Let $X$ be a smooth complex projective variety. Let $M\_{Higgs}(X, P)$ be the coarse moduli which universally corepresents the functor:


$$M^{\#}\_{Higgs}(X, P): Sch/\mathbb{C}\longrightarrow \mathcal{Set}$$
which assigns to any scheme $S$ the set of isomorphism classes of semi-stable Higgs bundles $(E, \theta)$ over $X\times S$ with Hilbert poly $P$. Here $P=nP\_0$, with $P\_0$ being the Hilbert polynomial of $\mathcal{O}\_X$ with respect to a fixed polarization $\mathcal{O}\_X(1)$.


Then we have the proper Hitchin map $$H\_1: M\_{Higgs}(X, P) \rightarrow \bigoplus\_{i=1}^nH^0(X, sym^i\Omega^1(X))$$


which maps $(E,\theta)$ to the characteristic polynomial of $\theta$. (Take the coefficients.)


Consider the Dolbeault Moduli space $M\_{Dol}(X, P)$, which universally corepresents $$M^{\#}\_{Dol}(X, P): Sch/\mathbb{C}\longrightarrow \mathcal{Set}$$
which assigns to any scheme $S$ the set of isomorphism classes of semi-stable Higgs bundles $(E, \theta)$ over $X\times S$ with Hilbert polynomial $P$ for which the Chern classes of $E$ vanish. 


Since $M\_{Dol}(X, P)$ is disjoint union of some of the connected components of $M\_{Higgs}(X, P)$, we should also have the following proper Hitchin map
$$H\_2: M\_{Dol}(X, P) \rightarrow \bigoplus\_{i=1}^nH^0(X, sym^i\Omega^1(X))$$


Can any one give some interesting, even small, geometric applications of the Hitchin map $H\_1$ or $H\_2$? Any answers and references are appreciated. 



Interesting geometric application of Hitchin Fibration
You may know this already, but I'm not sure if you'll get many other answers. A nice result of Brunebarbe, Klingler and Totaro [Symmetric differentials and the fundamental group, Duke 2013] is that if the fundamental group of a smooth complex projective variety $X$ has a finite dimensional representation with infinite image, then $X$ carries a nonzero holomorphic symmetric differential form, i.e. $H^0(X, sym^i\Omega\_X^1)\not=0$ for some $i>0$. As I recall, one step of their proof uses a special case, due to me, that the same conclusion holds if $\pi\_1(X)$ has a nonrigid representation. The latter assumption amounts to saying that the "Betti" moduli space 
$$M\_{Betti}(X,n)=Hom(\pi\_1(X),GL\_n(\mathbb{C}))//GL\_n(\mathbb{C})$$
is an infinite set. The proof of the special case is easy, modulo Simpson's work. Since $M\_{Betti}(X,n)$ is affine, and infinite, it must be noncompact in the classical topology. But $M\_{Betti}(X,n)^{an}$ is homeomorphic to $M\_{Dol}(X,n)^{an}$ (the moduli space of rank $n$ Higgs bundles with the usual conditions). Since the Hitchin map $H\_2$ is proper, the non compactness of $M\_{Dol}(X,n)^{an}$ forces $H\_2$ to have infinite image, so $sym^i\Omega\_X$ must have sections for some $i$ in the range $i=1,\ldots, n$.


