<p>We investigate the positivity properties of the direct image <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2068_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$f_{}(K_{X/Y} \otimes L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>f</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow> <mi>X</mi> <mo stretchy="false">/</mo> <mi>Y</mi> </mrow> </msub> <mo>⊗</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the adjoint line bundle associated with a nef and big line bundle <i>L</i>, under a smooth fibration <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2068_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$f: X\rightarrow Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> between projective varieties. We show that the vector bundle <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2068_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$f_{}(K_{X/Y} \otimes L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>f</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow> <mi>X</mi> <mo stretchy="false">/</mo> <mi>Y</mi> </mrow> </msub> <mo>⊗</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2068_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$\textit{Viehweg}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="italic">Viehweg</mi> </math></EquationSource> </InlineEquation>-big.</p>

On the direct image of the adjoint nef and big line bundles

期刊论文
发表日期：

2025年7月7日

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Yuta Watanabe,
Yongpan Zou

摘要

We investigate the positivity properties of the direct image $f_{*}(K_{X/Y} \otimes L)$ $f_{*} (K_{X / Y} \otimes L)$ of the adjoint line bundle associated with a nef and big line bundle L, under a smooth fibration $f: X\rightarrow Y$ $f : X \to Y$ between projective varieties. We show that the vector bundle $f_{*}(K_{X/Y} \otimes L)$ $f_{*} (K_{X / Y} \otimes L)$ is $\textit{Viehweg}$ $Viehweg$ -big.