Algebra & Number Theory

Deformations of smooth complete toric varieties: obstructions and the cup product

Nathan Ilten and Charles Turo

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Let X be a complete -factorial toric variety. We explicitly describe the space H2(X,𝒯X) and the cup product map H1(X,𝒯X)×H1(X,𝒯X)H2(X,𝒯X) in combinatorial terms. Using this, we give an example of a smooth projective toric threefold for which the cup product map does not vanish, showing that in general, smooth complete toric varieties may have obstructed deformations.

Article information

Algebra Number Theory, Volume 14, Number 4 (2020), 907-926.

Received: 2 January 2019
Revised: 25 November 2019
Accepted: 6 February 2020
First available in Project Euclid: 30 June 2020

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 14B12: Local deformation theory, Artin approximation, etc. [See also 13B40, 13D10] 14D15: Formal methods; deformations [See also 13D10, 14B07, 32Gxx]

deformation theory toric varieties cup product


Ilten, Nathan; Turo, Charles. Deformations of smooth complete toric varieties: obstructions and the cup product. Algebra Number Theory 14 (2020), no. 4, 907--926. doi:10.2140/ant.2020.14.907.

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  • K. Altmann, “Computation of the vector space $T^1$ for affine toric varieties”, J. Pure Appl. Algebra 95:3 (1994), 239–259.
  • K. Altmann, “Minkowski sums and homogeneous deformations of toric varieties”, Tohoku Math. J. $(2)$ 47:2 (1995), 151–184.
  • K. Altmann, “Infinitesimal deformations and obstructions for toric singularities”, J. Pure Appl. Algebra 119:3 (1997), 211–235.
  • K. Altmann, “The versal deformation of an isolated toric Gorenstein singularity”, Invent. Math. 128:3 (1997), 443–479.
  • S. Bosch, Algebraic geometry and commutative algebra, Springer, 2013.
  • T. Coates, A. Corti, S. Galkin, V. Golyshev, and A. Kasprzyk, “Mirror symmetry and Fano manifolds”, pp. 285–300 in European Congress of Math. (Kraków, 2012), edited by R. Latała et al., Eur. Math. Soc., Zürich, 2013.
  • D. A. Cox, J. B. Little, and H. K. Schenck, Toric varieties, Graduate Studies in Math. 124, Amer. Math. Soc., Providence, RI, 2011.
  • M. Filip, “The Gerstenhaber product ${\rm HH}^2(A)\times {\rm HH}^2(A)\to {\rm HH}^3(A)$ of affine toric varieties”, preprint, 2018.
  • W. Fulton, Introduction to toric varieties, Ann. of Math. Studies 131, Princeton Univ. Press, 1993.
  • R. Godement, Topologie algébrique et théorie des faisceaux, Actualités Sci. Indust. 1252, Hermann, Paris, 1958.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, 1977.
  • N. O. Ilten, “Deformations of smooth toric surfaces”, Manuscripta Math. 134:1-2 (2011), 123–137.
  • N. Ilten and H. Süss, “K-stability for Fano manifolds with torus action of complexity 1”, Duke Math. J. 166:1 (2017), 177–204.
  • N. O. Ilten and R. Vollmert, “Deformations of rational $T$-varieties”, J. Algebraic Geom. 21:3 (2012), 531–562.
  • K. Jaczewski, “Generalized Euler sequence and toric varieties”, pp. 227–247 in Classification of algebraic varieties (L'Aquila, 1992), edited by C. Ciliberto et al., Contemp. Math. 162, Amer. Math. Soc., Providence, RI, 1994.
  • A. R. Mavlyutov, “Deformations of Calabi–Yau hypersurfaces arising from deformations of toric varieties”, Invent. Math. 157:3 (2004), 621–633.
  • A. Mavlyutov, “Deformations of toric varieties via Minkowski sum decompositions of polyhedral complexes”, preprint, 2009.
  • A. Petracci, “Homogeneous deformations of toric pairs”, preprint, 2018.
  • Y. Rollin and C. Tipler, “Deformations of extremal toric manifolds”, J. Geom. Anal. 24:4 (2014), 1929–1958.
  • G. Tian, “Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric”, pp. 629–646 in Mathematical aspects of string theory (San Diego, CA, 1986), edited by S.-T. Yau, Adv. Ser. Math. Phys. 1, World Sci., Singapore, 1987.
  • A. N. Todorov, “The Weil–Petersson geometry of the moduli space of ${\rm SU}(n\geq 3)$ (Calabi–Yau) manifolds, I”, Comm. Math. Phys. 126:2 (1989), 325–346.
  • R. Vakil, “Murphy's law in algebraic geometry: badly-behaved deformation spaces”, Invent. Math. 164:3 (2006), 569–590.