A partial converse to the Andreotti–Grauert theorem
Let $X$ be a smooth projective manifold with $\dim _{\mathbb{C}}X=n$ . We show that if a line bundle $L$ is $(n-1)$ -ample, then it is $(n-1)$ -positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold $X$ is uniruled if and only if...
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| Veröffentlicht in: | Compositio mathematica 2019-01, Vol.155 (1), p.89-99 |
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| container_title | Compositio mathematica |
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| creator | Yang, Xiaokui |
| description | Let
$X$
be a smooth projective manifold with
$\dim _{\mathbb{C}}X=n$
. We show that if a line bundle
$L$
is
$(n-1)$
-ample, then it is
$(n-1)$
-positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold
$X$
is uniruled if and only if there exists a Hermitian metric
$\unicode[STIX]{x1D714}$
on
$X$
such that its Ricci curvature
$\text{Ric}(\unicode[STIX]{x1D714})$
has at least one positive eigenvalue everywhere. |
| doi_str_mv | 10.1112/S0010437X18007509 |
| format | Article |
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$X$
be a smooth projective manifold with
$\dim _{\mathbb{C}}X=n$
. We show that if a line bundle
$L$
is
$(n-1)$
-ample, then it is
$(n-1)$
-positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold
$X$
is uniruled if and only if there exists a Hermitian metric
$\unicode[STIX]{x1D714}$
on
$X$
such that its Ricci curvature
$\text{Ric}(\unicode[STIX]{x1D714})$
has at least one positive eigenvalue everywhere.</description><identifier>ISSN: 0010-437X</identifier><identifier>EISSN: 1570-5846</identifier><identifier>DOI: 10.1112/S0010437X18007509</identifier><language>eng</language><publisher>London, UK: London Mathematical Society</publisher><subject>Eigenvalues ; Manifolds ; Theorems</subject><ispartof>Compositio mathematica, 2019-01, Vol.155 (1), p.89-99</ispartof><rights>The Author 2018</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c383t-572d328a56ecadf652ff1baf7be236627f321e9271f818b0d628280cc7e850b83</citedby><cites>FETCH-LOGICAL-c383t-572d328a56ecadf652ff1baf7be236627f321e9271f818b0d628280cc7e850b83</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.cambridge.org/core/product/identifier/S0010437X18007509/type/journal_article$$EHTML$$P50$$Gcambridge$$H</linktohtml><link.rule.ids>164,314,780,784,27924,27925,55628</link.rule.ids></links><search><creatorcontrib>Yang, Xiaokui</creatorcontrib><title>A partial converse to the Andreotti–Grauert theorem</title><title>Compositio mathematica</title><addtitle>Compositio Math</addtitle><description>Let
$X$
be a smooth projective manifold with
$\dim _{\mathbb{C}}X=n$
. We show that if a line bundle
$L$
is
$(n-1)$
-ample, then it is
$(n-1)$
-positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold
$X$
is uniruled if and only if there exists a Hermitian metric
$\unicode[STIX]{x1D714}$
on
$X$
such that its Ricci curvature
$\text{Ric}(\unicode[STIX]{x1D714})$
has at least one positive eigenvalue everywhere.</description><subject>Eigenvalues</subject><subject>Manifolds</subject><subject>Theorems</subject><issn>0010-437X</issn><issn>1570-5846</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNp1kM9Kw0AQxhdRsFYfwFvAc3RmN_unx1K0CgUPKngLm2RWU9pu3d0I3nwH39AnMaEFD-JpYL7v983wMXaOcImI_OoBAKEQ-hkNgJYwOWAjlBpyaQp1yEaDnA_6MTuJcQkA3HAzYnKabW1IrV1ltd-8U4iUJZ-lV8qmmyaQT6n9_vyaB9tRSMPeB1qfsiNnV5HO9nPMnm6uH2e3-eJ-fjebLvJaGJFyqXkjuLFSUW0bpyR3DivrdEVcKMW1ExxpwjU6g6aCRg1PQV1rMhIqI8bsYpe7Df6to5jKpe_Cpj9ZciGVnuhCY-_CnasOPsZArtyGdm3DR4lQDu2Uf9rpGbFn7LoKbfNCv9H_Uz-EH2ZX</recordid><startdate>20190101</startdate><enddate>20190101</enddate><creator>Yang, 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Xiaokui</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c383t-572d328a56ecadf652ff1baf7be236627f321e9271f818b0d628280cc7e850b83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Eigenvalues</topic><topic>Manifolds</topic><topic>Theorems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Yang, Xiaokui</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest 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Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Yang, Xiaokui</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A partial converse to the Andreotti–Grauert theorem</atitle><jtitle>Compositio mathematica</jtitle><addtitle>Compositio Math</addtitle><date>2019-01-01</date><risdate>2019</risdate><volume>155</volume><issue>1</issue><spage>89</spage><epage>99</epage><pages>89-99</pages><issn>0010-437X</issn><eissn>1570-5846</eissn><abstract>Let
$X$
be a smooth projective manifold with
$\dim _{\mathbb{C}}X=n$
. We show that if a line bundle
$L$
is
$(n-1)$
-ample, then it is
$(n-1)$
-positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold
$X$
is uniruled if and only if there exists a Hermitian metric
$\unicode[STIX]{x1D714}$
on
$X$
such that its Ricci curvature
$\text{Ric}(\unicode[STIX]{x1D714})$
has at least one positive eigenvalue everywhere.</abstract><cop>London, UK</cop><pub>London Mathematical Society</pub><doi>10.1112/S0010437X18007509</doi><tpages>11</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0010-437X |
| ispartof | Compositio mathematica, 2019-01, Vol.155 (1), p.89-99 |
| issn | 0010-437X 1570-5846 |
| language | eng |
| recordid | cdi_proquest_journals_2356797471 |
| source | EZB-FREE-00999 freely available EZB journals; Cambridge University Press Journals Complete |
| subjects | Eigenvalues Manifolds Theorems |
| title | A partial converse to the Andreotti–Grauert theorem |
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