We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds.

中文翻译：

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法诺流形和切线束的稳定性

我们确定了某个类别的两个轨道变种中Fano变种的切线束的稳定性/不稳定性，这是由Pasquier在2009年进行分类的。因此，我们证明了其中一些变种允许不稳定的切线束，这证明了Fano流形切线束稳定性的猜想。