
Mathematics 1999
Hirzebruch genera of manifolds with torus actionAbstract: A quasitoric manifold is a smooth 2nmanifold 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 nonsingular 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 codimensionone faces of the polytope to primitive vectors of an integer lattice. We calculate the \chi_ygenus 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 wellknown facts in the theory of toric varieties.
