Stability of the canonical extension of tangent bundles on Picard-rank-1 Fano varieties

TL;DR

This paper investigates the stability properties of the canonical extension of tangent bundles on Picard-rank-1 Fano varieties, revealing conditions under which stability of the tangent bundle implies stability of the extension, with implications for moduli spaces.

Contribution

It establishes a link between tangent bundle stability and the stability of its canonical extension on specific Fano varieties, a novel insight in algebraic geometry.

Findings

Stability of tangent bundle implies (semi)stability of the canonical extension under certain divisibility conditions.

Canonical extensions on moduli spaces of stable bundles are at least semistable.

In some cases, the canonical extension is proven to be stable.

Abstract

We consider slope stability of the canonical extension of the tangent bundle by the trivial line bundle and with the extension class c_1(T_X) on Picard-rank-1 Fano varieties. In cases where the index divides the dimension or the dimension plus one, we show that stability of the tangent bundle implies (semi)stability of the canonical extension. One consequence of our result is that the canonical extensions on moduli spaces of stable vector bundles with a fixed determinant on a curve are at least semistable, and stable in some cases.