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sheaf Sheaf of kahler differentialKähler differentials for complex manifold

Let $X \subset \mathbb{C}^n$ be an analytic set which means that it is the zero locus of holomorphic functions $f_1,f_2,...,f_n$$f_1,f_2,\dotsc,f_n$ on $\mathbb{C}^n$ and let suppose that there is a singular locus denoted Sing$\mathrm{Sing}$. Let $\mathcal{O}_X(U)$ be the holomorphic functions on an open set $U$ of $X$ and we denote by $\mathcal{O}_X$ the sheaf of the holomorhic function on $X$. weWe can define the sheaf of kahler differentialKähler differentials as follow: consider for every $U$ open set $U$ in $X$ the $\mathcal{O}_X(U)$-module generated by the set of symbols $\{df : f \in \mathcal{O}_X(U)\}$ which verifiessatisfies the relations $$ d(fg)=gdf+fdg$$ $$ d(af+bg)=ad(f)+bd(g)$$\begin{gather*} d(fg)=gdf+fdg \\ d(af+bg)=ad(f)+bd(g). \end{gather*} thisThis is the sheaf of kahler differentialKähler differentials, denoted $\Omega_X^1$. whenWhen $\Omega_X^1$ is restricted to $X \backslash Sing$$X \setminus \mathrm{Sing}$ it is free of rank $n$ where $n$ is the dimension of X.

Question: doesIs the vector bundle generated by this sheaf on $X \backslash Sing$ is$X \backslash \mathrm{Sing}$ isomorphic to the cotangent bundle on $X \backslash Sing$ is $X \setminus \mathrm{Sing}$? If not, what is the sheaf we should define on $X$ to get the cotangent bundle on $X \backslash Sing$ $X \backslash \mathrm{Sing}$?

sheaf of kahler differential for complex manifold

Let $X \subset \mathbb{C}^n$ be an analytic set which means that it is the zero locus of holomorphic functions $f_1,f_2,...,f_n$ on $\mathbb{C}^n$ and let suppose that there is a singular locus denoted Sing. Let $\mathcal{O}_X(U)$ be the holomorphic functions on an open set $U$ of $X$ and we denote by $\mathcal{O}_X$ the sheaf of the holomorhic function on $X$. we can define the sheaf of kahler differential as follow: consider for every $U$ open set $X$ the $\mathcal{O}_X(U)$-module generated by the set of symbols $\{df : f \in \mathcal{O}_X(U)\}$ which verifies the relations $$ d(fg)=gdf+fdg$$ $$ d(af+bg)=ad(f)+bd(g)$$ this is the sheaf of kahler differential denoted $\Omega_X^1$. when $\Omega_X^1$ is restricted to $X \backslash Sing$ it is free of rank $n$ where $n$ is the dimension of X.

Question: does the vector bundle generated by this sheaf on $X \backslash Sing$ is isomorphic to the cotangent bundle on $X \backslash Sing$ is ? If not what is the sheaf we should define on $X$ to get the cotangent bundle on $X \backslash Sing$ ?

Sheaf of Kähler differentials for complex manifold

Let $X \subset \mathbb{C}^n$ be an analytic set which means that it is the zero locus of holomorphic functions $f_1,f_2,\dotsc,f_n$ on $\mathbb{C}^n$ and suppose that there is a singular locus denoted $\mathrm{Sing}$. Let $\mathcal{O}_X(U)$ be the holomorphic functions on an open set $U$ of $X$ and denote by $\mathcal{O}_X$ the sheaf of the holomorhic function on $X$. We can define the sheaf of Kähler differentials as follow: consider for every open set $U$ in $X$ the $\mathcal{O}_X(U)$-module generated by the set of symbols $\{df : f \in \mathcal{O}_X(U)\}$ which satisfies the relations \begin{gather*} d(fg)=gdf+fdg \\ d(af+bg)=ad(f)+bd(g). \end{gather*} This is the sheaf of Kähler differentials, denoted $\Omega_X^1$. When $\Omega_X^1$ is restricted to $X \setminus \mathrm{Sing}$ it is free of rank $n$ where $n$ is the dimension of X.

Question: Is the vector bundle generated by this sheaf on $X \backslash \mathrm{Sing}$ isomorphic to the cotangent bundle on $X \setminus \mathrm{Sing}$? If not, what is the sheaf we should define on $X$ to get the cotangent bundle on $X \backslash \mathrm{Sing}$?

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sheaf of kahler differential for complex manifold

Let $X \subset \mathbb{C}^n$ be an analytic set which means that it is the zero locus of holomorphic functions $f_1,f_2,...,f_n$ on $\mathbb{C}^n$ and let suppose that there is a singular locus denoted Sing. Let $\mathcal{O}_X(U)$ be the holomorphic functions on an open set $U$ of $X$ and we denote by $\mathcal{O}_X$ the sheaf of the holomorhic function on $X$. we can define the sheaf of kahler differential as follow: consider for every $U$ open set $X$ the $\mathcal{O}_X(U)$-module generated by the set of symbols $\{df : f \in \mathcal{O}_X(U)\}$ which verifies the relations $$ d(fg)=gdf+fdg$$ $$ d(af+bg)=ad(f)+bd(g)$$ this is the sheaf of kahler differential denoted $\Omega_X^1$. when $\Omega_X^1$ is restricted to $X \backslash Sing$ it is free of rank $n$ where $n$ is the dimension of X.

Question: does the vector bundle generated by this sheaf on $X \backslash Sing$ is isomorphic to the cotangent bundle on $X \backslash Sing$ is ? If not what is the sheaf we should define on $X$ to get the cotangent bundle on $X \backslash Sing$ ?