Twistor geometry of the flag manifold

Amedeo Altavilla; Edoardo Ballico; Maria Chiara Brambilla; Simon Salamon

doi:10.1007/s00209-022-03161-x

Twistor geometry of the flag manifold

Amedeo Altavilla, Edoardo Ballico, Maria Chiara Brambilla, Simon Salamon

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3 Citations (Scopus)

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Abstract

A study is made of algebraic curves and surfaces in the flag manifold F=SU(3)/T2, and their configuration relative to the twistor projection π from F to the complex projective plane P2, defined with the help of an anti-holomorphic involution j. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space P3 of the 4-dimensional sphere S4. Deformations of twistor fibers project to real surfaces in P2, whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in F of bidegree (1,1). These surfaces define orthogonal complex structures on specified dense open subsets of P2 relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree (1,1) are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in F.

Original language	English
Article number	24
Number of pages	43
Journal	MATHEMATISCHE ZEITSCHRIFT
Volume	303
Issue number	1
Early online date	16 Dec 2022
DOIs	https://doi.org/10.1007/s00209-022-03161-x
Publication status	Published - 2023

Keywords

math.DG
math.AG
math.CV
Primary: 32L25, 14M15, Secondary: 53C15, 53C28, 14J10, 15A21

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10.1007/s00209-022-03161-x

Twistor geometry of the_ALTAVILLA_Accepted14September2022_GREEN AAMAccepted author manuscript, 1.25 MB

https://arxiv.org/abs/2112.11100

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note = "34 pages; 7 figures. *** Comments are welcome! *** Funding Information: The authors were partially supported as follows:AA by GNSAGA and the INdAM project {\textquoteleft}Teoria delle funzioni ipercomplesse e applicazioni{\textquoteright}, MCB by GNSAGA and by the PRIN project {\textquoteleft}Geometria delle variet{\`a} algebriche{\textquoteright}, SS by the Simons Foundation (#488635, Simon Salamon). Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

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N2 - A study is made of algebraic curves and surfaces in the flag manifold F=SU(3)/T2, and their configuration relative to the twistor projection π from F to the complex projective plane P2, defined with the help of an anti-holomorphic involution j. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space P3 of the 4-dimensional sphere S4. Deformations of twistor fibers project to real surfaces in P2, whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in F of bidegree (1,1). These surfaces define orthogonal complex structures on specified dense open subsets of P2 relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree (1,1) are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in F.

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Twistor geometry of the flag manifold

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