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

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