Toric vector bundles: GAGA and Hodge theory

Jonas Stelzig

doi:10.1007/s00209-020-02605-6

Abstract

We prove a GAGA-style result for toric vector bundles with smooth base and give an algebraic construction of the Frölicher approximating vector bundle that has recently been introduced by Dan Popovici using analytic techniques. Both results make use of the Rees-bundle construction.

Highlights

A toric variety over a field k is an algebraic variety X over k with a Gnm-action that has a dense open orbit on which the group acts transitively
Toric varieties and vector bundles are an important source of examples in algebraic geometry
Just as normal toric varieties can be studied by combinatorial data, toric vector bundles on a given normal toric variety X have been classified in terms of linear-algebra-data, c.f. [10,11,13,14,18]

Summary

Introduction

A toric variety over a field k is an algebraic variety X over k with a Gnm-action that has a dense open orbit on which the group acts transitively. A vector bundle on such X is called toric if it is equipped with a Gnm-action s.t. the projection is an equivariant map. Just as normal toric varieties can be studied by combinatorial data, toric vector bundles (and more general classes of equivariant sheaves) on a given normal toric variety X have been classified in terms of linear-algebra-data (roughly as vector spaces and filtrations with certain compatibility conditions), c.f. One obtains an analytification functor: toric vector bundles on X −→ holomorphic toric vector bundles on X an. The first main result of this article is that for smooth toric varieties, this functor is an equivalence of categories: Theorem A For a smooth toric variety X over C, analytification induces an equivalence of categories between algebraic toric vector bundles on X and holomorphic toric vector bundles on X an. There is the known GAGA-principle by Serre [22], asserting an equivalence of the categories of coherent sheaves on a complex projective variety and its analytification

B Jonas Stelzig

Preliminaries

Definitions

Toric vector bundles on affine and projective spaces

Toric GAGA

The Frölicher approximating bundle