Tohoku Mathematical Journal

The polyhedral Hodge number $h^{2,1}$ and vanishing of obstructions

Klaus Altmann and Duco van Straten

Full-text: Open access


We prove a vanishing theorem for the Hodge number $h^{2,1}$ of projective toric varieties provided by a certain class of polytopes. We explain how this Hodge number also gives information about the deformation theory of the toric Gorenstein singularity derived from the same polytope. In particular, the vanishing theorem for $h^{2,1}$ implies that these deformations are unobstructed.

Article information

Tohoku Math. J. (2), Volume 52, Number 4 (2000), 579-602.

First available in Project Euclid: 3 May 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]


Altmann, Klaus; van Straten, Duco. The polyhedral Hodge number $h^{2,1}$ and vanishing of obstructions. Tohoku Math. J. (2) 52 (2000), no. 4, 579--602. doi:10.2748/tmj/1178207756.

Export citation


  • [Al] K. ALTMANN, The versal Deformation of an isolated toric Gorenstein singularity, Invent. Math. 128 (1997), 443-479.
  • [AH] K. ALTMANN AND L. HILLE, Strong exceptional sequences provided by quivers, Algebras and Repre sentation Theory 2 (1999), 1-17.
  • [AvS] K. ALTMANN AND D. VANSTRATEN, Quiver polytope varieties and their deformations, Inpreparation
  • [AS] K. ALTMANN AND A. B. SLETSJ0E, Andre-Quillen cohomology of monoid algebras, J. Algebra 21 (1998), 708-718.
  • [Ba] V. BATYREV, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), 493-535.
  • [BCKvS] V. BATYREV, I. CIOCAN-FONTANINE, B. KIM AND D. VAN STRATEN, Mirror symmetry and ton degenerations of partial flag manifolds, Acta Math. 184 (2000), 1-39.
  • [BC] K. BEHNKE AND J. A. CHRISTOPHERSEN, Hypersurface sections and obstructions (rational surfac singularities), Compositio Math. 77 (1991), 233-268.
  • [Br] M. BRION, The structure of the polytope algebra, Tohoku Math. J. 49 (1997), 1-32
  • [Da] V. I. DANILOV, The geometry of toric varieties, Russian Math. Surveys 33 (1978), 97-154
  • [Fu] W. FULTON, Introductionto Toric varieties, Annals of Mathematics Studies 131, Princeton, NJ, 1993
  • [GM1] S. I. GELFAND AND YU. I. MANIN, Methods of Homological Algebra, Springer-Verlag, Berlin, 1996
  • [GM2] S. I. GELFAND AND YU. I. MANIN, Homological Algebra, Encyclopaedia Math. Sci. 38, Algebra V, Springer-Verlag, Berlin, 1994.
  • [La] V. LAKSHMIBAI, Degeneration of flag varieties to toric varieties, C. R. Acad. Sci.Paris Ser. I Math. 32 (1995), 1229-1234.
  • [Lo] J. -L. LODAY, Cyclic Homology, Grundlehren Math. Wiss. 301, Springer-Verlag, New York, 1992
  • [St] B. STURMFELS, Grobner Bases and Convex Polytopes, Univ. Lecture Ser. 8, Amer. Math. Soc, Provi dence, RI, 1996.
  • [Od] T. ODA, Convex bodies and algebraic geometry, An introduction to the theory of toric varieties, Ergeb Math. Grenzgeb. (3), 15, Springer-Verlag, Berlin-New York, 1988.