Jie Liu, Wenhao Ou, Juanyong Wang, Xiaokui Yang, Guolei Zhong
{"title":"具有严格 nef 相对反 log 典范除数的代数纤维空间","authors":"Jie Liu, Wenhao Ou, Juanyong Wang, Xiaokui Yang, 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>. We prove that <span></span><math>\n <semantics>\n <mi>f</mi>\n <annotation>$f$</annotation>\n </semantics></math> is a locally trivial fibration with rationally connected fibres, and the base <span></span><math>\n <semantics>\n <mi>Y</mi>\n <annotation>$Y$</annotation>\n </semantics></math> is a canonically polarized hyperbolic manifold. In particular, when <span></span><math>\n <semantics>\n <mi>Y</mi>\n <annotation>$Y$</annotation>\n </semantics></math> is a single point, we establish that <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> is rationally connected. 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, Wenhao Ou, Juanyong Wang, Xiaokui Yang, 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>. We prove that <span></span><math>\\n <semantics>\\n <mi>f</mi>\\n <annotation>$f$</annotation>\\n </semantics></math> is a locally trivial fibration with rationally connected fibres, and the base <span></span><math>\\n <semantics>\\n <mi>Y</mi>\\n <annotation>$Y$</annotation>\\n </semantics></math> is a canonically polarized hyperbolic manifold. In particular, when <span></span><math>\\n <semantics>\\n <mi>Y</mi>\\n <annotation>$Y$</annotation>\\n </semantics></math> is a single point, we establish that <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> is rationally connected. 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\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12942\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12942","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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。
Algebraic fibre spaces with strictly nef relative anti-log canonical divisor
Let be a projective klt pair, and a fibration to a smooth projective variety with strictly nef relative anti-log canonical divisor . We prove that is a locally trivial fibration with rationally connected fibres, and the base is a canonically polarized hyperbolic manifold. In particular, when is a single point, we establish that is rationally connected. Moreover, when and is strictly nef, we prove that is ample, which confirms the singular version of a conjecture by Campana and Peternell for threefolds.
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.