Deformations of Rational T-Varieties

Abstract

We show how to construct certain homogeneous deformations for rational normal varieties with codimension one torus action. This can then be used to construct homogeneous deformations of any toric variety in arbitrary degree. For locally trivial deformations coming from this construction, we calculate the image of the Kodaira-Spencer map. We then show that for a smooth complete toric variety, our homogeneous deformations span the space of first-order deformations.

Deformations of Extremal Toric Manifolds

“…We will restrict ourselves to the study of the action of T ⊂ H on the space of infinitesimal deformations. In [15], Ilten and Vollmert gave a simple description for generators of the vector space H 1 (X, Θ X ).…”

“…From theorem 6.2. [15], for each ρ ∈ Ω(R) and each C a connected component of Γ ρ (−R), the element π(C, ρ, R) is given as a cocycle by derivations defined on intersections of an open cover of X. Each of these derivation is proportional to the derivation ∂(R, ρ) that takes…”

“…To conclude, from theorem 6.5. of [15], the elements π(C, ρ, R) span H 1 (X, Θ X )(R). Now we can investigate polystable points under the action of the torus.…”

“…This picture is particularly clear thanks to the theory of T-varieties due to Altmann, Ilten and Vollmert. Relying on [15], the stable deformations can be determined explicitly.…”

“…Let Σ (1) be the set of rays in Σ, identified with primitive generators of these rays. Following Ilten and Vollmert [15], we can compute from the fan Σ a finite set N * def (Σ) ⊂ N * (cf. §3.2) which is the set of weights of the torus action on H 1 (X, Θ X ).…”