Parabolic Hitchin maps and their generic fibers

Abstract

We set up a BNR correspondence for moduli spaces of Higgs bundles over a curve with a parabolic structure over any algebraically closed field. This leads to a concrete description of generic fibers of the associated strongly parabolic Hitchin map. We also show that the global nilpotent cone is equi-dimensional with half dimension of the total space. As a result, we prove the flatness and surjectivity of this map and the existence of very stable parabolic vector bundles.

Appendix

In this appendix, we discuss singularities of generic spectral curves, along with ramifications. Since we may work over positive characteristics, a little bit more work is needed to use the Jacobian criterion. We assume $D=x$ and if $\text {char}(k)=2$, rank $r\ge 3$.

Lemma 8

For a generic choice of $a\in \mathcal {H}_{P}$, the corresponding spectral curve $X_{a}$ is integral, totally ramified over x, and smooth elsewhere.

Proof

Since being integral is an open condition, similar as in [5, Remark 3.1], we only need to show there exists $a\in \mathcal {H}_P$, such that $X_{a}$ is integral.

Take $h_P((\mathcal {E},\theta ))=\lambda ^{r}+a_{r}=0$ with $a_{r}\in H^{0}(X, \omega ^{\otimes r}((r-\gamma _{r})\cdot x))$. The spectral curve $X_a$ is integral if $a_{r}$ is not an m-th power of some element in $\oplus _{i=1}^{r}H^{0}(X,\omega _X^{\otimes i}((i-\gamma _{i})x))$, for $m>1$. This is true for generic $a_{r}$.

Since smoothness outside x is an open condition, it is sufficient to find such a spectral curve.

When $\text {char}(k)\not \mid r$, we take $h_P((\mathcal {E},\theta ))=\lambda ^{r}+a_{r}=0$. Due to the weak Bertini theorem, we can choose $a_{r}$ with only simple roots outside x. Applying the Jacobian criterion, $X_{a}$ is what we want.

When $\text {char}(k)\mid r$, we take the following equation:

$$\begin{aligned} h_P((\mathcal {E},\theta ))=\lambda ^{r}+a_{r-1}\lambda +a_{r}=0. \end{aligned}$$

Then consider the following equations:

$$\begin{aligned}\left\{ \begin{array}{rlcl} \lambda ^{r}+a_{r-1}\lambda +&{}a_{r}&{}=&{}0\\ &{}a_{r-1}&{}=&{}0\\ a'_{r-1}\lambda +&{}a'_{r}&{}=&{}0 \end{array}\right. .\end{aligned}$$

If $\text {char}(k)=2$, we assume $r\ge 3$. Then in this case, i.e., $\text {char}(k)\mid r$, r is always greater than 2. By the weak Bertini theorem, we can choose $a_{r-1}$ with only simple roots outside of x. Take $s\in H^0(X,\omega ((1+\gamma _r-\gamma _{r-1})x))$ with zeros outside of $zero(a_{r-1})$, we can find $a_{r}=a_{r-1}\otimes s$ such that $zero(a_{r-1})\supset zero(a_{r})$, and $zero(a_{r-1})$ are simple zeros of $a_r$, then $X_{a}$ is smooth outside of x.⁴$\square$

Lemma 9

For a generic $a\in \mathcal {H}$, the corresponding spectral curve $X_{a}$ is smooth and $\pi _a:X_{a}\rightarrow X$ is unramified over x.

Proof

Smoothness is easy to show under the genericity condition. We will prove the second statement. The ramification divisor of $\pi _a$ is defined by the resultant. It is a divisor in the linear system of the line bundle $R:=\omega _X(x)^{\otimes r(r-1)}$. Considering the following morphism given by the resultant

$$\begin{aligned} \text {Res}: \mathcal {H}\rightarrow H^{0}(X,R),\quad a\mapsto \text {Res}(a),\end{aligned}$$

we have the codimension 1 sub-space

$$\begin{aligned}W:=H^{0}(X,R(-x))\subset H^{0}(X,R),\end{aligned}$$

such that $\text {Res}(a)\in W$ if and only if $\pi _a$ is ramified over x.

${\text {Res}}$ is a polynomial map so the image is a sub-variety. To prove our statement, we only need to find a particular a so that $\pi _a$ is unramified over x.

Consider the characteristic polynomial of the form $\lambda ^r+a_r$. In a neighbourhood of x, we can write it as $\lambda ^r+b_r\cdot (\frac{dt}{t})^{\otimes r}$. Here $\frac{dt}{t}$ is the trivialization of $\omega (x)$ near x. By the Jacobian criterion, $\pi _a$ is unramified over x if $b_r\in \mathcal {O}_{X,x}$ is indecomposable. Taking $b_r=t$ and extending $t\cdot (\frac{dt}{t})^{\otimes r}$ to a global section s, we find $a=\lambda ^r+s$ such that $\pi _a$ is unramified at x. $\square$