EQUIVARIANT TORIC GEOMETRY AND EULER–MACLAURIN FORMULAE – AN OVERVIEW –

Sylvain E. Cappell, Laurenţiu G. Maxim, Jörg Schürmann, Julius L. Shaneson

Research output: Contribution to journalArticlepeer-review

Abstract

We survey recent developments in the study of torus equivariant motivic Chern and Hirzebruch characteristic classes of projective toric varieties, with applications to calculating equivariant Hirzebruch genera of torus-invariant Cartier divisors in terms of torus characters, as well as to general Euler–Maclaurin type formulae for full-dimensional simple lattice polytopes. We present recent results by the authors, emphasizing the main ideas and some key examples. This in-cludes global formulae for equivariant Hirzebruch classes in the simplicial context proved by localization at the torus fixed points, weighted versions of a classical formula of Brion, as well as of the Molien formula of Brion–Vergne. Our Euler– Maclaurin type formulae provide generalizations to arbitrary coherent sheaf co-efficients of the Euler–Maclaurin formulae of Cappell–Shaneson, Brion–Vergne, Guillemin, etc., via the equivariant Hirzebruch–Riemann–Roch formalism. Our approach, based on motivic characteristic classes, allows us, e.g., to obtain such Euler–Maclaurin formulae also for (the interior of) a face. We obtain such results also in the weighted context, and for Minkovski summands of the given full-dimensional lattice polytope.

Original languageEnglish (US)
Pages (from-to)105-128
Number of pages24
JournalREVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES
Volume69
Issue number2
DOIs
StatePublished - 2024

Keywords

  • equivariant Hirzebruch–Riemann– Roch
  • equivariant motivic Chern and Hirzebruch classes
  • Euler–Maclaurin for-mulae
  • lattice points
  • lattice polytopes
  • Lefschetz–Riemann–Roch
  • localization
  • toric varieties

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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