Equivariant toric geometry and Euler-Maclaurin formulae -- an overview

TL;DR

This paper surveys recent advances in equivariant toric geometry, focusing on motivic characteristic classes and Euler-Maclaurin formulae, with applications to lattice polytopes and torus-invariant divisors.

Contribution

It introduces new Euler-Maclaurin formulae for toric varieties using motivic characteristic classes, extending classical results to broader contexts.

Findings

Global formulas for equivariant Hirzebruch classes via localization

Weighted versions of classical Brion and Molien formulas

Generalized Euler-Maclaurin formulae for arbitrary sheaf coefficients

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 includes global formulae for equivariant Hirzebruch classes in the simplicial context proved by localization at the torus fixed points, a 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 coefficients 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.