Stable vector bundle and Hitchin map

Question

Let $E$ be a stable vector bundle over a curve $X$. $K_X$ the canonical bundle of $X$. $W$ the base of the Hitchin map.

Is the Hitchin map $H:H^0(E\otimes E^*\otimes K_X)\rightarrow W$ surjective? If not, is there any conterexample?

Answer 1

There is one important case where $H$ is surjective, namely when $E$ is very stable. This means, by definition, that any nilpotent homomorphism $E\rightarrow E\otimes K$ is zero; or, equivalently, that $H^{-1}(0)=\{0\} $. It follows on one hand that $H$ is generically finite, hence dominant for dimension reasons, and on the other hand that $H$ descends to $\bar{H}:\mathbb{P}(H^0(\mathcal{E}nd(E)\otimes K))\rightarrow \mathbb{P}_w(W)$, where $\mathbb{P}_w(W)$ is the quotient of $W\smallsetminus\{0\} $ by $\mathbb{C}^*$ acting on $W=\bigoplus H^0(K^{i})$ by $\ t\cdot (\omega _i)=(t^{i}\omega _i)$. Then $\bar{H}$ is proper, hence surjective, and therefore $H$ is surjective.