TY - JOUR
T1 - The signature of a toric variety
AU - Leung, Naichung Conan
AU - Reiner, Victor
PY - 2002
Y1 - 2002
N2 - 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 .
AB - 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 .
UR - http://www.scopus.com/inward/record.url?scp=0036487164&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0036487164&partnerID=8YFLogxK
U2 - 10.1215/S0012-7094-02-11123-5
DO - 10.1215/S0012-7094-02-11123-5
M3 - Article
AN - SCOPUS:0036487164
SN - 0012-7094
VL - 111
SP - 253
EP - 286
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 2
ER -