<p>In this article, we will prove the Generalized Nonvanishing Conjecture holds for threefolds with either the Kodaira dimension <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3727_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{\kappa (X)> 0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">κ</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">X</mi> <mo mathvariant="bold" stretchy="false">)</mo> <mo mathvariant="bold">></mo> <mn mathvariant="bold">0</mn> </mrow> </math></EquationSource> </InlineEquation> or the irregularity <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3727_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="194" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{h^1(X,\mathcal {O}_X)=q(X)>0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="bold-italic">h</mi> <mn mathvariant="bold">1</mn> </msup> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">X</mi> <mo mathvariant="bold">,</mo> <msub> <mi mathvariant="bold-script">O</mi> <mi mathvariant="bold-italic">X</mi> </msub> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mo mathvariant="bold">=</mo> <mi mathvariant="bold-italic">q</mi> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">X</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mo mathvariant="bold">></mo> <mn mathvariant="bold">0</mn> </mrow> </math></EquationSource> </InlineEquation>. As a result, we can prove that for an algebraic fiber space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3727_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{f:X\rightarrow Y}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">f</mi> <mo mathvariant="bold">:</mo> <mi mathvariant="bold-italic">X</mi> <mo mathvariant="bold">→</mo> <mi mathvariant="bold-italic">Y</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3727_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{\dim }X=7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo mathvariant="bold">dim</mo> </mrow> <mi>X</mi> <mo>=</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>, the Iitaka Conjecture holds if either the <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3727_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{X}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> </math></EquationSource> </InlineEquation> has non-negative Kodaira dimension, or if the general fiber or <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3727_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ \varvec{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">F</mi> </mrow> </math></EquationSource> </InlineEquation> has positive Kodaira dimension, or if <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3727_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{Y}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">Y</mi> </mrow> </math></EquationSource> </InlineEquation> is not a threefold with <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3727_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="125" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{\kappa }(Y)=q(Y)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">κ</mi> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Generalized nonvanishing conjecture and Iitaka conjecture

期刊论文
发表日期：

2025年4月11日

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Chi-Kang Chang

摘要

In this article, we will prove the Generalized Nonvanishing Conjecture holds for threefolds with either the Kodaira dimension $\varvec{\kappa (X)> 0}$ $κ (X) > 0$ or the irregularity $\varvec{h^1(X,\mathcal {O}_X)=q(X)>0}$ $h^{1} (X, O_{X}) = q (X) > 0$ . As a result, we can prove that for an algebraic fiber space $\varvec{f:X\rightarrow Y}$ $f : X \to Y$ with $\varvec{\dim }X=7$ $dim X = 7$ , the Iitaka Conjecture holds if either the $\varvec{X}$ $X$ has non-negative Kodaira dimension, or if the general fiber or $ \varvec{F}$ $F$ has positive Kodaira dimension, or if $\varvec{Y}$ $Y$ is not a threefold with $\varvec{\kappa }(Y)=q(Y)=0$ $κ (Y) = q (Y) = 0$ .