We prove the following version of the Campana’s orbifold conjecture: Let X be a complex non-singular projective variety of dimension n. Let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$D_1,\ldots ,D_{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>D</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> be <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {Z}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>-linearly independent effective divisors in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{Div}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Div</mtext> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="159" /> </InlineMediaObject> <EquationSource Format="TEX">$D:=D_1+\cdots +D_{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>:</mo> <mo>=</mo> <msub> <mi>D</mi> <mn>1</mn> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>D</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> be a normal crossing divisor of X such that (X, D) is of general type. Assume furthermore that they are numerically parallel. Let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation> be an orbifold divisor with <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{Supp}(\Delta ) = \textrm{Supp}(D)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Supp</mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mtext>Supp</mtext> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">$f: \mathbb {C} \rightarrow (X, \Delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be an orbifold entire curve. Then there exists a positive integer <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> such that if f has multiplicity at least <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> along <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq11.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$D_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2152_Article_IEq12.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le i\le n+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, then f must be algebraically degenerate.

Campana’s Orbifold Conjecture for Numerically Equivalent Divisors

期刊论文
发表日期：

2025年8月15日

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Min Ru,
Julie Tzu-Yueh Wang

摘要

We prove the following version of the Campana’s orbifold conjecture: Let X be a complex non-singular projective variety of dimension n. Let $D_1,\ldots ,D_{n+1}$ $D_{1}, \dots, D_{n + 1}$ be ${\mathbb {Z}}$ $Z$ -linearly independent effective divisors in $\textrm{Div}(X)$ $Div (X)$ and $D:=D_1+\cdots +D_{n+1}$ $D : = D_{1} + \dots + D_{n + 1}$ be a normal crossing divisor of X such that (X, D) is of general type. Assume furthermore that they are numerically parallel. Let $\Delta $ $Δ$ be an orbifold divisor with $\textrm{Supp}(\Delta ) = \textrm{Supp}(D)$ $Supp (Δ) = Supp (D)$ and let $f: \mathbb {C} \rightarrow (X, \Delta )$ $f : C \to (X, Δ)$ be an orbifold entire curve. Then there exists a positive integer $\ell $ $ℓ$ such that if f has multiplicity at least $\ell $ $ℓ$ along $D_i$ $D_{i}$ , $1\le i\le n+1$ $1 \leq i \leq n + 1$ , then f must be algebraically degenerate.