1 February 2002 The signature of a toric variety
Naichung Conan Leung, Victor Reiner
Duke Math. J. 111(2): 253-286 (1 February 2002). DOI: 10.1215/S0012-7094-02-11123-5


We identify a combinatorial quantity (the alternating sum of the h-vector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by R. Charney and M. Davis in their work, which in particular showed that its nonnegativity is closely related to a conjecture of H. Hopf on the Euler characteristic of a nonpositively curved manifold.

We prove positive (or nonnegative) lower bounds for this quantity under geometric hypotheses on the polytope and, in particular, resolve a special case of their conjecture. These hypotheses lead to ampleness (or weaker conditions) for certain line bundles on toric divisors, and then the lower bounds follow from calculations using the Hirzebruch signature formula.

Moreover, we show that under these hypotheses on the polytope, the ith L-class of the corresponding toric variety is (−1)i times an effective class for any i.


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Naichung Conan Leung. Victor Reiner. "The signature of a toric variety." Duke Math. J. 111 (2) 253 - 286, 1 February 2002. https://doi.org/10.1215/S0012-7094-02-11123-5


Published: 1 February 2002
First available in Project Euclid: 18 June 2004

zbMATH: 1062.14067
MathSciNet: MR1882135
Digital Object Identifier: 10.1215/S0012-7094-02-11123-5

Primary: 14M25
Secondary: 52B05 , 52B20

Rights: Copyright © 2002 Duke University Press


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Vol.111 • No. 2 • 1 February 2002
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