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 Publish in OALib Journal ISSN: 2333-9721 APC: Only $99  Views Downloads  Relative Articles Hirzebruch genera of manifolds equipped with a Hamiltonian circle action Calculation of Hirzebruch genera for manifolds acted on by the group Z/p via invariants of the action Elliptic genera, torus manifolds and multi-fans On rigid Hirzebruch genera Todd genera of complex torus manifolds Chern numbers of manifolds with torus action Calibrated Fibrations on Complete Manifolds via Torus Action Calculations of the Hirzebruch$χ_y\$ genera of symmetric products by the holomorphic Lefschetz formula Elliptic formal group laws, integral Hirzebruch genera and Krichever genera K-Stability for Fano Manifolds with Torus Action of Complexity One More...
Mathematics  1999

# Hirzebruch genera of manifolds with torus action

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Abstract:

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.

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