Let $X$ be a smooth projective curve over $\mathbb{C}$, $\mathcal{M}$ be the moduli space of semistable higgs bundles, and $h: \mathcal{M}\rightarrow W=\bigoplus H^0(X,K_X^{\otimes{i}})$ be the Hitchin map.
I want to show that the natural map $H^0(W,\mathcal{O}_W)\rightarrow H^0(\mathcal{M},\mathcal{O_M})$ is a bijection.

my try :
Let $Y=\{y\in W \mid h^{-1}(y)$ is not irreducible$\}$. Then, $\operatorname{codim}(W\setminus Y)\geq2$ and $h^{-1}(W\setminus Y)$ is irreducible. $U=W\setminus Y,V=h^{-1}(W\setminus Y)$.
Since $h$ is proper, $(h\mid_{V})_{*}(\mathcal{O}_{\mathcal{M}}\mid_V)$ is a coherent $\mathcal{O}_V$ module. So, if $U$ is affine, any $\psi\in H^0(V,\mathcal{O}_V)=H^0(U,(h_*\mathcal{O_M})\mid_U)$ is integral over $H^0(U, \mathcal{O}_U)$. Considering $U$ is a normal scheme, I get $\psi\in H^0(U,\mathcal{O}_U)$.
But in general case why $\psi \in H^0(U,\mathcal{O}_U)$ ? and $\psi\in K(W)$ (the function field of $W$) ?

Any advice and comments will be appriciated.Thanks in advance.

Let $X$ be a smooth projective curve over $\mathbb{C}$, $\mathcal{M}$ be the moduli space of semistable higgs bundles, and $h: \mathcal{M}\rightarrow W=\bigoplus H^0(X,K_X^{\otimes{i}})$ be the Hitchin map.
I want to show that the natural map $H^0(W,\mathcal{O}_W)\rightarrow H^0(\mathcal{M},\mathcal{O_M})$ is a bijection.

my try :
Let $Y=\{y\in W \mid h^{-1}(y)$ is not irreducible$\}$. Then, $\operatorname{codim}(W\setminus Y)\geq2$ and $h^{-1}(W\setminus Y)$ is irreducible. $U=W\setminus Y,V=h^{-1}(W\setminus Y)$.
Since $h$ is proper, $(h\mid_{V})_{*}(\mathcal{O}_{\mathcal{M}}\mid_V)$ is a coherent $\mathcal{O}_U$ module. So, if $U$ is affine, any $\psi\in H^0(V,\mathcal{O}_V)=H^0(U,(h_*\mathcal{O_M})\mid_U)$ is integral over $H^0(U, \mathcal{O}_U)$. Considering $U$ is a normal scheme, I get $\psi\in H^0(U,\mathcal{O}_U)$.
But in general case why $\psi \in H^0(U,\mathcal{O}_U)$ ? and $\psi\in K(W)$ (the function field of $W$) ?

Any advice and comments will be appriciated.Thanks in advance.