Hirzebruch genera of manifolds with torus action

T. E. Panov

Izvestiya. Mathematics , Tome 65 (2001) no. 3, pp. 543-556.

Résumé

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 $\mathbb C^n$ and the orbit space is diffeomorphic, as a 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 those 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. Buchstaber and Ray have recently shown that the stably complex structure on a quasitoric manifold is determined 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 the integer lattice. We calculate the $\chi_y$-genus of a quasitoric manifold with a fixed stably complex structure in terms of the corresponding combinatorial data. In particular, this gives explicit formulae for the classical Todd genus and the signature. We also compare our results with well-known facts in the theory of toric varieties.

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