Characteristic Classes of Singular Toric Varieties

Abstract

Abstract. In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the context of complete toric varieties fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As important special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of the torus-orbit decomposition and the motivic properties of the motivic Chern and resp. homology Hirzebruch classes to express the latter in terms of dualizing sheaves and resp. the (dual) Todd classes of closures of orbits. This method even applies to torus-invariant subspaces of a given toric variety. The obtained formula is then applied to weighted lattice point counting in lattice polytopes and their subcomplexes, yielding generalized Pick-type formulae. Secondly, in the case of simplicial toric varieties, we compute our characteristic classes by using the Lefschetz-Riemann-Roch theorem of Edidin-Graham in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties (which specialize to the mock Chern, mock Todd and mock L-classes of such varieties) and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. Finally, the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.

Characteristic classes of mixed Hodge modules and applications

“…By using toric geometry, formula (3.32) adapted to the notations of this section yields the following (see [28] …”

“…The results of this section hold more generally for torus-invariant closed algebraic subsets of X Σ which are known to also have Du Bois singularities. For more applications and examples, e.g., generalized Pick-type formulae for full-dimensional lattice polytopes, see [28].…”

Equivariant Hirzebruch class for singular varieties

“…The case when X is the complement of a simple normal crossing divisor D ⊂ M is of particular interest, and it is worth to give an explicit formula in terms of logarithmic forms. A different formula connecting Hirzebruch class with the sheaf of logarithmic forms was given in [MS14,§2].…”

“…The nonequivariant version of Corollary 11.2 appeared in [MS14]. The Euler-Maclaurin formula for the Todd class of a closed orbit was given by many authors, see e.g.…”

Motivic and derived motivic Hirzebruch classes

“…We get the following corollary from [3] and [21]: (1) If X is a toric variety, then T 0 * (X) can be replaced by Baum-Fulton-MacPherson's Todd class td * (X). (2) If X is a simplicial projective toric variety, then T 0 * (X) can be replaced by Baum-Fulton-MacPherson's Todd class td * (X) and furthermore T 1 * (X) can be replaced by Capell-Shaneson's homology L-class L * (X).…”