Compact connected components in relative character varieties of punctured spheres

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.


Introduction
Let Σ g,s be an oriented surface of genus g with s punctures, let Γ g,s denote its fundamental group and let G be a non compact semi-simple Lie group. The character variety X(Σ g,s , G) is roughly the space of morphisms from Γ g,s to G up to conjugation by G. Character varieties of closed surfaces (i.e. without puncture) carry a natural symplectic structure (constructed by Atiyah-Bott [AB83] and Goldman [Gol84]), which is preserved by the action of the mapping class group of the surface. The study of these character varieties, from a topological, algebraic, geometric or dynamical point of view, has attracted a lot of interest in the last decades.
The fundamental group of a surface with punctures is a free group, so its character varieties carry an action of the outer automorphism group of this free group. These character varieties, however, do not reflect all the geometric properties of the surface. Indeed, two punctured surfaces with the same Euler characteristic (such as Σ 0,3 and Σ 1,1 ) have the same fundamental group, hence the same character varieties.
One can nevertheless endow these character varieties with some extra structure that truly depends on the topology of Σ g,s . Indeed, Goldman's construction has been extended to define a Poisson structure on X(Σ g,s , G) ( [GHJW97,Law09]), which is preserved by the action of the mapping class group of Σ g,s . On the smooth locus, this Poisson structure defines a foliation by symplectic leaves. These leaves are the so called relative character varieties, and consist of representations of Γ g,s into G for which the conjugacy classes of the images of the loops going around the punctures are fixed. Given an element h ∈ G s , we will denote by X h (Σ g,s , G) the relative character variety whose representations map the loop around the i th puncture to the conjugacy class of h i (here i ∈ {1, ..., s}). In many aspects, these relative character varieties appear as the analogs for punctured surfaces of character varieties.
Deroin and the first author studied in [DT19] the relative character varieties of the punctured sphere Σ 0,s into PSL(2, R) PSU(1, 1). For all s ≥ 3, they exhibit representations with surprising properties that they call supra-maximal representations, and prove that these representations form compact connected components of certain relative character varieties. Prior to their work, such compact components were only known to exist for s = 4 [BG99].
The existence of such compact components may come out as a surprise, given the great flexibility of surface group representations. Indeed, Deroin-Tholozan's supra-maximal representations have several counter-intuitive properties: • Though these representations are typically Zariski dense, the image of any simple closed curve has a fixed point in the hyperbolic plane H 2 .
• More generally, the eigenvalues of the image of a closed curve c are bounded by a constant that depends only on the number of self-intersections of c.
• For any such representation ρ and any identification of Σ 0,s with a punctured Riemann sphere, there exists a ρ-equivariant holomorphic map from Σ 0,s to H 2 .
The goal of the present paper is to construct such components for a larger class of Lie groups.
To this end, we will exploit a very powerful tool for understanding the topology of relative character varieties: the non-Abelian Hodge correspondence. When we fix a Riemann surface structure on the closed surface Σ g,0 , this correspondence associates to every irreducible linear representation of Γ g,0 a stable Higgs bundle over the Riemann surface. One can use tools from complex geometry to understand topological properties of the moduli space of Higgs bundles, and infer results on character varieties (see [Hit92]). The non-Abelian Hodge correspondence has been extended by Simpson and others [Sim90,BGPMiR20] to a correspondence between representations of punctured surfaces and parabolic Higgs bundles.
Most of the theorems (if not all) about the topology of (relative) character varieties of surfaces can be recovered and improved using non-Abelian Hodge theory. As a matter of fact, shortly after Deroin-Tholozan's results, Mondello used non-Abelian Hodge theory to give a complete description of relative character varieties of punctured surfaces into PSL(2, R) [Mon16], showing in particular that Deroin-Tholozan's compact components could be recovered via Higgs bundle techniques.
Here, we use non-Abelian Hodge theory to construct compact components in relative character varieties of punctured spheres into Lie groups of Hermitian type. We will prove the following theorem: Remark 1.1. Most of the paper focuses on representations into SU(p, q). We discuss Sp(2n, R) and SO * (2n) in Section 6.2. Note that the only family of classical Hermitian Lie groups which our theorem does not cover is SO(2, n). See Section 6.2.3.
The starting point of the proof is to convert this theorem into a result about moduli spaces of parabolic SU(p, q)-Higgs bundles. The general theory of Higgs bundles corresponding to representations into real Lie groups started with Hitchin's paper [Hit87a], and was fully sorted out by Garcia-Prada and Mundet i Riera [GPMiR04], while the parabolic version of the theory was developped by Biquard, Garcia-Prada and Mundet i Riera [BGPMiR20]. Parabolic SU(p, q)-Higgs bundles have been specifically studied in [GPLM09], where the authors chose to put aside the punctured sphere case.
Let us identify Σ 0,s with the punctured Riemann sphere CP 1 \{x 1 , . . . , x s }. Very briefly, parabolic Higgs bundles on the punctured Riemann sphere corresponding to representations into SU(p, q) are the data of (U • , V • , γ, δ) where U • and V • are holomorphic vector bundles over CP 1 of respective rank p and q with a parabolic structure at each puncture (which involves a choice of weights between 0 and 1) and γ and δ are meromorphic sections of Hom(U , V ) ⊗ K and Hom(V , U ) ⊗ K respectively, with at most simple poles at the punctures and satisfying certain constraints depending on the weights (here K is the canonical bundle on CP 1 ).
One associates to representations into Hermitian Lie groups a topological invariant called the Toledo invariant (see Paragraph 2.1.1), which is of main importance in the topology of character varieties. Under some open condition on the weights, we show that, if the Toledo invariant of the corresponding representation is small, then δ must vanish for stable Higgs bundles. Though this condition is very restrictive, we prove that it can be satisfied for s > 2 + p q + q p , giving rise to compact components in relative character varieties. The underlying holomorphic bundles U and V in these components are both direct sums of a fixed line bundle. This fact permits a parametrization of these components by certain quiver varieties called Kronecker varieties. For more generic weights, the parabolic structure of the Higgs bundles give rise to a decoration of those quivers that we call feathered Kronecker varieties and construct using Geometric Invariant Theory.
This construction only works for s large enough. However, restricting these representations to subsurfaces, one can prove the existence of compact components for all s (see Section 6.3).
Finally, let us point out that the representations we obtain have the same kind of properties as supramaximal representations into PSL(2, R).

Theorem 2.
Let G be one of the classical Hermitian Lie groups SU(p, q), Sp(2n, R) and SO * (2n). There exists an open set in Hom(Γ 0,s , G) consisting of representations ρ satisfying the following properties: • For every homotopy class of simple closed curve c in Σ 0,s , the complex eigenvalues of ρ(c) have modulus 1.
• More generally, for every k ≥ 0, there is a constant C(k) such that, if c is the homotopy class of a closed curve with at most k self interserctions, then the eigenvalues of ρ(c) have modulus less than C(k).
• For every identification of Σ 0,s with a punctured Riemann sphere, there is a ρ-equivariant holomorphic map from Σ 0,s to the symmetric space of G.

Structure of the article
Section 2 introduces some background notations and recalls briefly the non-Abelian Hodge correspondence between relative character varieties and parabolic Higgs bundles. In Section 3, we recall some classical facts about Geometric Invariant Theory and its application to the construction of certain quiver varieties. Section 4 contains the core of our work. We give a condition for a component of the moduli space of parabolic SU(p, q)-Higgs bundles to be compact and prove that, for some specific choices of weights, this component is indeed non empty and isomorphic to a Kronecker variety. In Section 5, we describe some properties of the representations in these components, proving Theorems 1 and 2 for SU(p, q), under the condition s > p q + q p . Finally, in Section 6, we discuss some extensions of our construction (to other Hermitian Lie groups, and to spheres with fewer punctures), concluding the proof of Theorems 1 and 2.

Toledo invariant.
The Lie group SU(p, q) is Hermitian, meaning that its symmetric space carries a SU(p, q)-invariant Kähler structure. The Kähler form induces a continuous cohomology class ω of degree 2 on SU(p, q) (seen as a topological group). Given a morphism ρ : Γ g,0 → SU(p, q), one can pull back ω by ρ to obtain an element of H 2 (Γ g,0 , R) H 2 (Σ g,0 , R) R. For a surface with punctures, this construction does not work directly since H 2 (Σ g,s , R) = {0}, but one can fix this issue using bounded cohomology [BIW10]. This eventually defines a function Tol : X(Γ g,s , G) → R called the Toledo invariant. The Toledo invariant is continuous and additive in the following sense: Proposition 2.4 (see [BIW10]). If b is a simple closed curve on Σ g,s separating Σ g,s into two surfaces Σ and Σ , then for every representation ρ : Γ g,s → SU(p, q), Tol(ρ) = Tol(ρ |π 1 (Σ ) ) + Tol(ρ |π 1 (Σ )| ) .
For closed surfaces, the Toledo invariant takes values into rZ for some rational number r (depending on the normalization of the cohomology class ω). For punctured surfaces, the image of Tol is not discrete, but the congruence of Tol(ρ) modulo rZ only depends on C(h 1 ), . . . , C(h s ) [BIW10]. In particular, the Toledo invariant is constant on connected components of relative character varieties. We will denote by X τ h (Σ g,s , SU(p, q)) the set of representations ρ ∈ X h (Σ g,s , SU(p, q)) such that Tol(ρ) = τ. We will sometimes call this space a relative component, though it may not be connected in general.

Parabolic Higgs bundles
The non-Abelian Hodge correspondence, developed by Hitchin, Simpson and many others [Hit87a,Sim88], gives a homeomorphism between the character variety of a closed Riemann surface and the moduli space of Higgs bundles over this surface. This correspondence was refined by Simpson [Sim90] and Biquard-García-Prada-Mundet i Riera [BGPMiR20] to a correspondence between relative character varieties of punctured Riemann surfaces and moduli spaces of parabolic Higgs bundles. In this subsection, we introduce parabolic Higgs bundles and state the non-Abelian Hodge correspondence in the case that interests us.

Parabolic vector bundles.
Let X be a closed Riemann surface and x a point on X. By a vector bundle over X, we always mean a holomorphic vector bundle. We denote vector bundles with curly letters and use the same capital letter to denote the corresponding sheaf of holomorphic sections.
Definition 2.5. Given a vector bundle E over X, a parabolic structure at x ∈ X is a filtration {E α } α∈R of the germs of meromorphic sections of E at x, such that Remark 2.6. Given a vector bundle E over X, one can always equip E with the trivial parabolic structure where E 0 is the space of germs of holomorphic sections of E at x and α is the integer part of α.
Given two vector bundles E and F with a parabolic structure at x ∈ X, their direct sum and tensor product are respectively defined to be the bundles E ⊕ F and E ⊗ F with the parabolic structure {(E ⊕ F) α } α∈R and If F ⊂ E is a subbundle and E has a parabolic structure at x, then F inherits a parabolic structure {F α } α∈R where F α are the germs of meromorphic section of F at x contained in E α .
Denote by E x the fiber of E over x. The evaluation map ev : has kernel E 1 = E(−x). In particular, ev gives an isomorphism E 0 /E 1 E x and identifies E α i ⊂ E 0 with a linear subspace E x,i ⊂ E x . In other words, a parabolic structure at x is equivalent to a weighted flag Setting k i = dim E x,i+1 /E x,i , the parabolic type of E at x is the n-tuple (α 1 , . . . ,α n ) ∈ [0, 1) n (where n is the rank of E) such that α 1 =α 1 = · · · =α k 1 , α 2 =α k 1 +1 = · · · =α k 1 +k 2 and so on. Definition 2.7. A morphism f : E → F of vector bundles with parabolic structure of type α and β at Given a punctured Riemann surface X (i.e. a Riemann surface with cusps) with closure X = X ∪{x 1 , . . . , x s }, a parabolic bundle of rank n over X will be a rank n vector bundle E over X together with a parabolic structure at each x i ∈ {x 1 , . . . , x s }. We denote by E • such a parabolic bundle. The parabolic type of E • is α = (α 1 , . . . , α s ) where α i = (α i 1 , . . . , α i n ) ∈ [0, 1) n is the parabolic type of E • at x i . Finally, we denote by Hom(E • , F • ) the bundle of strongly parabolic morphisms from E • to F • , and write End( The dual of a parabolic bundle E • is the parabolic bundle E ∨ where O is the trivial bundle equipped with the trivial parabolic structure. Note that the underlying bundle of E ∨ • is E ∨ (−D) where D is the effective divisor x 1 + · · · + x s .
Definition 2.8. The parabolic degree of a parabolic bundle E • of type α over X is the number Example 2.9 (Parabolic line bundles over CP 1 ). Recall that CP 1 carries a unique line bundle of degree l for all l ∈ Z, denoted O(l). Let X = CP 1 \ {x 1 , . . . , x s } be a punctured Riemann sphere. A parabolic line bundle over X is thus given by a parabolic structure over some O(l) at each x i . Since a flag in O(l) x i is necessarily trivial, such a parabolic structure is simply given by its type α i ∈ [0, 1). We will denote this parabolic line 1} be the integer part of α i + β i and set |ε| = s i=1 ε i . The tensor product L • ⊗ M • is thus the line bundle of degree l + m + |ε| whose parabolic weight at

Parabolic SU(p, q)-Higgs bundles.
Recall that X = X \ {x 1 , . . . , x s }. We denote by D the effective divisor s i=1 x i . Let K denote the canonical bundle of X. Definition 2.10. A parabolic Higgs bundle over X is a pair (E • , Φ) where E • is a parabolic bundle and Φ is a holomorphic section of K(D) ⊗ End(E • ).
Remark 2.11. In this definition, the Higgs field Φ is a strongly parabolic morphism from E • to E • ⊗ K(D). Stricto sensu, this is the definition of a strongly parabolic Higgs bundle. In this paper, however, we limit ourselves to strongly parabolic Higgs bundles, so we omit this precision. -With respect to the splitting E • = U • ⊕ V • , the Higgs field Φ takes the form has trivial parabolic structure and is isomorphic to the trivial bundle O.
. It is polystable if it is the direct sum of stable parabolic Higgs bundles of degree 0.
The parabolic type of a parabolic SU(p, q)-Higgs bundle (U • ⊕ V • , γ ⊕ δ) is the pair (α, β) where α and β are respectively the parabolic types of U • and V • .
Note that the parabolic type of the parabolic line bundle det(U • ) ⊗ det(V • ) at x i is given by the quantity for all i ∈ {1, . . . , s}. Elements of W (s, p, q) will be called SU(p, q)-multiweights.

Moduli spaces.
Moduli spaces of semistable Higgs bundles can be constructed using Geometric Invariant Theory. This was done by Yokogawa in a fairly general setting which includes parabolic Higgs bundles [Yok95,Yok93]. In our setting, we have the following There is a continuous map ϕ : Remark 2.15. Using the equality deg(U We now consider the case where α i < β i for all i ∈ {1, . . . , s}. This implies that α i = 1 − β i ∈ (0, 1 2 ), and deg(U ) + deg(V ) = −s so d = deg(U ) − deg(V ) = 2l + s. One sees that if the parity of d is different from the one of s, then M(α, β, d) = ∅. Consider now the Higgs field. Since α i < β i , we have ) .
This component corresponds to a lift to SU(1, 1) of a supra-maximal component in PSL(2, R) PU(1, 1) studied by Deroin and the first author in [DT19]. The goal of the present paper is to extend this phenomenon to SU(p, q).
Definition 2.17. Given a rank n parabolic Higgs bundle (E • , Φ) over X, one can associate a vector This defines the Hitchin fibration The following was proven by Hitchin [Hit87b] for closed Riemann surfaces and extended by Yokogawa [Yok93] to the parabolic case: Theorem 2.18. The Hitchin fibration is proper.
The definition of a SU(p, q)-multiweight implies that each h k belongs to SU(p, q).
Theorem 2.19 (Non-Abelian Hodge correspondence). There exist a homeomorphism NAH from the moduli space M(α, β) to the relative character variety X h (Σ 0,s , SU(p, q)). Through this correspondence, stable Higgs bundles correspond to irreducible representations.
This correspondence is highly transcendental in the sense that it relies on the resolution of a geometric PDE on Hermitian metrics on the bundle. It is thus quite difficult to understand geometric properties of a representation associated to a given Higgs bundle. There are nevertheless a few attributes that transit well through the correspondence, among which is the Toledo invariant.

Then (with the proper normalization of the Toledo invariant) one has
Remark 2.21. This Proposition is proved in [GPLM09], where a different normalization of the Toledo invariant is used. The one we use here is not standard, but it will simplify notations.
Recall that α (resp. β ) denotes the sum of the weights α i j (resp. β i j ) for all j and i. By definition, we have It follows that the non Abelian Hodge correspondence restricts to a homeomorphism

Parabolic Higgs bundles over the Riemann sphere.
In this paper, we consider the special case of X = CP 1 \ {x 1 , . . . , x s }. Let us briefly recall the classification of holomorphic vector bundles over CP 1 .
Recall that we denote by O(d) the unique line bundle of degree d over CP 1 .

Theorem 2.22 (Birkhoff-Grothendieck). Any rank n vector bundle
The degree vector defines a map

Geometric Invariant Theory and quiver varieties
In this section, we recall a few facts about Mumford's Geometric Invariant Theory [Mum65] and apply it to the construction of certain quiver varieties called Kronecker varieties, as well as a mix between these and spaces of flag configurations, that we call feathered Kronecker varieties. These varieties will arise in our construction of compact components in the next sections.

GIT quotients of complex varieties
Let Y be a smooth quasi-projective variety with an algebraic action of a complex reductive algebraic group G. The goal of Geometric Invariant Theory (often abreviated in GIT) is to define a good notion of quotient of Y under the action of G in the category of complex algebraic varieties. We briefly recall here Mumford's construction, which relies on the choice of an ample G-linearized line bundle. We restrict for simplicity to the case where Y is projective over an affine base. (This will be sufficient for our applications.) We do not want to assume the action to be faithful. Let us denote by Z the kernel of the action, that is, the subgroup of G that acts trivially on Y .
Definition 3.1. A G-linearized line bundle L is a line bundle over Y together with an algebraic G-action on the total space of L that lifts the action on Y , and such that Z acts trivially on L.
Let L be a G-linearized ample line bundle and k an integer. The group G acts on the space of sections of with the structure of a graded algebra.
The main discovery of Mumford is that this algebraic construction has a geometric interpretation: the (closed) points of Y L //G "almost" parametrize the G-orbits in Y .
• polystable if it is semistable and its G-orbit is closed in the subset of semistable points, • stable if it is polystable and its stabilizer is finite modulo Z, is clearly open in Y for the Zariski topology (in particular, it is dense when non-empty). Let Y ss (L)//G denote the quotient of Y ss (L) by the equivalence relation where x ∼ y if the closures of the orbits of x and y in Y ss (L) intersect.
The semistability of a point x can be checked by looking at the action of one parameter subgroups of G, thanks to the Hilbert-Mumford criterion. This criterion is often stated for projective varieties, but it has been extended to other settings. We present here an ad hoc version adapted from [GHH15, Corollary 1.1].
Let λ : C * → G be a 1-parameter subgroup and x a point in Y . Let x 0 denote the limit lim t→0 λ(t) · x, when it exists. Then x 0 is fixed by λ(C * ) and λ(C * ) thus acts linearly on the fiber L x 0 of L at x 0 .
for all v ∈ L x 0 and all t ∈ C * . If the limit lim t→0 λ(t) · x does not exist, we set µ L (λ, x) = +∞.
This stability criterion will prove particularly handy when we look at GIT quotients of products. Indeed, we have the following proposition:

1≤i≤n be a family of quasi-projective varieties with an algebraic G-action and a
Then for every x = (x 1 , . . . , x n ) in Y and every λ : C * → G, we have

Kronecker varieties
Let p, q, r be positive integers. The actions of GL(p, C) and GL(q, C) on C p and C q induce a linear action of the group Let χ : G p,q → C * denote the character defined by where p = p gcd(p,q) and q = q gcd(p,q) . This character induces an action of G p,q on the trivial bundle This provides E(p, q, r) × C with the structure of a G p,q -linearized line bundle, that we denote by L χ .
Remark 3.8. Since every line bundle over E(p, q, r) is trivial and since every character on G p,q with kernel containing Z p,q is a power of χ, every G p,q -linearized line bundle over E(p, q, r) is a power of L χ .
We can now define a GIT quotient of the semistable locus of E(p, q, r) under the action of G p,q . This quotient is a particular case of quiver variety sometimes refered to as a Kronecker moduli space (see for instance [Wei13]). Here we call it a Kronecker variety.
Definition 3.9. The Kronecker variety R(p, q, r) is the GIT quotient Note that the only G p,q -invariant regular functions on E(p, q, r) are constant (see [Kin94]). Thus R(p, q, r) is a (reduced) projective scheme over C, that is, a projective variety.
Remark 3.10. The Kronecker varieties R(p, q, r) are a particular case of quiver varieties. More precisely, R(p, q, r) is the representation variety of the Kronecker quiver : We will now apply the Hilbert-Mumford criterion (Theorem 3.6) to characterize (semi)stable points of E(p, q, r). Let λ : C * → G p,q be a 1-parameter subgroup. Then we have λ(e t ) = (exp(tu), exp(tv)) where u and v are diagonalizable endomorphisms of C p and C q respectively, with integral eigenvalues. Let m 1 > · · · > m k and n 1 > · · · > n l denote the eigenvalues of u and v respectively, and let F i and H j denote the eigenspaces of u and v associated respectively to the eigenvalues m i and n j , so that with respect to the decompositions C p = F 1 ⊕ · · · ⊕ F k and C q = H 1 ⊕ · · · ⊕ H l . We then define decreasing filtrations of C p and C q associated to λ.
Definition 3.11. The filtrations (U n ) n∈Z and (V n ) n∈Z associated to λ are defined by for all n ∈ Z and all j ∈ {1, . . . , r}. In that case, we have where q = q gcd(p,q) and p = p gcd(p,q) .
Proof of Proposition 3.12. The following computation is a particular case of a more general result of King for quiver varieties [Kin94]. We give it here for completeness.
Writing the A j with respect to diagonalization bases for u and v, one easily verifies that λ(t) · A converges when t goes to 0 if and only if A j (U n (λ)) ⊂ V n (λ) for all j ∈ {1, . . . , r} and all n ∈ Z. Otherwise, there is a vector w in some F m a and some j ∈ {1, . . . , r} such that A j (w) has a non-zero component in H n b for some If λ(t) · A converges as t → 0, then the action of λ(t) on the fiber of L χ at the limit is simply given by multiplication by χ −1 (λ(t)) and we thus obtain Let us take as a convention that m 0 = n 0 = b > max(m 1 , n 1 ) and m k+1 = n l+1 = a < min(m k , n l ). Note that we have dim( Applying an Abel transform to (1), we get Proposition 3.12 easily implies King's characterization of semi-stable points in E(p, q, r) : It is stable if, moreover, the equality only holds when U and V are reduced to {0} or when U = C p and V = C q .
• If p q + q p ≥ r, then R(p, q, r) is empty except when r = 1 or 2 and p = q. • If p q + q p < r, then E(p, q, r) contains stable points and R(p, q, r) has positive dimension.
• If p and q are coprime, then every semistable point in E is stable.
The cases r ≤ 2 and p = q are special and are often excluded of the study of Kronecker varieties. However, they are relevant here. The following proposition deals with them.
• The Kronecker variety R(p, p, 1) is reduced to a single point. There is no stable point. A point of The proof will rely on the following linear algebra lemma: Conversely, assume that A + tB has a kernel for all t. For all n ∈ N >0 , let u n be a non-zero vector in the kernel of A + 1 n B. Up to scaling u n and extracting a subsequence, we can assume that u n converges to a non-zero vector u in the kernel of A. Set V = Span(u n ) n∈N and W = A(V ). Since u n is in the kernel of A + 1 n B, Bu n is colinear to Au n for all n, hence B(V ) ⊂ W . Finally, since V is closed (as a linear subspace in finite dimension), it contains the vector u. Thus A : V → W is not injective and dim W < dim V , contradicting semistability.
If A ∈ End(C p ) is not invertible, then the dimension of the image of A is strictly less than p, which contradicts stability. If A is invertible, then dim A(U ) = dim U for every subspace U of C p , hence A is semistable and not stable. Since any invertible matrice is similar to the identity, there is a unique semistable orbit. Hence R(1, p, p) is reduced to a point.

Flag configurations
Another classical application of GIT is the construction of configuration spaces of flags (see for instance [FMS20]).
Recall that a complete flag F in C p is a nested sequence of linear subspaces Remark 3.18. All the content of this subsection and the next one would easily extend to partial flags (i.e. decreasing sequences of subspaces). We chose to restrict to complete flags to avoid proliferation of indices. For the rest of this section, "flag" will mean "complete flag".
Let Gr i (C p ) denote the Grassmanian of i-dimensional subspaces of C p , and define Then the map ι = (ι 1 , . . . , the pullback to Gr i (C p ) of the unique line bundle of degree a on P i C p by the Plücker embedding. The following is classical.
for some (a 1 , . . . , a p−1 ) ∈ Z p−1 . This bundle is ample if and only if all the a i are positive.
The group GL(p, C) acts on each Gr i (C p ) with kernel C * Id p . There is a canonical lift of this action to the total space of O i (1), such that λId p acts by multiplication by λ −i in each fiber. This action, however, is not a GL(p, C) linearization according to Definition 3.1 since it does not have C * Id p in its kernel. One can nonetheless linearize the action on O i (p). More precisely, if we simply denote by g(v) the action of g on a point v ∈ O i (p) induced by the canonical action of GL(p, C) on O i (1), then we get a new action by setting with kernel C * Id p , thus providing O i (p) with a GL(p, C)-linearization. The computation of the Hilbert-Mumford weights in this setting is classical.
Proposition 3.20. Let λ be a 1-parameter subgroup of GL(p, C) and let U n (λ) be the filtration defined in Definition 3.11. For F in Gr i (C p ), we have Now, let s be a positive integer. We denote respectively by F s and H s the spaces F (C p ) s and F (C q ) s . The group G p,q = GL(p, C) × GL(q, C) acts on F s × H s with kernel C * Id p × C * Id q . One can endow F s × H s with many G p,q -linearized ample line bundles via its embedding into More precisely, for any family of positive integers a = (a We want to express this formula in terms of "induced weights" on the filtration associated to λ. In order to do so, let us choose (η If U (resp. V ) is a subspace of C p (resp. C q ), we denote by |η j (U ∩ F j )| (resp. |ζ j (V ∩ H j )|) the sum of the weights of the weighted flag induced on U (resp. V ) by (F j , η j ) (resp. (H j , ζ j )), i.e.
Finally, we set Proposition 3.21. For (F, H) ∈ F s × H s and λ : C * → G p,q a 1-parameter subgroup, we have Proof. Replacing µ(λ, F j i ) and µ(λ, H j i ) in (2) by the formula given in Proposition 3.20, we obtain i and applying an Abel transform, one obtains the result.

Feathered Kronecker varieties
Let us define In this section, we consider GIT quotients of E ♠ (p, q, r, s) with different choices of G p,q -linearized line bundles.
Let η = (η j i ) 1≤j≤s,1≤i≤p and ζ = (ζ j i ) 1≤j≤s,1≤i≤q be tuples of real numbers such that η j 1 < · · · < η j p and ζ j 1 < · · · < ζ j q for all j. We first define a numerical stability condition on E ♠ (p, q, r, s) depending on η and ζ. We will then prove that this stability condition is equivalent to Mumford's stability for a suitable choice of ample G p,q -linearized line bundle.
It is (η, ζ)-stable if, moreover, equality only holds for U = V = {0} or U = C p and V = C q . The feathered Kronecker variety R ♠ (p, q, r, s, η, ζ) is the largest Hausdorff quotient of the set of (η, ζ)-semistable points of E ♠ (p, q, r, s) under the action of G p,q .
The following proposition asserts that these spaces are indeed GIT quotients of E ♠ (p, q, r, s). In particular, R ♠ (p, q, r, s, η, ζ) is homeomorphic to a projective variety.
Proof. Let us first remark the following: one can find rational weights (η , ζ ) close to (η, ζ) such that the (η , ζ )-(semi)stable points are exactly the (η, ζ)-(semi)stable points. Indeed, all the (semi)stability conditions in Definition 3.22 for all points in E ♠ (p, q, r, s) are linear equations and inequalities in (η, ζ) with integral coefficients bounded by some constant. There are thus only finitely many such conditions, and one can find a rational point (η , ζ ) close to (η, ζ) which satisfies the same exact conditions.
We can therefore assume without loss of generality that the weights (η, ζ) are rational. Let k be a positive integer such that ( k p η, k q ζ) is integral. Set Let L a,b denote the G p,q -linearized line bundle over F s × H s defined in Section 3.3. Let also L χ be the G p,q -linearized line bundle over E(p, q, r) defined in Section 3.2. Finally let L k,a,b denote the G p,q -linearized line bundle on E ♠ (p, q, r, s) defined by where p E and p F ×H respectively denote the projections to the factors E(p, q, r) and F s × H s . We now write µ k,a,b for µ L k,a,b .
Let (A, F, H) be a point in E ♠ (p, q, r, s). Given a one parameter subgroup λ : C * → G p,q , putting together Propositions 3.12, 3.20 and 3.7, we obtain that µ k,a,b (A, F, H) = +∞ unless A j (U n (λ)) ⊂ V n (λ) for all n and all j, in which case If (A, F, H) is not L k,a,b -semistable, then there exists a one parameter subgroup λ such that µ k,a,b (λ, (A, F, H)) < 0. Thus, for some n ∈ Z, Define λ(e t ) = (exp(tp U ), exp(tp V )), where p U and p V are respectively projectors on U and V . Then Thus µ k,a,b (λ(A, F, H)) = ( * ) < 0, contradicting L k,a,b -stability. This proves the equivalence between (η, ζ)-semistability and L k,a,b -semistability. Weakening the inequalities in the proof gives the equivalence between (η, ζ)-stability and L k,a,b -stability.
Despite being given by an explicit formula, the (η, ζ)-stability criterion is not very enlightening in general. For future use, let us specialize to the case where η and ζ are small.

It is stable if it is semistable and equality in (3) only holds for
In particular, if p and q are coprime, then R ♠ (p, q, r, s, η, ζ) fibers over R(p, q, r) with fibers isomorphic to The first part of the proposition follows.
Assume now that p and q are coprime. Then we cannot have q dim U − p dim V = 0 for proper subspaces U and V . Hence, for (η, ζ) small enough, (A, F, H) is (η, ζ)-stable if and only if A ∈ E(p, q, r) is stable. We thus have E stable ♠ (p, q, r, s, η, ζ) = E(p, q, r) stable × F s × H s and the projection on the first factor induces a morphism from R ♠ (p, q, r, s, η, ζ) to R(p, q, r). The fact that this map is a fibration with fibers isomorphic to F s × H s follows from the fact that the stabilizer of any stable point A ∈ E(p, q, r) is the subgroup C * (Id p , Id q ), which acts trivially on E ♠ (p, q, r, s).

Compact components 4.1. A compactness criterion
In this subsection, we find a sufficient criterion for a relative component M(α, β, d) of parabolic SU(p, q)-Higgs bundles over X = CP 1 \ {x 1 , . . . , x s } to be compact.
Remark 4.2. Since we only consider strongly parabolic Higgs bundles, the assumption that α j p ≤ β j 1 would be sufficient to deduce the vanishing of δ. We wrote a strict inequality because we will use later that condition (1) is open. The condition α j p < β j 1 implies that δ does not have poles at the puncture x j , i.e. δ is a holomorphic section of Hom(V , U ⊗ K). Let N and I ⊗ K be the subsheaves of V and U ⊗ K respectively given by the kernel and the image of δ, so that δ induces an exact sequence of sheaves In particular, we have The bundles N • and V • ⊕ I • are γ ⊕ δ-invariant. By semistability, they must have non-positive parabolic degree. To write this more explicitly, let us define In particular A(p) = α and B(q) = β . The parabolic weights on N • (resp. I • ) at the point x j are larger or equal to β j 1 , . . . , β j n (resp. α j 1 , . . . , α j q−n ). The stability condition thus implies Adding the two lines in (6) and using Equation (5), we get , we can rewrite this as Finally, by definition of A(q − n) and B(q − n), we have We thus obtain Considering the interval then Proposition 4.1 gives the following compactness criterion: Therefore, δ vanishes identically by Proposition 4.1. In particular, the Higgs field γ ⊕ δ is nilpotent and the Hitchin map sends M(α, β, d) to 0. The properness of the Hitchin map (see Theorem 2.18) thus implies that M(α, β, d) is compact.

A non trivial example
When ε < 1, the interval J α,β has length at least 1 and thus contains an integer. It is not easy, however, to construct a stable Higgs bundle with deg(U ) − deg(V ) = d for generic (α, β). Here, we restrict to more specific choices of weights that are "constant" at each puncture, for which computations are easier. This will allow us to construct examples of moduli spaces M(α, β, d) that are compact and isomorphic to the Kronecker moduli space R(p, q, s − 2). In the next subsection, we will look at generic choices of weights close to these constant weights.
Remark 4.6. The interval s+p p+q , (p+q−1)s+p p+q has length 1 − 2 p+q s which is at least 1 as soon as s ≥ 3 and p or q is greater than 1. In that case this inteval always contains an integer. On the other hand, when p = q = 1, it is reduced to { s+1 2 }, which is an integer if and only if s is odd (this corresponds to Example 2.16).
Proof of Lemma 4.5. The condition a ∈ s+p p+q , Since a is an integer, it implies a ≤ s. Write (p+q)a−p = ks+r with 0 ≤ r < s and define k j = k+1 for j ∈ {1, . . . , r} and k j = k for j ∈ {r +1, . . . , s}, so that s j=1 k j = (p + q)a − p. The inequality (9) implies that 1 ≤ k j ≤ p + q − 1 for all j. We now choose (ε j ) 1≤j≤s ∈ R >0 satisfying the following conditions: Let us first show that these conditions can be simultaneously satisfied. If k ≤ q −1, then min k j q , p+q−k j p = k j q , and the conditions (10) can be simultaneously satisfied since Similarly, if k ≥ q, we have min k j q , p+q−k j p = p+q−k j p , and the conditions (10) can be simultaneously satisfied since With the k j and ε j chosen as above, let us set Then pα j + qβ j = k j ∈ N, 0 < α j < β j < 1 for all j ∈ {1, . . . , s} and ε = s j=1 ε j ∈ 2pq−2p−2q 2pq−p−q , 1 , so (α, β) is a constant SU(p, q)-multiweight and satisfies conditions (a) and (b).
Remark 4.7. In the proofs of Lemmas 4.4 and 4.5, the assumption that (α, β) is constant simplifies the notations a lot. However, one could easily extend the proof to SU(p, q)-multiweights in a neighborhood of a constant multiweight satisfying the required hypotheses.
For the rest of this section, we fix an integer a and a constant multiweight (α, β) stafisfying the conditions of Lemma 4.5. We take According to Lemmas 4.4 and 4.5, every semistable parabolic SU(p, q)-Higgs bundle in M(α, β, d) is of the form (U • ⊕ V • , γ ⊕ 0), where the parabolic structure on U • and V • at x j is given by trivial flags with weights α j and β j respectively, and for some γ ∈ H 0 (K(D) ⊗ Hom(U , V )). 1 We are left with understanding which choices of γ actually produce a (semi)stable Higgs bundle.
Note that H 0 (O(s − 3)) has dimension s − 2. Recall from Section 3.2 that E(p, q, r) = Hom(C p , C q ) r , so fixing a basis of H 0 (O(s − 3)), we eventually obtain an isomorphism Lemma 4.8. Let A be a point in E(p, q, s − 2). The following are equivalent: • the parabolic SU(p, q)-Higgs bundle (U ⊕ V , γ A ⊕ 0) is semistable (resp. stable), • A is a semistable (resp. stable) point of the space E(p, q, s − 2) in the sense of Theorem 3.14.
Proof. Let U and V be subbundles of U and V respectively. Remark first that with equalities if and only if U = U ⊗ O(−a + 1) and V = V ⊗ O(−a) for some subspaces U and V of C p and C q respectively. In that case, moreover, we have that Conversely, assume that A is semistable. Let U and V be two subbundles of U and V of respective rank m and n such that U ⊕ V is invariant by γ ⊕ 0 (i.e. such that γ(V ) ⊂ U ). Two cases appear: . . , s}. By the previous computation we have which is non-positive by semistability of A.
The same computation as before gives If p and q are greater than 1, recall that we imposed the condition Plugging this in (11), we get since p + q ≤ pq for p and q ≥ 2. If p = 1, then we simply use ε > 0, and (11) gives Similarly, if q = 1, we get This proves that (U • ⊕ V • , γ A ⊕ 0) is semistable if and only if A ∈ E(p, q, s − 2) is semistable. Repeating the proofs with strict inequalities instead of inequalities, one obtains the equivalence between the stability of (U • ⊕ V • , γ A ⊕ 0) and that of A. Proof. We set once and for all U = C p ⊗ O(−a + 1) and V = C q ⊗ O(−a), and we let U • and V • denote the parabolic bundles with underlying bundles U and V and parabolic structure at x j given by trivial flags and respective weights α j and β j . Then the map identifies E(p, q, s − 2) with an algebraic family of SU(p, q)-Higgs bundles. Since M(α, β) is a coarse moduli space of semistable parabolic Higgs bundles and since semistability in the sense of parabolic Higgs bundles and in the sense of Theorem 3.14 are equivalent by Lemma 4.8, we obtain an algebraic map ϕ : E ss (p, q, s − 2) → M(α, β, d). By Lemmas 4.5 and 4.4, this map is surjective. Finally, the automorphisms of U = C p ⊗ O(−a + 1) and V = C q ⊗ O(−a) are given respectively by the actions of GL(p, C) and GL(q, C) on the factors C p and C q . One easily deduces that (U • ⊕ V • , γ A ⊕ 0) and (U • ⊕ V • , γ A ⊕ 0) are isomorphic if and only if A and A are in the same G p,q -orbit. This shows that the map ϕ factors through a bijection from R(p, q, s − 2) to M(α, β, d). Since M(α, β, d) is a normal variety (see [Yok93]), this bijection is an isomorphism.

Pertubation of the constant weights
In this subsection, we remark that the example constructed previously can be perturbed to obtain non-empty compact components for generic multiweights.
Remark 4.12. The assumption that the perturbations of the weights η j and ζ j are strictly increasing is only here so that the associated parabolic structures are given by complete flags. As in Section 3.4, this is merely to avoid a proliferation of indices.
Proof of Proposition 4.11. First, let us associate to a point in E ♠ (p, q, s − 2, s) a parabolic SU(p, q)-Higgs bundle of type (α , β ). To do so, let us fix a basis of H 0 (O(s − 3)) so as to identify E(p, q, s − 2) with Hom(C p , C q ⊗ H 0 (O(s − 3))).
Set U = C p ⊗ O(−a + 1) and V = C q ⊗ O(−a). The canonical bases of C p and C q define bases of U x and V x at any point x, up to scaling. This induces canonical isomorphisms between the space of complete flags in C p (resp. C q ) and the space of complete flags on U x (resp. V x ). Given a point (F, H) ∈ F s × H s , we now see the flag F j (resp. H j ) as a flag on U x j (resp. V x j ). We denote by U • (F, α ) and V • (H, β ) the parabolic bundles with underlying bundles U and V respectively, and parabolic structure at x j given by the complete flags F j and H j and the weights α j and β j . Let us also denote respectively by U • (α) and V • (β) the parabolic bundles with underlying bundle U and V and parabolic structure at x j given by the trivial flags and the constant weights α j and β j .
3 One can easily verify that everytime we ask for (α , β ) "close enough" to (α, β), there actually is an explicit such choice which does not depend on (A, F, H), so that in the end we are indeed describing a neighborhood of (α, β).

Properties of our compact relative components
In this section we return to the "representation variety" side of the non-Abelian Hodge correspondence where we summarize the results obtained so far and prove Theorems 1 and 2 for SU(p, q), under the condition that p q + q p < s − 2.
Recall also from Subsection 2.2.4 that the non-Abelian Hodge correspondence gives a homeomorphism Now, let us fix an integer a ∈ s+p p+q , (p+q−1)s+p p+q and let (α, β) be a constant multi-weight satisfying the hypotheses of Lemma 4.5. Through the non-Abelian Hodge correspondence, the results of the previous section translate into the following theorem: is compact, non-empty, and contains an irreducible representation.
It would be interesting to replace "irreducible" by "Zariski dense" in the above theorem. Though we strongly suspect that each of our compact components contains a Zariski dense representation, we do not know of a quick argument to rule out all other possible Zariski closures in general. We can nevertheless make the improvement for "almost all" weights (α , β ), thanks to the following remark: has non-empty interior in the (absolute) character variety X Σ 0,s , SU(p, q) .
Proof. Let ρ 0 be a representation in X (Σ 0,s , SU(p, q)). We simply need to remark that, for every representation ρ : Γ 0,s → SU(p, q) in a neighborhood of ρ 0 and every j ∈ {1, . . . , s}, ρ(c j ) has all its eigenvalues of modulus 1. Indeed, ρ 0 (c j ) has two eigenspaces of respective dimension p and q in restriction to which the pseudo-Hermitian form h of C p,q is respectively positive definite and negative definite. In particular, the eigendirections of ρ 0 (c j ) are "far" from the isotropic cone of equation h = 0. Thus, for ρ close enough to ρ 0 , ρ(c j ) does not have any isotropic eigenvector either. Since ρ(c j ) preserves h, every non-isotropic eigenvector has a corresponding eigvenvalue of modulus 1.
In conclusion, every ρ in a neighborhood of ρ 0 belongs to X h(α ,β ) (Σ 0,s , SU(p, q)) for some (α , β ) in a neighborhood of (α, β). Moreover, the Toledo invariant of ρ has the form d + α − β for some d ∈ Z. By continuity of the Toledo invariant, d = d for ρ close enough to ρ 0 . Hence ρ belongs to SU(p, q) . Now, the set of Zariski dense representations has full measure in X(Γ 0,s , SU(p, q)) (see [Win02]). We thus obtain the corollary: Corollary 5.3. There is a set W of full measure in W (α, β) such that, for all (α , β ) ∈ W , the relative component contains a Zariski dense representation.
This concludes the proof of Theorem 1 for SU(p, q), under the assumption that p q + q p < s − 2.

Proof of Theorem 2
Let us now describe some properties of the representations contained in these compact components and prove Theorem 2. Assume s > 2 + p q + q p , and let (α, β), W (α, β) and Ω be as in the previous subsection.
Theorem 5.4. Every representation ρ ∈ Ω has the following properties: (i) For every identification of Σ 0,s with a punctured Riemann sphere, there is a ρ-equivariant holomorphic map from Σ 0,s to the symmetric space of SU(p, q).
(ii) For every homotopy class of simple closed curve c in Σ 0,s , the complex eigenvalues of ρ(c) have modulus 1.
(iii) More generally, for every k ≥ 0, there is a constant C(k) such that, if c is the homotopy class of a closed curve with at most k self-intersections, then the eigenvalues of ρ(c) have modulus less than C(k).
We first describe the geometry of the symmetric space associated to SU(p, q). Recall that C p,q is the usual (p + q)-dimensional complex vector space equipped with a signature (p, q) Hermitian product h. Let Y be the Grassmanian of p-dimensional (complex) linear subspaces of C p,q on which the restriction of h is positive definite. Given P ∈ Y , the stabilizer of P in SU(p, q) is a maximal compact subgroup conjugated to K := S(U(p) × U(q)), identifying Y with the symmetric space SU(p, q)/K.
The Grassmanian Y carries two natural tautological bundles. Consider the trivial bundle and define the rank p vector subbundle U ⊂ E by U P = P . Similarly, one has the subbundle V ⊂ E defined by V P = P ⊥ . In particular, E = U ⊕ V . The tangent space T P Y is identified with Hom C (P , P ⊥ ), so the tangent bundle T Y is isomorphic to the bundle Hom C (U , V ). Its complexification is thus given by where T 1,0 Y and T 0,1 Y are respectively the holomorphic and anti-holomorphic tangent bundle of Y .
The non-Abelian Hodge correspondence relies on the existence, for any irreducible representation ρ : Γ → SU(p, q) and Riemann surface structure X on Σ, of a unique ρ-equivariant harmonic map f : X → Y . More precisely, the SU(p, q)-Higgs bundle (U ⊕ V , γ ⊕ δ) on X associated to ρ is Properties (ii) and (iii) follow from (i) using the contraction property of holomorphic maps. We first introduce the Kobayashi metric on Y . By a theorem of Harish-Chandra, Y is biholomorphic to a bounded domain in C n , and the Kobayashi distance d K between two points x, y ∈ Y can be defined as Here, d is the unit disk in C equipped with the Poincaré distance d P .
The Kobayashi metric is a SU(p, q) invariant Finsler metric on Y which is thus bi-Lipschitz to the symmetric Riemannian metric d Y . Moreover, by definition, any holomorphic map f : Given an element g ∈ SU(p, q), define the translation length of g by where d Y is the distance associated to the Killing metric. Up to a multiplicative constant, we have where the λ i are the eigenvalues of g.
We are now ready to prove (ii) and (iii). Take ρ : Γ 0,s → SU(p, q) a representation in Ω and let γ ∈ Γ 0,s be a homotopy class of simple closed curve on Σ 0,s . For any complex structure X on Σ 0,s , Property (i) gives the existence of a ρ-equivariant holomorphic map f : X → Y . Given a biholomorphism X d, and x ∈ d a point on (a lift of) the geodesic homotopic to γ, we deduce where l X (γ) is the hyperbolic length of the geodesic representing γ and C is a constant.
This inequality being true for any complex structure on Σ 0,s , we obtain that, up to a multiplicative constant, the translation length of ρ(γ) is less than the infimum of the hyperbolic length of γ over the Teichmüller space of Σ 0,s .
It is well-known that this infimum is 0 when γ is simple, proving (ii). When γ has k self-intersections, this infimum is less than a constant depending only on k, implying (iii).
This concludes the proof of Theorem 5.4 and that of Theorem 2 for SU(p, q) with the assumption p q + q p < s − 2.

More compact components
In this section, we give ways of constructing more compact components : by tensorizing with a line bundle, by restricting to representations into other Hermitian Lie groups, and by restricting to subsurfaces. These examples show how far we are from a classification of compact connected components.

Tensor product with a line bundle
There is a notion of projective equivalence for parabolic SU(p, q)-Higgs bundles. Under the non-Abelian Hodge correspondence, projectively equivalent polystable SU(p, q)-Higgs bundles correspond to representations ρ : Γ → SU(p, q) with the same image in PU(p, q) = SU(p, q)/Z, where Z = e 2ikπ p+q Id, k = 0, . . . , p + q − 1 Z p+q is the center of SU(p, q). In particular, given a representation ρ : Γ → SU(p, q), the set of representations ρ : Γ → SU(p, q) with the same image in PU(p, q) is parametrized by H 1 (Σ 0,s , Z p+q ).
That is, two projectively equivalent parabolic SU(p, q)-Higgs bundles differ by a (parabolic) (p + q) th root of the trivial bundle. If we denote by J X the group of degree 0 parabolic line bundles over X, the set of (p + q) th roots of the trivial bundle is the subgroup of (p + q) torsion points and is denoted J X [p + q].
As a result, given a compact conponent M(α, β, d), one gets a family of compact components These components correspond to different lifts of the same representations in PU(p, q). Note that the condition α 1 i < β q i we imposed when looking for compact components is generally not preserved by taking the tensor product with L • ∈ J X [p + q]. This shows that the conditions of our compactness criterion (Corollary 4.3) are too restrictive to account for all compact components.

Other Hermitian Lie groups
We extend here our construction to other classical Hermitian Lie groups, namely Sp(2p, R) and SO * (2p). Sp(2p, R).

Compact components in
Let (V , ω) be a real vector space of dimension 2p together with a symplectic form ω. The group Sp(2p, R) is the group of linear transformations of V preserving ω.
The group Sp(2p, R) acts on the complexification V C of V which is naturally equipped with the Hermitian form h defined by h(u, v) = iω C (u, v), where ω C is the C-linear extension of ω, and one can check that h has signature (p, p). In particular, (V C , h) C p,p and we get an embedding Sp(2p, R) → SU(p, p).
Via this embedding, the relative Sp(2p, R) character varieties are identified with closed subsets of the SU(p, p) character varieties. We will prove the following lemma: Lemma 6.4. Assume s ≥ 5 is odd. Then we can find a subset Ω ⊂ X(Σ 0,s , SU(p, p)) as in Section 5 which contains a Sp(2p, R) representation.
Let us explain first why the lemma implies Theorems 1 and 2 for X(Σ 0,s , Sp(2p, R)) with s odd. Indeed, the intersection of Ω ∩ X (Σ 0,s , Sp(2p, R)) is an open subset of the Sp(2p, R) character variety which thus contains a Zariski dense representation. Moreover, the intersection of X(Σ 0,s , Sp(2p, R)) with a compact component of a relative SU(p, p) character variety is a compact component of a relative Sp(2p, R) character variety. Hence Ω ∩ X(Σ 0,s , Sp(2p, R)) is a union of compact relative components, proving Theorem 1. Finally, properties of Theorem 2 are satisfied by all the representations in Ω, including those in X(Σ 0,s , Sp(2p, R)) (for the third property, it is important that the embedding Sp(2p, R) → SU(p, p) induces a holomorphic embedding of the symmetric spaces).
Proof of Lemma 6.4. We transit once again through the non-Abelian Hodge correspondence. One can indeed detect which parabolic SU(p, p)-Higgs bundles give rise to a representation in Sp(2p, R).
Take a = s+1 2 , and set α j = 1−ε j 2 and β j = 1+ε j 2 where the ε j ∈ (0, 1) satisfy One easily verifies that these multiweights satisfy the hypotheses of Lemma 4.5 (with p = q). Let L • be the parabolic line bundle O(− s−1 2 + s j=1 α j x j ). Then L ∨ • = O(− s+1 2 + s j=1 β j x j ). By Lemmas 4.4 and 4.5, every parabolic SU(p, p)-Higgs bundle in M(α, β, p) has the form (U • ⊕ V • , γ ⊕ 0), where U • = L • ⊗ C p and V • = L ∨ • ⊗ C p . The standard bilinear pairing of C p given by (z, w) → z 1 w 1 + · · · + z p w p induces an isomorphism between V • and U ∨ • . Finally, if A = (A 1 , . . . , A s−2 ) is a tuple of endomorphisms of C p which are symmetric (with respect to the standard bilinear pairing), then the associated section Let Ω be the subset of the SU(p, p) character variety given by where W (α, β) is an open neighborhood of (α, β) in W (p, p, s).
Then Ω contains a representation in Sp(2p, R).
Note that, for the above choice of (α, β), the relative component X p h(α,β) (Σ 0,s , Sp(2p, R)) could be described as a GIT quotient of the space S(p, s − 2) of s − 2-tuples of symmetric matrices of size p by the action of GL(p, C) given by g · S = (gS 1 g T , . . . , gS s−2 g T ) .
One could also define a "feathered version" of this space to describe the nearby relative components. We don't know if these spaces have been studied in the litterature.

Compact components in SO * (2p).
Recall that C p,p is the complex vector space C 2p equipped with the signature (p, p) Hermitian form h defined by h(z, w) = z 1 w 1 + · · · + z p w p − z p+1 w p+1 − · · · − z 2p w 2p .
Consider the C-bilinear quadratic form q on C 2q defined by q(z) = p k=1 z k z k+p . The group SO * (2p) is the group of linear transofrmations of C 2p preserving both h and q. In particular, SO * (2p) ⊂ SU(p, p).
Similarly to the Sp(2p, R) case, one can see the SO * (2p) character variety as a closed subset of the SU(p, p) character variety and Theorems 1 and 2 for SO * (2p) and s odd will follow from the following lemma: Lemma 6.5. Assume s ≥ 5 is odd. Then we can find a subset Ω ⊂ X(Σ 0,s , SU(p, p)) as in Section 5 which contains a SO * (2p) representation.
Proof. The proof is very similar to the Sp(2p, R) case and we only highlight the main differences between the two cases.
A parabolic SO * (2p)-Higgs bundle is a SU(p, p)-Higgs bundle (U • ⊕ V • , γ ⊕ δ) such that • )) and δ ∈ H 0 (K(D) ⊗ Hom(U ∨ • , U • )) are antiself-adjoint. If we make the same choices of (α, β) as in the proof of Lemma 6.4, SO * (2p)-Higgs bundles in M(α, β, p) will be of the form (U • ⊕ U ∨ • , γ A ⊕ 0), where A = (A 1 , . . . , A s−2 ) is a tuple of antisymmetric matrices of size p. We are thus left with the problem of finding a semistable point in E(p, p, s − 2) consisting of antisymmetric matrices. This is slightly more elaborate than the case of symmetric matrices. If p is even, any choice of antisymmetric matrices A 1 , . . . , A s with A 1 invertible will give a semistable point. If p is odd, on the other side, every antisymmetric matrix has a non-trivial kernel. In particular, there is no stable point in E(p, p, 2) consisting of antisymmetric matrices by Lemma 3.17, so one needs at least 3 matrices.
We leave to the reader the verification that A 1 , A 2 and A 3 are antisymmetric.
We claim that (A 1 , A 2 , A 3 ) is a semistable point of E(p, p, 3). By contradiction, assume that there exists V and W ⊂ C p such that A i (V ) ⊂ W for i = 1, 2, 3 and dim W < dim V . Since the kernel of A 1 is Ce 0 we have that V contains e 0 and dim W = dim V − 1. We can thus write V = Ce 0 ⊕ V where V ⊂ e ⊥ 0 . Since e 0 is also the kernel of A 2 , we have W = A 1 (V ) = A 2 (V ).
Remark that A 1 and A 2 preserve e ⊥ 0 . Let A 1 and A 2 be the induced endomorphisms of e ⊥ 0 . Then V is invariant by A 1 −1 A 2 and it is thus spanned by eigenvectors of A 1 −1 A 2 . Moreover, we also have A 3 (V ) ⊂ W ⊂ e ⊥ 0 . Since A 3 (e 2k ), e 0 = 1, we must have V = Span(e 2i−1 ) i∈I for some subset I of {1, . . . , p }. But we then have A 3 (V ) ⊂ A 1 (V ) = Span(e 2i ) i∈I , which contradicts the condition on A 3 .
There is a well-defined restriction map Res : which is clearly open.
Let Ω be an open domain in X(Σ 0,s 0 , G) foliated by compact relative components. Let Ω be the subset of Ω consisting of representations ρ such that ρ(b) is diagonalizable with distinct eigenvalues. Note that Ω is open and non-empty since the map ρ → ρ(b) is open. Theorem 1 in full generality now follows from the following proposition: Proposition 6.6. The domain Res(Ω ) ⊂ X(Σ 0,s , G) is foliated by compact connected components.
We want to prove that the connected component of ρ 0 in X h (Σ 0,s , G) is compact and contained in Res(Ω ).
Since ρ 0 (b) is diagonalizable, its conjugation orbit is closed. The set K is therefore closed in X h (Γ 0,s , SU(p, q)) ∩ Ω, hence compact. It is moreover contained in Ω .
By definition of K, the restriction map Res sends K to → X h (Γ 0,s , G). Let us prove that its image contains the connected component of [ρ 0 ]. Let [ρ 1 ] be a point in this connected component and let [ρ t ] 0≤t≤1 be a continuous path from [ρ 0 ] to [ρ 1 ]. For each t, we know that ρ t (b) belongs to C(ρ 0 (b)). Since ρ 0 (b) has distinct eigenvalues, this implies that ρ t (b) and ρ 0 (b) are conjugate. Let g t ∈ G be such that g t ρ (b)g −1 t = ρ 0 (b) . We can assume that g t varies continuously with t. Define ρ t : Γ 0,s 0 → SU(p, q) such that ρ t |Γ = g t ρ g −1 t and ρ t |Γ = ρ 0 .
Since Ω contains the connected component of ρ 0 in its relative character variety, we deduce that ρ t ∈ Ω for all t. Moreover, ρ t (b) is conjugate to ρ 0 (b). Hence [ρ t ] belongs to K. Finally, ρ t |Σ is conjugate to ρ t . Thus [ρ t ] belongs to Res(K) ⊂ Res(Ω ).
Finally, to prove Theorem 2 in full generality, we prove the following: Proposition 6.7. Every representation ρ ∈ Res(Ω) satisfies the properties of Theorem 5.4.
Proof. Properties (ii) and (iii) immediately pass to the restriction of a representation to a subsurface. Property (i) is slightly more subtle. A representation j : Γ 0,s → Isom + (H 2 ) is called Fuchsian if it is discrete and faithful and j(Γ 0,s )\H 2 is biholomorphic to a punctured Riemann sphere.
For any Fuchsian representation j of π 1 (Σ ) Γ 0,s , one can find a sequence of Fuchsian representations (j n ) n∈N of Γ 0,s 0 such that (j n |Σ ) n∈N converges to j. Now let ρ be a representation in Ω. By Theorem 5.4, for every n ∈ N, there exists a (j n , ρ)-equivariant holomorphic map f n from H 2 to the symmetric space X of G. All this maps are 1-Lipschitz with respect to the Kobayashi metric of X. After extracting a subsequence, (f n ) n∈N converges to a (j, ρ |Σ )-equivariant holomorphic map from H 2 to X. Since j can be any Fuchsian representation, this shows that ρ |Σ satisfies Property (i).