T-Varieties and T-Deformations

Université de Nantes, June 11-14 2012

There is a well-known correspondence between normal varieties with an effective action by a torus with dense orbit (toric varieties) and certain combinatorial data (rational polyhedral fans). Altmann, Hausen, and Süß have generalized this as follows: a normal variety X together with an effective action by a torus T with general orbit of codimension k (a complexity-k T-variety) corresponds to a k-dimensional variety (a sort of quotient of X by T) together with some combinatorial data (a p-divisor in the affine case, or more generally, a divisorial fan).

In this lecture series, I will describe this correspondence, and then show how it can be used to combinatorially construct equivariant deformations of complexity-one rational T-varieties. Since any toric variety may be considered as a complexity-one T-variety by restricting to the action of a subtorus, this construction in particular yields deformations of toric varieties. If the toric variety at hand is smooth and projective, these deformations will in fact span the vector space of first-order infinitesimal deformations.

No prior background in toric or algebraic geometry is necessary.

Tentative Outline

  1. Lecture I:
    • Introduction
    • Cones, fans, and polyhedra
    • Affine and non-affine toric varieties
    • Toric Morphisms
    • Toric Bouquets
  2. Lecture II:
    • p-divisors
    • Affine T-varieties
    • Orbits of Affine T-varieties
    • Downgrades of affine toric varieties
  3. Lecture III:
    • Admissible Minkowski decompositions
    • Deformations of affine rational complexity-one T-varieties
    • Local smoothing criteria
    • Non-affine T-varieties
  4. Lecture IV:
    • Minkowski decompositions of polyhedral complexes
    • Deformations of general rational complexity-one T-varieties
    • Kodaira-Spencer calculations
    • Global smoothing criteria

References

  1. The Geometry of T-Varieties. With K. Altmann, L. Petersen, H. Süß, R. Vollmert. (To appear in Contributions to Algebraic Geometry, IMPANGA Lecture Notes) [arXiv:1102.5760]
  2. Deformations of Rational T-Varieties. With Robert Vollmert. (To appear in the Journal of Algebraic Geometry) [arXiv:0903.1393]
  3. Deformations of Smooth Toric Surfaces. (Manuscripta Mathematica 134 (2011) pp. 123-137) [arXiv:0902.0529]