In this paper, we study holomorphic vector bundles on the homogeneous varieties <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_10022_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$G_2/P_1\cong \mathbb {Q}^5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>P</mi> <mn>1</mn> </msub> <mo>≅</mo> <msup> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> <mn>5</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_10022_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$G_2/P_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>P</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. We prove that if a rank 2 vector bundle E on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_10022_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">$G_2/P_i~(i=1,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>P</mi> <mi>i</mi> </msub> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is uniform with respect to the special family of lines, then E is either a direct sum of line bundles or an indecomposable 2-bundle, which is unique up to twist. As a consequence, we give a new characterization of the Cayley bundles on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_10022_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Q}^5$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> <mn>5</mn> </msup> </math></EquationSource> </InlineEquation>.

Uniform bundles on the homogeneous varieties of type $G_2$

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发表日期：

2025年10月3日

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Xinyi Fang

摘要

In this paper, we study holomorphic vector bundles on the homogeneous varieties $G_2/P_1\cong \mathbb {Q}^5$ $G_{2} / P_{1} ≅ Q^{5}$ and $G_2/P_2$ $G_{2} / P_{2}$ . We prove that if a rank 2 vector bundle E on $G_2/P_i~(i=1,2)$ $G_{2} / P_{i} (i = 1, 2)$ is uniform with respect to the special family of lines, then E is either a direct sum of line bundles or an indecomposable 2-bundle, which is unique up to twist. As a consequence, we give a new characterization of the Cayley bundles on $\mathbb {Q}^5$ $Q^{5}$ .

Uniform bundles on the homogeneous varieties of type \(G_2\)

摘要