Existence of embeddings of smooth varieties into linear algebraic groups
We prove that every smooth affine variety of dimension d d embeds into every simple algebraic group of dimension at least 2 d + 2 2d+2 . We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and b...
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| Veröffentlicht in: | Journal of algebraic geometry 2023-01, Vol.32 (4), p.729-786 |
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| container_title | Journal of algebraic geometry |
| container_volume | 32 |
| creator | Feller, Peter van Santen, Immanuel |
| description | We prove that every smooth affine variety of dimension
d
d
embeds into every simple algebraic group of dimension at least
2
d
+
2
2d+2
. We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and build upon parametric transversality results for flexible affine varieties due to Kaliman. By adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show that our result is optimal up to a possible improvement of the bound to
2
d
+
1
2d+1
.
In order to study the limits of our embedding method, we use rational homology group calculations of homogeneous spaces and we establish a domination result for rational homology of complex smooth varieties. |
| doi_str_mv | 10.1090/jag/793 |
| format | Article |
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d
d
embeds into every simple algebraic group of dimension at least
2
d
+
2
2d+2
. We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and build upon parametric transversality results for flexible affine varieties due to Kaliman. By adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show that our result is optimal up to a possible improvement of the bound to
2
d
+
1
2d+1
.
In order to study the limits of our embedding method, we use rational homology group calculations of homogeneous spaces and we establish a domination result for rational homology of complex smooth varieties.</description><identifier>ISSN: 1056-3911</identifier><identifier>EISSN: 1534-7486</identifier><identifier>DOI: 10.1090/jag/793</identifier><language>eng</language><ispartof>Journal of algebraic geometry, 2023-01, Vol.32 (4), p.729-786</ispartof><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c258t-2c17415e9ed57a2be34f09fbe748f2b743a57363f213f3dc475f98c3f0e448ce3</citedby><cites>FETCH-LOGICAL-c258t-2c17415e9ed57a2be34f09fbe748f2b743a57363f213f3dc475f98c3f0e448ce3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,777,781,27905,27906</link.rule.ids></links><search><creatorcontrib>Feller, Peter</creatorcontrib><creatorcontrib>van Santen, Immanuel</creatorcontrib><title>Existence of embeddings of smooth varieties into linear algebraic groups</title><title>Journal of algebraic geometry</title><description>We prove that every smooth affine variety of dimension
d
d
embeds into every simple algebraic group of dimension at least
2
d
+
2
2d+2
. We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and build upon parametric transversality results for flexible affine varieties due to Kaliman. By adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show that our result is optimal up to a possible improvement of the bound to
2
d
+
1
2d+1
.
In order to study the limits of our embedding method, we use rational homology group calculations of homogeneous spaces and we establish a domination result for rational homology of complex smooth varieties.</description><issn>1056-3911</issn><issn>1534-7486</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNotkEFLwzAYhoM4cG7iX8jNU12SL2mao4zphIEXPZc0_VIz2mYkVfTf2-FO7_teHl4eQu45e-TMsM3Rdhtt4IosuQJZaFmV13NnqizAcH5DbnM-MiY4V3JJ9rufkCccHdLoKQ4Ntm0Yu3xeeYhx-qTfNgWcAmYaxinSPoxoE7V9h02ywdEuxa9TXpOFt33Gu0uuyMfz7n27Lw5vL6_bp0PhhKqmQjiuJVdosFXaigZBemZ8g_NPLxotwSoNJXjBwUPrpFbeVA48Qykrh7AiD_9cl2LOCX19SmGw6bfmrD4LqGcB9SwA_gCzWk6-</recordid><startdate>20230101</startdate><enddate>20230101</enddate><creator>Feller, Peter</creator><creator>van Santen, Immanuel</creator><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20230101</creationdate><title>Existence of embeddings of smooth varieties into linear algebraic groups</title><author>Feller, Peter ; van Santen, Immanuel</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c258t-2c17415e9ed57a2be34f09fbe748f2b743a57363f213f3dc475f98c3f0e448ce3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Feller, Peter</creatorcontrib><creatorcontrib>van Santen, Immanuel</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of algebraic geometry</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Feller, Peter</au><au>van Santen, Immanuel</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Existence of embeddings of smooth varieties into linear algebraic groups</atitle><jtitle>Journal of algebraic geometry</jtitle><date>2023-01-01</date><risdate>2023</risdate><volume>32</volume><issue>4</issue><spage>729</spage><epage>786</epage><pages>729-786</pages><issn>1056-3911</issn><eissn>1534-7486</eissn><abstract>We prove that every smooth affine variety of dimension
d
d
embeds into every simple algebraic group of dimension at least
2
d
+
2
2d+2
. We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and build upon parametric transversality results for flexible affine varieties due to Kaliman. By adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show that our result is optimal up to a possible improvement of the bound to
2
d
+
1
2d+1
.
In order to study the limits of our embedding method, we use rational homology group calculations of homogeneous spaces and we establish a domination result for rational homology of complex smooth varieties.</abstract><doi>10.1090/jag/793</doi><tpages>58</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1056-3911 |
| ispartof | Journal of algebraic geometry, 2023-01, Vol.32 (4), p.729-786 |
| issn | 1056-3911 1534-7486 |
| language | eng |
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| source | American Mathematical Society Publications |
| title | Existence of embeddings of smooth varieties into linear algebraic groups |
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