Witten classes (with negative contact orders) of \(({\underline{X}}/Z)\) ( X ̲ / Z ) . As applications, we study the Gromov–Witten classes of blowups and blowdowns.