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Equivariant Hirzebruch class for singular varieties

Selecta Mathematica volume 22, pages 1413–1454 (2016)Cite this article

Abstract

We develop an equivariant version of the Hirzebruch class for singular varieties. When the group acting is a torus we apply localization theorem of Atiyah–Bott and Berline–Vergne. The localized Hirzebruch class is an invariant of a singularity germ. The singularities of toric varieties and Schubert varieties are of special interest. We prove certain positivity results for simplicial toric varieties. The positivity for Schubert varieties is illustrated by many examples, but it remains mysterious.

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Notes

  1. The original formulation of the localization theorem is stronger. Here we apply it for the localization in the dimension ideal. For an arbitrary prime ideal \(\mathfrak {p}\subset K_\mathbb {T}(pt)=Rep(\mathbb {T})\) there is a maximal group \(H\subset \mathbb {T}\), such that \(K_\mathbb {T}(X)_{\mathfrak {p}}\mathop {\rightarrow }\limits ^{\sim }K_\mathbb {T}(X^H)_{\mathfrak {p}}\). For a smaller ideal \(\mathfrak {p}\) we have to take a smaller subgroup H.

  2. I thank Oleg Karpenkov for driving my attention to this method of summation.

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Authors and Affiliations

  1. Department of Mathematics, Warsaw University, Banacha 2, 02-097, Warsaw, Poland

    Andrzej Weber

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  1. Andrzej Weber
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Correspondence to Andrzej Weber.

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Supported by NCN Grant 2013/08/A/ST1/00804.

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Weber, A. Equivariant Hirzebruch class for singular varieties. Sel. Math. New Ser. 22, 1413–1454 (2016). https://doi.org/10.1007/s00029-015-0214-x

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  • DOI: https://doi.org/10.1007/s00029-015-0214-x

Keywords

  • Characteristic classes of singular varieties
  • Hirzebruch class
  • Equivariant cohomology
  • Toric varieties
  • Schubert varieties

Mathematics Subject Classification

  • 14C17
  • 14E15
  • 14F43
  • 19L47
  • 55N91
  • 14M25
  • 14M15