Equivariant Abelian principal bundles on nonsingular toric varieties

Abstract

We give a classification of the equivariant principal G-bundles on a nonsingular toric variety when G is a closed Abelian subgroup of GL k (C). We prove that any such bundle splits, that is, admits a reduction of structure group to the intersection of G with a torus. We give an explicit parametrization of the isomorphism classes of such bundles for a large family of G when X is complete.2010 Mathematics Subject Classification. 32L05,14M25.

A classification of equivariant principal bundles over nonsingular toric varieties

“…This concludes the proof. The following generalizes a similar result for abelian structure groups in [1].…”

“…Moreover since det ξ τ (t)P (τ, σ)ξ σ (t) −1 = det P (τ, σ) for every η ∈ σ τ Ξ(1). Comparing (4.2) with (4.1), we have η(ξ τ i ) = η(ξ σ γ(i) ) for every i and every η in σ τ Ξ (1). this association sends an isomorphism class of bundles to an isomorphism class of data.…”

“…As an application of the above description, we prove that if G is nilpotent then the principal G-bundle admits an equivariant reduction of structure group to a torus (meaning the bundle is split). For G abelian, this splitting result appeared in [1].…”

“…Some of the basic lemmas regarding local action functions in section 2 have appeared before in [1]. We include them here for the convenience of the reader.…”

“…Then any K-equivariant holomorphic principal G-bundle E over C k is equivariantly isomorphic to C k × E(0) where the latter has diagonal K-action. Some of the basic lemmas regarding local action functions in section 2 have appeared before in [1]. We include them here for the convenience of the reader.…”