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Complete intersections in spherical varieties

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Abstract

Let G be a complex reductive algebraic group. We study complete intersections in a spherical homogeneous space G / H defined by a generic collection of sections from G-invariant linear systems. Whenever nonempty, all such complete intersections are smooth varieties. We compute their arithmetic genus as well as some of their \(h^{p,0}\) numbers. The answers are given in terms of the moment polytopes and Newton–Okounkov polytopes associated to G-invariant linear systems. We also give a necessary and sufficient condition on a collection of linear systems so that the corresponding generic complete intersection is nonempty. This criterion applies to arbitrary quasi-projective varieties (i.e., not necessarily spherical homogeneous spaces). When the spherical homogeneous space under consideration is a complex torus \((\mathbb {C}^*)^n\), our results specialize to well-known results from the Newton polyhedra theory and toric varieties.

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Notes

  1. David Bernstein is the younger brother of Joseph Bernstein. He discovered the famous formula for the number of solutions in \((\mathbb {C}^*)^n\) of a generic system of n polynomial equations with fixed Newton polytopes [2]. This amazing formula inspired much activity that eventually lead to the creation of Newton polyhedra theory and the theory of Newton–Okounkov bodies.

  2. In [20, 21], instead of our notation \(N(\Delta )\), \(N^\circ (\Delta )\) and \(N'(\Delta )\) respectively, the notation \(T(\Delta )\), \(B^+(\Delta )\) and \(B(\Delta )\) is used.

  3. As the referee pointed out, in [7] Brion obtains more general vanishing results for the so-called log-homogeneous varieties. From this he deduces some results on the mixed Hodge structures of complete intersections in log-homogeneous varieties.

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Acknowledgments

We would like to thank Michel Brion for telling us about some references regarding moment polytopes of G-varieties, as well as the referee for some corrections and useful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA

    Kiumars Kaveh

  2. Department of Mathematics, University of Toronto, Toronto, Canada

    A. G. Khovanskii

  3. Moscow Independent University, Moscow, Russia

    A. G. Khovanskii

Authors
  1. Kiumars Kaveh
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  2. A. G. Khovanskii
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Correspondence to Kiumars Kaveh.

Additional information

Dedicated to Joseph Bernstein on the occasion of his 70th birthday.

The first author is partially supported by a National Science Foundation Grant (Grant ID: 1200581).

The second author is partially supported by the Canadian Grant No. 156833-12.

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Kaveh, K., Khovanskii, A.G. Complete intersections in spherical varieties. Sel. Math. New Ser. 22, 2099–2141 (2016). https://doi.org/10.1007/s00029-016-0271-9

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