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The Generalized Roof F(1, 2,n): Hodge Structures and Derived Categories

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Abstract

We consider generalized homogeneous roofs, i.e. quotients of simply connected, semisimple Lie groups by a parabolic subgroup, which admit two projective bundle structures. Given a general hyperplane section on such a variety, we study the zero loci of its pushforwards along the projective bundle structures and we discuss their properties at the level of Hodge structures. In the case of the flag variety F(1,2,n) with its projections to ℙn− 1 and G(2,n), we construct a derived embedding of the relevant zero loci by methods based on the study of B-brane categories in the context of a gauged linear sigma model.

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Acknowledgments

We would like to thank Sergio Cacciatori, Jacopo Gandini, Riccardo Moschetti and Fabio Tanturri for sharing valuable insight. We would also like to express our gratitude to Akihiro Kanemitsu for helpful comments to the first version of this paper, and for pointing out the reference [30]. We thank the anonymous referee for pointing out a flaw in the proof of the main theorem, and for the careful and thorough reading. EF and GM are members of the INDAM-GNSAGA. EF, GM and MR are partially supported by PRIN2017 “2017YRA3LK”, GM and MR are partially supported by PRIN2020 “2020KKWT53”. MK is supported by the project Narodowe Centrum Nauki 2018/31/B/ST1/02857.

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Authors and Affiliations

  1. Dipartimento di Matematica “Guido Castelnuovo”, Sapienza - Università di Roma, Piazzale Aldo Moro 5, 00185, Rome, Italy

    Enrico Fatighenti

  2. Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warsaw, Poland

    Michał Kapustka

  3. Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

    Giovanni Mongardi & Marco Rampazzo

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  1. Enrico Fatighenti
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  2. Michał Kapustka
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  3. Giovanni Mongardi
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  4. Marco Rampazzo
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Correspondence to Marco Rampazzo.

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Fatighenti, E., Kapustka, M., Mongardi, G. et al. The Generalized Roof F(1, 2,n): Hodge Structures and Derived Categories. Algebr Represent Theor 26, 2313–2342 (2023). https://doi.org/10.1007/s10468-022-10173-y

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