Parabolic Higgs bundles on the projecitive line, quiver varieties and Deligne-Simpson problem

Parabolic bundles over curve was introduced by Metha and Seshadri in [16] as objects corresponding to unitary representations of fundamental group of punctured Riemann surface. The moduli theory of parabolic bundles is specially useful in the finite dimensional proof(in the sense of Beauville in [2]) of Verlinde formula. Verlinde formula can be seen as the dimension of nonabelian theta functions on curve. In [17], [23], [24], [25] the authors reduces the computation of Verlinde formula to computation of nonabelian parabolic theta functions on the projective line $\mathbb{P}^{1}$, using the method of degeneration of moduli spaces.

Grothendieck proved that any vector bundle on $\mathbb{P}^{1}$ is a direct sum of line bundles. So the classification of vector bundles on $\mathbb{P}^{1}$ is clear. However, if we consider parabolic bundles(parabolic Higgs bundles) on $\mathbb{P}^{1}$, things would be not so easy and become interesting. In this paper, we use $\mathbf{M}_{P}$ to denote the moduli spaces of rank $r$, degree $0$ semistable parabolic bundles on $\mathbb{P}^{1}$ and use $\mathbf{Higgs}^{\circ}_{P}$ to denote the moduli space of homologically trivial semistable parabolic Higgs bundles. Despite the use in study of Verlinde formula, the moduli space of parabolic bundles(parabolic Higgs bundles) on $\mathbb{P}^{1}$ can be related with certain quiver variety.

A quiver is a finite oriented graph. One can define representations of quiver and the moduli spaces of quiver representations are called quiver varieties. Quiver varieties is well studied and useful in representation theory. The relation between quiver varieties and moduli spaces of parabolic Higgs bundles was firstly stated by Godinho and Mandini in [9]. They establish an isomorphism between the moduli space of rank two homologically trivial parabolic Higgs bundles on $\mathbb{P}^{1}$ and quiver variety of certain star-shaped quiver. Later in [7], Fisher and Rayan give an similar isomorphism in rank $r$ case, but all parabolic structures are given by a one dimensional subspace and they did not fix weights of the parabolic bundles.

One main result in our paper is that we establish an isomorphism between the moduli space of parabolic bundles on $\mathbb{P}^{1}$ and quiver variety of certain shar-shaped quiver, in any rank and any parabolic structure case. For a star-shaped quiver $Q$, we use $\mathpzc{R}_{\chi}(\mathbf{v})$ and $\mathfrak{M}_{\chi}(\mathbf{v})$ to denote quiver varieties associated to $Q$(please refer to Section 4 for details). Then we have

As for the parabolic bundle case, the relation between moduli stack of parabolic bundles and moduli stack of quiver representations has been mentioned by Soibelman in [21]. He study the very good property of these two moduli stacks.

The third topic in our paper is (additive) Deligne-Simpson problem. Deligne-Simpson problem can be formulated as follows: Given $n$ conjugacy classes $\mathfrak{c}_{i}\subset\mathfrak{gl}_{r}$, can we find $A_{i}\in\mathfrak{c}_{i}$ so that $\sum_{i=1}^{n}A_{i}=0$? This problem is related to the existence of certain Fuchsian system on the $\mathbb{P}^{1}$. Deligne-Simpson problem was studied by Simpson, Kostov, Crawley-Boevey and so on in [20], [13], [14], [15], [6] and [21]. Especially, in [6] Crawley-Boevey relates solutions of Deligne-Simpson with certain star-quiver representations and gives a complete criterion to the existence of solution of Deligne-Simpson problem.

Inspired by Crawley-Boevey’s work and the isomorphism in Theorem 1.1, we relate the solutions of Deligne-Simpson problem to parabolic Higgs bundles on $\mathbb{P}^{1}$. In terms of parabolic Higgs bundle, one can study its characteristic polynomial and associated spectral curve(see subsection 2.3). In this way, we can solve the nilpotent case of Deligne-Simpson problem in a geometrical way.

To be precise, if each conjugacy class $\mathfrak{c}_{i}$ is given by the conjugacy class of a nilpotent matrix $N_{i}$ with rank $\gamma_{i}$(we call it the nilpotent case of Deligne-Simpson problem in this case), then we have

The isomorphism in Theorem 1.1 enables us translate properties of two moduli spaces interchangeably. For example, we have a parabolic Hitchin map on the moduli space of parabolic Higgs bundle, then we can define a parabolic Hitchin map on the quiver variety. As a result, we have

Here we take the definition of algebraically completely integrable system from [10]. In the proof of Theorem 1.4, we use properties from both $\mathbf{Higgs}^{\circ}_{P}$ and $\mathfrak{M}_{\chi}(\mathbf{v})$.

This paper is organized as follows:

In Section 2, we recall the definition and some properites of parabolic bundles, homologically trivial parabolic Higgs bundles. Then under the condition of choice of weights(condition (2.1)), we construct these moduli spaces explicitly. We also study the parabolic Hithchin map in this case, recall some results in [22].

In Section 3, we establish the connection between homologically trivial parabolic Higgs bundles and solutions of nilpotent case of Deligne-Simpson problem, then use the parabolic Hitchin map and parabolic BNR correspondence(Theorem 2.13) to solve the nilpotent case of Deligne-Simpson problem.

In Section 4, we firstly recall the definition of quiver varieties and then in the case of star-shaped quiver, we construct the quiver varieties explicitly. Compare with results in Section 2, we can prove Theorem 1.1. Next we recall the definition of Poisson varieties, algebraically completely integrable system and then prove Theorem 1.4.

For given $n$ conjugacy classes $\mathfrak{c}_{i}\subset\mathfrak{gl}_{r}$, can we find $A_{i}\in\mathfrak{c}_{i}$ so that $\sum_{i}A_{i}=0$? This problem is called the (additive version of) Deligne-Simpson problem. If the matrices $\{A_{i}\}$ have no common proper invariant subspaces, then we say that the solution $\{A_{i}\}$ is irreducible. One of the source of Deligne-Simpson is the linear system of differential equations defined on the Riemann’s sphere:

where the $n\times n$ matrix $A(t)$ is meromorphic on $\mathbb{CP}^{1}$, with poles at $x_{1},\cdots,x_{n}$. We say the system is Fuchsian if its poles are logarithmic. A Fuchisian system admits the following presentation

where $A_{i}\in\mathfrak{gl}_{r}$ and $\sum_{i}A_{i}=0$. So the existence of a Fuchsian on $\mathbb{C}\mathbb{P}^{1}$ is equivalent to the existence of a solution of Deligne-Simpson problem. Please refer to [15] for details.

The (additive) Deligne-Simpson problem was studied by Kostov in [13], [14] and he gives criterion for the existence of the solution for some special case, including nilpotent case. Later in [6], Crawley-Boevey gives a criterion for each case via quiver representations. In [21], Soibelman gives a sufficient condition for the moduli stack of parabolic vector bundles over $\mathbb{P}^{1}$ to be very good and use this property to study the space of solutions to Deligne-Simpson problem.

Here we use a more geometrical way to study Deligne-Simpson problem in nilpotent case, using parabolic Higgs bundles over $\mathbb{P}^{1}$ and the parabolic Hitchin map.

Now we assume that the conjugacy class $\mathfrak{c}_{i}$ given by the conjugacy class of a nilpotent matrix $N_{i}$. We assume that $\text{rk}(N_{i}^{j})=\gamma_{i}^{j}$, then $\{\gamma_{i}^{j}\}$ determines the conjugacy class $\mathfrak{c}_{i}$. Notice that we have $\gamma_{i}^{j}-\gamma_{i}^{j+1}\geq\gamma_{i}^{k}-\gamma_{i}^{k+1}$ for any $j\leq k$. We consider rank $r$, degree $0$ homologically trivial parabolic Higgs bundles with type

over $\mathbb{P}^{1}$, where the weights satisfy condition (2.1) and $\gamma_{j}(x_{i})=\sum_{k=j+1}^{\sigma_{x_{i}}}n_{k}(x_{i})=\gamma_{i}^{j}$. The moduli space is denoted by $\mathbf{Higgs}^{\circ}_{P}$.

For any closed point in $\mathbf{Higgs}^{\circ}_{P}$, assume that one of the corresponding homologiacally trivial parabolic Higgs bundle is $(E,\phi)$. We choose a basis $dz/(z-x_{i})$ for $\omega_{\mathbb{P}^{1}}(D_{I})|_{x_{i}}$ and fix an isomorphism $\text{H}^{0}(\mathbb{P}^{1},E)\cong k^{\oplus r}$. Then by Remark 2.9, $(E,\phi)$ is equivalent to $n$ linear transformations $A_{i}:k^{\oplus r}\rightarrow k^{\oplus r}$, so that $A_{i}$ preserves the filtration on $k^{\oplus r}$ induced by the parabolic structure of $E$ strongly and $\sum_{i}A_{i}=0$. However, at first sight, we can not control the conjugacy class of $A_{i}$, for example, we do not know whether $\text{rk}A_{i}=\gamma_{i}^{1}$.

So we consider the parabolic Hitchin map $h_{P}:\mathbf{Higgs}_{P}\rightarrow\mathbf{H}_{P}$. Recall that

The image of $(E,\phi)$ under $h_{P}$ is given by sections $\alpha_{j}=\text{Tr}(\wedge^{j}\phi)\in\text{H}^{0}(\mathbb{P}^{1},(\omega_{\mathbb{P}^{1}}(D_{I}))^{\otimes j})$. Now the zero orders of $\{\alpha_{j}\}$ at $x_{i}$ determine the conjugacy class of $A_{i}$ and $\text{rk}(A_{i}^{j})=\gamma_{i}^{j}$ if and only if the zero order of $\alpha_{j}$ at $x_{i}$ is exactly $\varepsilon_{j}(x_{i})$.

To find such sections $\{\alpha_{j}\}$ in $\mathbf{H}_{P}$, we need conditions

for every $j$. By the definition of $\varepsilon_{j}(x_{i})$, we find out that these conditions is equivalent to

Recall that $\varepsilon_{r}(x_{i})=\text{max}\{n_{j}(x_{i})\}$ then this condition is equivalent to

Now we assume that condition (3.1) holds, then

By the argument in the Appendix of [22], in this case, there exists $\alpha=\{\alpha_{j}\}\in\mathbf{H}_{P}$, so that the spectral curve $C_{\alpha}$ is integral. Notice that the condition that $C_{\alpha}$ is integral and the condition $\alpha_{j}$ has zero order $\varepsilon_{j}(x_{i})$ at $x_{i}$ are both open conditions. So if we assume that condition (3.1) holds, we can choose $\alpha\in\mathbf{H}_{P}$, so that $\alpha$ satisfies conditions in Theorem 2.13, and the $\alpha_{j}$ has zero order $\varepsilon_{j}(x_{i})$ at $x_{i}$. We choose $(E,\phi)\in(h_{P})^{-1}(\alpha)$, if $E$ is homologically trivial, we have a solution for the Delingne-Simpson problem has a solution. Moreover, this solution is actually irreducible, otherwise we would have a proper parabolic sub-Higgs bundle of $(E,\phi)$, which will give a factor of the characteristic polynomial of $(E,\phi)$ and makes $C_{\alpha}$ not integral.

If we use $\tilde{C}_{\alpha}$ to denote the normalization of $C_{\alpha}$, and $\tilde{\pi}_{\alpha}:\tilde{C}_{\alpha}\rightarrow\mathbb{P}^{1}$ to be the projection. By Theorem 2.13, we have

Thus the homologically trivial parabolic Higgs bundles forms a nonempty affine open subset scheme of $(h_{P})^{-1}(\alpha)$. In summary, we have

In this section we work over the field $\mathbb{C}$. A quiver is a finite oriented graph. Let $Q=(\text{I},\text{E})$ be a quiver where I is the vertex set, E is the set of oriented edges. Given a dimension vector $\mathbf{v}=(v_{i})\in\mathbb{Z}_{\geq 0}^{\text{I}}$, a representation of $Q$ with dimension $\mathbf{v}$ is a collection of vector spaces $\{V_{i}\}_{i\in\text{I}}$ with $\text{dim}V_{i}=v_{i}$, and a collection of linear maps $\{\phi_{ij}:V_{i}\rightarrow V_{j}\}_{(i\rightarrow j)\in\text{E}}$. All such representations can be gathered in a linear space:

with an action of a group $GL(\mathbf{v}):=\prod_{i\in\text{I}}GL(v_{i})$. Notice that the diagonal $\Delta:\mathbb{G}_{m}\rightarrow GL(\mathbf{v})$ acts trivially. Two representations in $\text{Rep}(Q,\mathbf{v})$ are isomorphic if and only if they are in same orbit.

One can construct the moduli space of representations of $Q$ by taking quotient space $\text{Rep}(Q,\mathbf{v})/\text{GL}(\mathbf{v})$. But this space is usually not Hausdorff. Instead one can do the GIT quotient $\mathpzc{R}_{0}(\mathbf{v}):=\mathbf{R}//GL(\mathbf{v})=\text{Spec}\mathbb{C}[\mathbf{R}]^{GL(\mathbf{v})}$, where $\mathbb{C}[\mathbf{R}]$ is the coordinate ring of $\mathbf{R}$. We have a description for this ring:

As a corollary, if $Q$ has no oriented cycles, then $\mathpzc{R}_{0}(\mathbf{v})$ is a point. To separate more orbits, King introduced the following definition([11]):