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On intersection cohomology with torus actions of complexity one

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Abstract

The purpose of this article is to investigate the intersection cohomology for algebraic varieties with torus action. Given an algebraic torus \({{\mathbb {T}}}\), one of our result determines the intersection cohomology Betti numbers of any normal projective \(\mathbb {T}\)-variety admitting an algebraic curve as global quotient. The calculation is expressed in terms of a combinatorial description involving a divisorial fan which is the analogous of the defining fan of a toric variety. Our main tool to obtain this computation is a description of the decomposition theorem in this context.

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Notes

  1. We are grateful to the referees for suggesting this reference.

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Acknowledgements

We would like to thank the anonymous referees for their valuable comments which allowed to improve the presentation. The authors also thank Javier Fernández de Bobadilla, Javier Elizondo, Daniel Juteau, Simon Riche, Kari Vilonen, and Geordie Williamson for useful discussions. The authors benefited from the support of the ERC Consolidator Grant NMST. The second author thanks the Max Planck Institut für Mathematik Bonn for support. This research is supported by ERCEA Consolidator Grant 615655 – NMST and also by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence. accreditation SEV-2013-0323. This work has been partially supported by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO).

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Authors and Affiliations

  1. Basque Center of Applied Mathematics (BCAM), Bilbao, Spain

    Marta Agustin Vicente

  2. Heinrich Heine Universität, Düsseldorf, Germany

    Kevin Langlois

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  1. Marta Agustin Vicente
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  2. Kevin Langlois
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Correspondence to Kevin Langlois.

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Agustin Vicente, M., Langlois, K. On intersection cohomology with torus actions of complexity one. Rev Mat Complut 31, 163–186 (2018). https://doi.org/10.1007/s13163-017-0232-7

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