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Elliptic classes of Schubert varieties

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Abstract

We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov–Libgober classes of Schubert varieties in general homogeneous spaces G/P. While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi–Tarasov–Varchenko we prove that weight functions represent elliptic classes of Schubert varieties.

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Notes

  1. Or by \(\hbar \) in physics literature, also sometimes by y in K-theory—to match the classical notion of Hirzebruch \(\chi _y\)–genus.

  2. These extra variables are probably related with the “dynamical variables”, a.k.a. “Kähler variables” of mathematical physics literature.

  3. The proof is parallel to that of [22, \(\S \)10].

  4. The discrepancy divisor is equal to \(-D\). We do not assume that \(\Delta \) is effective.

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Acknowledgements

S.K. is supported by the NSF Grant DMS 1802328, R.R. is supported by a Simons foundation grant. A.W. is supported by the research project of the Polish National Research Center 2016/23/G/ST1/04282 (Beethoven 2, German-Polish joint project).

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

    Shrawan Kumar & Richárd Rimányi

  2. Institute of Mathematics, University of Warsaw, Warsaw, Poland

    Andrzej Weber

Authors
  1. Shrawan Kumar
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  2. Richárd Rimányi
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  3. Andrzej Weber
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Correspondence to Richárd Rimányi.

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Communicated by Jean-Yves Welschinger.

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Kumar, S., Rimányi, R. & Weber, A. Elliptic classes of Schubert varieties. Math. Ann. 378, 703–728 (2020). https://doi.org/10.1007/s00208-020-02043-z

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