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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.

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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 $i=1,\ldots, n$.

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  • $\begingroup$ Dear Prof. Arapura, I only saw your result in your paper. But not the one in [Symmetric differentials and the fundamental group, Duke 2013]. Thank you for you answer. $\endgroup$ – Feng Hao Jul 15 '17 at 15:09
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If $X$ is a smooth projective curve with genus $g \geq 2$ and $M_{Higgs}(X,P)$ is smooth, the map: $$ H_1 : M_{Higgs}(X,P) \longrightarrow \bigoplus_{i} H^0(X, Sym^i \Omega_X)$$ gives $M_{Higgs}(X,P)$ a structure of complete algebraic integrable system. I think this result is due to Beauville. See : https://projecteuclid.org/download/pdf_1/euclid.acta/1485890603. As far as I know, it is quite difficult to provide examples of complete algebraic integrable systems.

EDIT : as Beauville comments below, the algebraic complete integrability of this moduli space was proved by Hitchin himself.

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    $\begingroup$ Actually this is due to Hitchin himself, and it was the main reason for him to considered this fibration. $\endgroup$ – abx Jul 15 '17 at 14:37

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