TY - JOUR
T1 - EQUIVARIANT TORIC GEOMETRY AND EULER–MACLAURIN FORMULAE – AN OVERVIEW –
AU - Cappell, Sylvain E.
AU - Maxim, Laurenţiu G.
AU - Schürmann, Jörg
AU - Shaneson, Julius L.
N1 - Publisher Copyright:
© 2024, Publishing House of the Romanian Academy. All rights reserved.
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
KW - equivariant Hirzebruch–Riemann– Roch
KW - equivariant motivic Chern and Hirzebruch classes
KW - Euler–Maclaurin for-mulae
KW - lattice points
KW - lattice polytopes
KW - Lefschetz–Riemann–Roch
KW - localization
KW - toric varieties
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U2 - 10.59277/RRMPA.2024.105.128
DO - 10.59277/RRMPA.2024.105.128
M3 - Article
AN - SCOPUS:85203074878
SN - 0035-3965
VL - 69
SP - 105
EP - 128
JO - REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES
JF - REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES
IS - 2
ER -