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Twistor geometry of the Flag manifold

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Abstract

A study is made of algebraic curves and surfaces in the flag manifold \(\mathbb {F}=SU(3)/T^2\), and their configuration relative to the twistor projection \(\pi \) from \(\mathbb {F}\) to the complex projective plane \(\mathbb {P}^{2}\), 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 \(\mathbb {P}^3\) of the 4-dimensional sphere \(S^4\). Deformations of twistor fibers project to real surfaces in \(\mathbb {P}^{2}\), whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in \(\mathbb {F}\) of bidegree \((1,1)\). These surfaces define orthogonal complex structures on specified dense open subsets of \(\mathbb {P}^{2}\) 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 \(\mathbb {F}\).

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Acknowledgements

The fourth author is grateful to Fran Burstall and Nick Shepherd-Barron for explaining relevant facts to him. We thank the referee for many helpful comments.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Bari ‘Aldo Moro’, via Edoardo Orabona, 4, 70125, Bari, Italy

    Amedeo Altavilla

  2. Dipartimento di Matematica, Università di Trento, Via Sommarive 14, Povo, 38123, Trento, Italia

    Edoardo Ballico

  3. Università Politecnica delle Marche, via Brecce Bianche, 60131, Ancona, Italia

    Maria Chiara Brambilla

  4. Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK

    Simon Salamon

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  1. Amedeo Altavilla
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  2. Edoardo Ballico
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  3. Maria Chiara Brambilla
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  4. Simon Salamon
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Correspondence to Amedeo Altavilla.

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The authors were partially supported as follows:AA by GNSAGA and the INdAM project ‘Teoria delle funzioni ipercomplesse e applicazioni’, MCB by GNSAGA and by the PRIN project ‘Geometria delle varietà algebriche’, SS by the Simons Foundation (#488635, Simon Salamon).

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Altavilla, A., Ballico, E., Brambilla, M.C. et al. Twistor geometry of the Flag manifold. Math. Z. 303, 24 (2023). https://doi.org/10.1007/s00209-022-03161-x

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