  A quasitoric manifold is a smooth 2n-manifold M^{2n} with an action of the
compact torus T^n such that the action is locally isomorphic to the standard
action of T^n on C^n and the orbit space is diffeomorphic, as manifold with
corners, to a simple polytope P^n. The name refers to the fact that topological
and combinatorial properties of quasitoric manifolds are similar to that of
non-singular algebraic toric varieties (or toric manifolds). Unlike toric
varieties, quasitoric manifolds may fail to be complex; however, they always
admit a stably (or weakly almost) complex structure, and their cobordism
classes generate the complex cobordism ring. As it have been recently shown by
Buchstaber and Ray, a stably complex structure on a quasitoric manifold is
defined in purely combinatorial terms, namely, by an orientation of the
polytope and a function from the set of codimension-one faces of the polytope
to primitive vectors of an integer lattice. We calculate the \chi_y-genus of a
quasitoric manifold with fixed stably complex structure in terms of the
corresponding combinatorial data. In particular, this gives explicit formulae
for the classical Todd genus and signature. We also relate our results with
well-known facts in the theory of toric varieties.
