具有严格 nef 相对反 log 典范除数的代数纤维空间

IF 1 2区 数学 Q1 MATHEMATICS
Jie Liu, Wenhao Ou, Juanyong Wang, Xiaokui Yang, Guolei Zhong
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Moreover, when <span></span><math>\n <semantics>\n <mrow>\n <mo>dim</mo>\n <mi>X</mi>\n <mo>=</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$\\dim X=3$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mo>−</mo>\n <mo>(</mo>\n <msub>\n <mi>K</mi>\n <mi>X</mi>\n </msub>\n <mo>+</mo>\n <mi>Δ</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$-(K_X+\\varDelta)$</annotation>\n </semantics></math> is strictly nef, we prove that <span></span><math>\n <semantics>\n <mrow>\n <mo>−</mo>\n <mo>(</mo>\n <msub>\n <mi>K</mi>\n <mi>X</mi>\n </msub>\n <mo>+</mo>\n <mi>Δ</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$-(K_X+\\varDelta)$</annotation>\n </semantics></math> is ample, which confirms the singular version of a conjecture by Campana and Peternell for threefolds.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"109 6","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Algebraic fibre spaces with strictly nef relative anti-log canonical divisor\",\"authors\":\"Jie Liu,&nbsp;Wenhao Ou,&nbsp;Juanyong Wang,&nbsp;Xiaokui Yang,&nbsp;Guolei Zhong\",\"doi\":\"10.1112/jlms.12942\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>,</mo>\\n <mi>Δ</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(X,\\\\varDelta)$</annotation>\\n </semantics></math> be a projective klt pair, and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n <mo>:</mo>\\n <mi>X</mi>\\n <mo>→</mo>\\n <mi>Y</mi>\\n </mrow>\\n <annotation>$f\\\\colon X\\\\rightarrow Y$</annotation>\\n </semantics></math> a fibration to a smooth projective variety <span></span><math>\\n <semantics>\\n <mi>Y</mi>\\n <annotation>$Y$</annotation>\\n </semantics></math> with strictly nef relative anti-log canonical divisor <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>−</mo>\\n <mo>(</mo>\\n <msub>\\n <mi>K</mi>\\n <mrow>\\n <mi>X</mi>\\n <mo>/</mo>\\n <mi>Y</mi>\\n </mrow>\\n </msub>\\n <mo>+</mo>\\n <mi>Δ</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$-(K_{X/Y}+\\\\varDelta)$</annotation>\\n </semantics></math>. 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引用次数: 0

摘要

让 ( X , Δ ) $(X,\varDelta)$ 是一个投影 klt 对,并且 f : X → Y $f\colon X\rightarrow Y$ 是一个光滑投影多元 Y $Y$ 的纤度,具有严格 nef 相对反逻辑正则除数 - ( K X / Y + Δ ) $-(K_{X/Y}+\varDelta)$ 。我们证明 f $f$ 是一个具有合理连接纤维的局部琐碎纤维,并且基 Y $Y$ 是一个典型极化双曲流形。特别是,当 Y $Y$ 是一个单点时,我们证明 X $X$ 是有理连接的。此外,当 dim X = 3 $\dim X=3$ 和 - ( K X + Δ ) $-(K_X+\varDelta)$ 是严格 nef 时,我们证明 - ( K X + Δ ) $-(K_X+\varDelta)$ 是充裕的,这证实了坎帕纳和佩特内尔对三维流形的猜想的奇异版本。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Algebraic fibre spaces with strictly nef relative anti-log canonical divisor

Let ( X , Δ ) $(X,\varDelta)$ be a projective klt pair, and f : X Y $f\colon X\rightarrow Y$ a fibration to a smooth projective variety Y $Y$ with strictly nef relative anti-log canonical divisor ( K X / Y + Δ ) $-(K_{X/Y}+\varDelta)$ . We prove that f $f$ is a locally trivial fibration with rationally connected fibres, and the base Y $Y$ is a canonically polarized hyperbolic manifold. In particular, when Y $Y$ is a single point, we establish that X $X$ is rationally connected. Moreover, when dim X = 3 $\dim X=3$ and ( K X + Δ ) $-(K_X+\varDelta)$ is strictly nef, we prove that ( K X + Δ ) $-(K_X+\varDelta)$ is ample, which confirms the singular version of a conjecture by Campana and Peternell for threefolds.

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来源期刊
CiteScore
1.90
自引率
0.00%
发文量
186
审稿时长
6-12 weeks
期刊介绍: The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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