Anticanonically balanced metrics and the Hilbert–Mumford criterion for the δm-invariant of Fujita–Odaka
We prove that the stability condition for Fano manifolds defined by Saito–Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein–Tian–Zhang, we obtain the following alg...
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| Veröffentlicht in: | Annals of global analysis and geometry 2023-09, Vol.64 (2), p.8, Article 8 |
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| creator | Hashimoto, Yoshinori |
| description | We prove that the stability condition for Fano manifolds defined by Saito–Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein–Tian–Zhang, we obtain the following algebro-geometric corollary: the
δ
m
-invariant of Fujita–Odaka satisfies
δ
m
>
1
if and only if the Fano manifold is stable in the sense of Saito–Takahashi, establishing a Hilbert–Mumford-type criterion for
δ
m
>
1
. We also extend this result to the Kähler–Ricci
g
-solitons and the coupled Kähler–Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations. |
| doi_str_mv | 10.1007/s10455-023-09911-2 |
| format | Article |
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δ
m
-invariant of Fujita–Odaka satisfies
δ
m
>
1
if and only if the Fano manifold is stable in the sense of Saito–Takahashi, establishing a Hilbert–Mumford-type criterion for
δ
m
>
1
. We also extend this result to the Kähler–Ricci
g
-solitons and the coupled Kähler–Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations.</description><identifier>ISSN: 0232-704X</identifier><identifier>EISSN: 1572-9060</identifier><identifier>DOI: 10.1007/s10455-023-09911-2</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Analysis ; Approximation ; Criteria ; Differential Geometry ; Geometry ; Global Analysis and Analysis on Manifolds ; Invariants ; Mathematical Physics ; Mathematics ; Mathematics and Statistics ; Solitary waves</subject><ispartof>Annals of global analysis and geometry, 2023-09, Vol.64 (2), p.8, Article 8</ispartof><rights>The Author(s), under exclusive licence to Springer Nature B.V. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c1852-d274a17baa1464a8c073a79ef885e72a7c414ec46685128bafcc5b98e7159863</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10455-023-09911-2$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10455-023-09911-2$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Hashimoto, Yoshinori</creatorcontrib><title>Anticanonically balanced metrics and the Hilbert–Mumford criterion for the δm-invariant of Fujita–Odaka</title><title>Annals of global analysis and geometry</title><addtitle>Ann Glob Anal Geom</addtitle><description>We prove that the stability condition for Fano manifolds defined by Saito–Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein–Tian–Zhang, we obtain the following algebro-geometric corollary: the
δ
m
-invariant of Fujita–Odaka satisfies
δ
m
>
1
if and only if the Fano manifold is stable in the sense of Saito–Takahashi, establishing a Hilbert–Mumford-type criterion for
δ
m
>
1
. We also extend this result to the Kähler–Ricci
g
-solitons and the coupled Kähler–Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations.</description><subject>Analysis</subject><subject>Approximation</subject><subject>Criteria</subject><subject>Differential Geometry</subject><subject>Geometry</subject><subject>Global Analysis and Analysis on Manifolds</subject><subject>Invariants</subject><subject>Mathematical Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Solitary 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Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><jtitle>Annals of global analysis and geometry</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Hashimoto, Yoshinori</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Anticanonically balanced metrics and the Hilbert–Mumford criterion for the δm-invariant of Fujita–Odaka</atitle><jtitle>Annals of global analysis and geometry</jtitle><stitle>Ann Glob Anal Geom</stitle><date>2023-09-01</date><risdate>2023</risdate><volume>64</volume><issue>2</issue><spage>8</spage><pages>8-</pages><artnum>8</artnum><issn>0232-704X</issn><eissn>1572-9060</eissn><abstract>We prove that the stability condition for Fano manifolds defined by Saito–Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein–Tian–Zhang, we obtain the following algebro-geometric corollary: the
δ
m
-invariant of Fujita–Odaka satisfies
δ
m
>
1
if and only if the Fano manifold is stable in the sense of Saito–Takahashi, establishing a Hilbert–Mumford-type criterion for
δ
m
>
1
. We also extend this result to the Kähler–Ricci
g
-solitons and the coupled Kähler–Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations.</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s10455-023-09911-2</doi></addata></record> |
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| subjects | Analysis Approximation Criteria Differential Geometry Geometry Global Analysis and Analysis on Manifolds Invariants Mathematical Physics Mathematics Mathematics and Statistics Solitary waves |
| title | Anticanonically balanced metrics and the Hilbert–Mumford criterion for the δm-invariant of Fujita–Odaka |
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