Stability of equivariant vector bundles over toric varieties

TL;DR

This paper provides a comprehensive analysis of the (semi)stability of tangent bundles on certain classes of complex toric varieties, using combinatorial criteria and constructing specific vector bundles.

Contribution

It offers a complete characterization of (semi)stability for tangent bundles on toric varieties with Picard number 2 and certain Fano 4-folds, and constructs new equivariant vector bundles.

Findings

Complete (semi)stability criteria for tangent bundles on specified toric varieties.

Construction of indecomposable rank 2 equivariant vector bundles.

Existence of stable rank 2 bundles with particular Chern classes.

Abstract

We give a complete answer to the question of (semi)stability of tangent bundle of any nonsingular projective complex toric variety with Picard number 2 by using combinatorial crietrion of (semi)stability of an equivariant sheaf. We also give a complete answer to the question of (semi)stability of tangent bundle of all toric Fano 4-folds with Picard number (\leq) 3 which are classified by Batyrev \cite{batyrev}. We have constructed a collection of equivariant indecomposable rank 2 vector bundles on Bott tower and pseudo-symmetric toric Fano varieties. Further in case of Bott tower, we have shown the existence of an equivariant stable rank 2 vector bundle with certain Chern classes with respect to a suitable polarization.