Rational versus real cohomology algebras of low-dimensional toric varieties

Author:
Eva Maria Feichtner

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1695-1704

MSC (2000):
Primary 14M25; Secondary 14F25, 52B20

DOI:
https://doi.org/10.1090/S0002-9939-02-06735-7

Published electronically:
October 1, 2002

MathSciNet review:
1955255

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Abstract: We show that the real cohomology algebra of a compact toric variety of complex dimension is determined, up to isomorphism, by the combinatorial data of its defining fan. Surprisingly enough, this is no longer the case when taking rational coefficients. Moreover, we show that neither the rational nor the real or complex cohomology algebras of compact quasi-smooth toric varieties are combinatorial invariants in general.

**[Bi]**L. Billera:*The algebra of continuous piecewise polynomial functions*; Adv. Math. 76 (1989), 170-183. MR**90g:13021****[Br]**M. Brion:*The structure of the polytope algebra*; Tôhoku Math. J. 49 (1997), 1-32. MR**98a:52019****[Da]**V.I. Danilov:*The geometry of toric varieties*; Russ. Math. Surv. 33 (1978), 97-154. MR**80g:14001****[Ew]**G. Ewald:*Combinatorial Convexity and Algebraic Geometry*; Graduate Texts in Mathematics 168, Springer-Verlag, 1996. MR**97i:52012****[FY]**S. Fischli, D. Yavin:*Which**-manifolds are toric varieties?*; Math. Z. 215 (1994), 179-185. MR**95g:57045****[Fu]**W. Fulton:*Introduction to Toric Varieties*; Annals of Mathematics Studies 131, Princeton University Press, 1993. MR**94g:14028****[GM]**M. Goresky, R. MacPherson:*Intersection homology theory*; Topology 19 (1980), 135-162. MR**82b:57010****[GH]**P. Griffiths, J. Harris:*Principles of Algebraic Geometry*; Wiley-Interscience, New York, 1978. MR**80b:14001****[McC]**M. McConnell:*The rational homology of toric varieties is not a combinatorial invariant*; Proc. Amer. Math. Soc. 105 (1989), 986-991. MR**89i:14042****[McM]**P. McMullen:*On simple polytopes*; Invent. Math. 113 (1993), 419-444. MR**94d:52015****[Od]**T. Oda:*Convex Bodies and Algebraic Geometry*; Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 15, Springer-Verlag, 1988. MR**88m:14038****[Sa]**M. Saito:*Hodge structures via filtered**-modules*; Astérisque 130 (1985), 342-351. MR**87b:32019****[Zi]**G.M. Ziegler:*Lectures on Polytopes*; Graduate Texts in Mathematics 152, Springer-Verlag, 1995. MR**96a:52011**

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Additional Information

**Eva Maria Feichtner**

Affiliation:
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland

Email:
feichtne@math.ethz.ch

DOI:
https://doi.org/10.1090/S0002-9939-02-06735-7

Keywords:
Toric varieties,
cohomology algebras,
simplicial fans,
Hodge-Riemann-Minkowski inequalities

Received by editor(s):
May 14, 1999

Received by editor(s) in revised form:
December 18, 2001, January 25, 2002, and January 30, 2002

Published electronically:
October 1, 2002

Communicated by:
John R. Stembridge

Article copyright:
© Copyright 2002
American Mathematical Society