Cycles representing the Todd class of a toric variety
Authors:
James Pommersheim and Hugh Thomas
Journal:
J. Amer. Math. Soc. 17 (2004), 983-994
MSC (2000):
Primary 14M25; Secondary 14C17, 52B20
DOI:
https://doi.org/10.1090/S0894-0347-04-00460-6
Published electronically:
May 25, 2004
MathSciNet review:
2083474
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product, we assign a rational number $\mu (\sigma )$ to each rational polyhedral cone $\sigma$ in the lattice, such that for any toric variety $X$ with fan $\Sigma$ in the lattice, we have \[ \operatorname {Td}(X)=\sum _{\sigma \in \Sigma } \mu (\sigma ) [V(\sigma )].\] This constitutes an improved answer to an old question of Danilov. In a similar way, beginning from the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli. Our construction is based on an intersection product on cycles of a simplicial toric variety developed by the second author. Important properties of the construction are established by showing a connection to the canonical representation of the Todd class of a simplicial toric variety as a product of torus-invariant divisors developed by the first author.
- Alexander Barvinok and James E. Pommersheim, An algorithmic theory of lattice points in polyhedra, New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97) Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 91–147. MR 1731815
- V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247 (Russian). MR 495499
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037
- Peter McMullen, Weakly continuous valuations on convex polytopes, Arch. Math. (Basel) 41 (1983), no. 6, 555–564. MR 731639, DOI https://doi.org/10.1007/BF01198585
- Robert Morelli, Pick’s theorem and the Todd class of a toric variety, Adv. Math. 100 (1993), no. 2, 183–231. MR 1234309, DOI https://doi.org/10.1006/aima.1993.1033
- James E. Pommersheim, Toric varieties, lattice points and Dedekind sums, Math. Ann. 295 (1993), no. 1, 1–24. MR 1198839, DOI https://doi.org/10.1007/BF01444874
- James E. Pommersheim, Products of cycles and the Todd class of a toric variety, J. Amer. Math. Soc. 9 (1996), no. 3, 813–826. MR 1358042, DOI https://doi.org/10.1090/S0894-0347-96-00209-3 [Tho]Tho H. Thomas, Cycle-level intersection theory for toric varieties, to appear in Canad. J. Math., available at http://www.arxiv.org/math.AG/0306144.
- Jarosław Włodarczyk, Decomposition of birational toric maps in blow-ups & blow-downs, Trans. Amer. Math. Soc. 349 (1997), no. 1, 373–411. MR 1370654, DOI https://doi.org/10.1090/S0002-9947-97-01701-7
Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14M25, 14C17, 52B20
Retrieve articles in all journals with MSC (2000): 14M25, 14C17, 52B20
Additional Information
James Pommersheim
Affiliation:
Department of Mathematics, Pomona College, Claremont, California 92037
Email:
jpommersheim@pomona.edu
Hugh Thomas
Affiliation:
Fields Institute, 222 College Street, Toronto ON, M5T 3J1 Canada
Address at time of publication:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3 Canada
MR Author ID:
649257
ORCID:
0000-0003-1177-9972
Email:
hthomas@fields.utoronto.ca
Keywords:
Toric variety,
Todd class,
polytopes,
counting lattice points
Received by editor(s):
October 25, 2003
Published electronically:
May 25, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.