A Berger type normal holonomy theorem for complex submanifolds
Identifieur interne : 000467 ( Main/Exploration ); précédent : 000466; suivant : 000468A Berger type normal holonomy theorem for complex submanifolds
Auteurs : Sergio Console [Italie] ; Antonio J. Di Scala [Italie] ; Carlos Olmos [Argentine, États-Unis]Source :
- Mathematische Annalen [ 0025-5831 ] ; 2011-09-01.
Abstract
Abstract: We prove a Berger type theorem for the normal holonomy $${\Phi^\perp}$$ (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space $${\mathbb{C} P^n}$$. Namely, if $${\Phi^\perp}$$ does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of $${\mathbb{C}^n}$$ the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the $${\mathbb{C} P^n}$$ case) and basic facts of complex submanifolds.
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DOI: 10.1007/s00208-010-0597-0
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Di Scala</name><affiliation wicri:level="3"><country xml:lang="fr">Italie</country><wicri:regionArea>Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino</wicri:regionArea><placeName><settlement type="city">Turin</settlement><region type="région" nuts="2">Piémont</region></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">Italie</country></affiliation></author><author><name sortKey="Olmos, Carlos" sort="Olmos, Carlos" uniqKey="Olmos C" first="Carlos" last="Olmos">Carlos Olmos</name><affiliation wicri:level="1"><country xml:lang="fr">Argentine</country><wicri:regionArea>FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000, Córdoba</wicri:regionArea><wicri:noRegion>Córdoba</wicri:noRegion></affiliation><affiliation wicri:level="1"><country wicri:rule="url">États-Unis</country></affiliation></author></analytic><monogr></monogr><series><title level="j">Mathematische Annalen</title><title level="j" type="abbrev">Math. Ann.</title><idno type="ISSN">0025-5831</idno><idno type="eISSN">1432-1807</idno><imprint><publisher>Springer-Verlag</publisher><pubPlace>Berlin/Heidelberg</pubPlace><date type="published" when="2011-09-01">2011-09-01</date><biblScope unit="volume">351</biblScope><biblScope unit="issue">1</biblScope><biblScope unit="page" from="187">187</biblScope><biblScope unit="page" to="214">214</biblScope></imprint><idno type="ISSN">0025-5831</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0025-5831</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: We prove a Berger type theorem for the normal holonomy $${\Phi^\perp}$$ (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space $${\mathbb{C} P^n}$$. Namely, if $${\Phi^\perp}$$ does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of $${\mathbb{C}^n}$$ the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the $${\mathbb{C} P^n}$$ case) and basic facts of complex submanifolds.</div></front></TEI><affiliations><list><country><li>Argentine</li><li>Italie</li><li>États-Unis</li></country><region><li>Piémont</li></region><settlement><li>Turin</li></settlement></list><tree><country name="Italie"><region name="Piémont"><name sortKey="Console, Sergio" sort="Console, Sergio" uniqKey="Console S" first="Sergio" last="Console">Sergio Console</name></region><name sortKey="Console, Sergio" sort="Console, Sergio" uniqKey="Console S" first="Sergio" last="Console">Sergio Console</name><name sortKey="Di Scala, Antonio J" sort="Di Scala, Antonio J" uniqKey="Di Scala A" first="Antonio J." last="Di Scala">Antonio J. 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