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Let $X$ be a smooth projective curve over $\mathbb{C}$ of genus $g\ge 3$, $M$ be a moduli space of stable vector bundles on $X$ of rank $n\ge 2$ and degree $d$, $\mathcal{M}$ be a moduli space of semistable Higgs bundles and $H:\mathcal{M} \rightarrow W=\bigoplus H^0(X,K_X^{\otimes i})$ be the Hitchin map( $K_X$ is the canonical bundle ).

I'm looking for the proof of the well-known fact,

  1. Codimension of $\mathcal{M} \setminus T^{*}M \ge2$
  2. The fibers of $H:\mathcal{M} \rightarrow W$ are equidimensional, and moreover the fibers of $H$ are Lagrangian.

It would be greatly appreciated if you could tell me some good reference. Many thanks.

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