Duke Mathematical Journal

The signature of a toric variety

Naichung Conan Leung and Victor Reiner

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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.

Article information

Duke Math. J., Volume 111, Number 2 (2002), 253-286.

First available in Project Euclid: 18 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 52B05: Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx] 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]


Leung, Naichung Conan; Reiner, Victor. The signature of a toric variety. Duke Math. J. 111 (2002), no. 2, 253--286. doi:10.1215/S0012-7094-02-11123-5. https://projecteuclid.org/euclid.dmj/1087575041

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