Hirzebruch–Milnor classes of complete intersections

Maxim, Laurenţiu; Saito, Morihiko; Schürmann, Jörg

doi:10.1016/j.aim.2013.04.001

Cited by 31 publications

(43 citation statements)

References 32 publications

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“…Finally, the following crucial result is proved in (greater generality in) [44][Prop.5.5] by using an alternative and more sophisticated definition of the Hirzebruch class transformations in terms of M. Saito's theory of mixed Hodge modules (as explained in [10,55]…”

Section: Motivic Chern and Homology Hirzebruch Classes Of Singular Vamentioning

confidence: 99%

Characteristic Classes of Singular Toric Varieties

Maxim

Schürmann

2014

Comm Pure Appl Math

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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.

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Section: Motivic Chern and Homology Hirzebruch Classes Of Singular Vamentioning

confidence: 99%

Characteristic Classes of Singular Toric Varieties

Maxim

Schürmann

2014

Comm Pure Appl Math

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“…homology Hirzebruch classes (see [30][Prop.5.1.2]): Proposition 3.8. Let X be a complex algebraic variety, and fix M ∈ D b MHM(X) with underlying Q-complex K. Let S = {S} be a complex algebraic stratification of X such that for any S ∈ S, S is smooth, S \ S is a union of strata, and the sheaves H i K| S are local systems on S for any i.…”

Section: 1] and [39][p428])mentioning

confidence: 99%

“…In this section, we explain how to compute the homology Hirzebruch classes of globally defined hypersurfaces in an algebraic manifold (but see also [30] for the global complete intersection case). The key technical result needed for this calculation is the specialization property of the Hirzebruch class transformation, obtained by the second author in [38].…”

Section: Hirzebruch-milnor Classesmentioning

confidence: 99%