The title of my talk should be: "Characteristic classes of singular toric varieties

I will discuss the computation of the homology Hirzebruch
characteristic classes of (possibly singular) toric varieties. I will
present two different perspectives for the computation of these
characteristic classes. First, by taking advantage of the torus-orbit
decomposition and the motivic properties of the homology Hirzebruch
classes, we express the latter in terms of the (dual) Todd classes of
closures of orbits. The obtained formula is then applied to  weighted
lattice point counting in lattice polytopes. Secondly, in the case of
simplicial toric varieties, we make use of the Lefschetz-Riemann-Roch
theorem in the context of the geometric quotient description of such
varieties. In this setting, we define mock Hirzebruch classes of
simplicial toric 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. This is joint work with Joerg Schuermann.