  We introduce a relative refined $\chi_y$-genus for sch\"on subvarieties of
algebraic tori. These are rational functions of degree minus the codimension
with coefficients in the ring of lattice polytopes. We prove that the relative
refined $\chi_y$ turns sufficiently generic intersections into products, and
that we can recover the ordinary $\chi_y$-genus by counting lattice points.
Applying the tropical Chern character to the relative refined $\chi_y$-genus we
obtain a refined tropicalization which is a tropical cycle having rational
functions with $\mathbb Q$-coefficients as weights. We prove that the
top-dimensional component of the refined tropicalization specializes to the
unrefined tropicalization up to sign when setting $y=0$ and show that we can
recover the $\chi_y$-genus by integrating the refined tropicalization with
respect to a Todd measure.

Algebraic Geometry