# Rank $1$ locally free quotient sheaves of the cotangent bundle restricted to a smooth generic curve

Let $$\mathbb{K}$$ be an algebraically closed field of characteristic $$0$$. Let $$X$$ be a smooth projective variety over $$\mathbb{K}$$ such that $$K_X$$ (the canonical bundle) is ample.

For $$m\gg0$$ there exists $$D_1,\dotsc,D_{n-1}\in|mK_X|$$ such that $$C=D_1\cap\dotsc\cap D_{n-1}$$ is a smooth curve.

Let $$\Omega^1_X$$ be the cotangent bundle of $$X$$, then $$\Omega^1_C$$ (the cotangent bundle of $$C$$) is a rank $$1$$ locally free quotient sheaf of $$\Omega^1_{X\vert C}$$.

Is $$\Omega^1_C$$ the unique (up to isomorphism) rank $$1$$ locally free quotient sheaf of $$\Omega^1_{X\vert C}$$?

If the answer is no, how can one classify (up to isomorphism) these quotient sheaves of $$\Omega^1_{X\vert C}$$?

This is not true in general. For instance, let $$X = C_1 \times C_2$$ be a product of two curves of genus larger than 1. Then $$\Omega_X = p_1^*\Omega_{C_1} \oplus p_2^*\Omega_{C_2}$$ is the sum of two line bundles, hence the same us true for its restriction to $$C$$.
• In what terms do you want to classify these? I think this is a particular case of the question of classifying line bundle quotients of a vector bundle $E$ and there is nothing specific about $\Omega_X$. Rephrased geometrically, this is the question of classifying sections of a projective bundle over a curve. Commented Jul 7 at 12:56