© Revista de Matemática: Teoría y Aplicaciones 2021 28(1) : 1–38 cimpa – ucr issn: 1409-2433 (Print), 2215-3373 (Online)

doi: https:// doi. org/ 10.15517/ rmta. v28i1. 42154

notes on coherent systems

notas sobre los sistemas coherentes

Alexander H.W. Schmitt^{∗}

Received: 2/jun/2020; Revised: 4/Nov/2020; Accepted: 12/Nov/2020

∗Freie Universität Berlin, Institut für Mathematik, Arnimallee 3, D-14195 Berlin, Germany. E-Mail: alexander.schmitt@fu-berlin.de

2 a.h.w. schmitt

Abstract

We present an alternative approach to semistability and moduli spaces for coherent systems associated with decorated vector bundles. In this approach, it seems possible to construct a Hitchin map. We relate some examples to classical problems from geometric invariant theory.

Keywords: coherent system; moduli space; Hitchin map; first fundamental theorem of invariant theory.

Resumen

En estas notas se presenta un nuevo enfoque para el estudio de las condiciones de semi-estabilidad, así como de los espacios de móduli, de los sistemas coherentes asociados a fibrados vectoriales con estructura adicional. Bajo este enfoque, se abre la posibilidad de definir un morfismo de Hitchin. Se muestra, además, la relación entre algunos ejemplos concretos con problemas clásicos presentes en la teoría geométrica de invariantes.

Palabras clave: sistema coherente; espacio de móduli; morfismo de Hitchin; primer teorema fundamental de la teoría de invariantes.

Mathematics Subject Classification: 14H40, 14D20, 13A50.

1 Introduction

The group G := GL_{r}(ℂ) is the group of linear automorphisms of the complex vector space Q := ℂ^{r}. So, tautologically, there is the action

G × Q ⟶ Q
(g, x) ⟼ g · x.

There are many actions which may be derived from it, e.g.,

1. the action

G × Mat_{r}(ℂ) ⟶ Mat_{r}(ℂ)
(g, m) ⟼ g · m · g^{−1}

of G on the vector space of (r × r)-matrices,

2. the action

G × Mat_{r}(ℂ)^{⊕2} ⟶ Mat_{r}(ℂ)
(g, m_{1}, m_{2}) ⟼ (g · m_{1} · g^{−1}, g · m_{2} · g^{−1})

of G on the vector space of pairs of (r × r)-matrices, 3. for 1 ≤ s ≤ r − 1, the action

G × Gr_{s}(Q) ⟶ Gr_{s}(Q)
(g, R) ⟼ g · R

of G on the Graßmannian variety Gr_{s}(Q) of s-dimensional sub vector spaces of Q, and

4. for a ≥ 2 and 1 ≤ s ≤ a · r − 1, the action

G × Gr_{s}(Q^{⊗a}) ⟶ Gr_{s}(Q^{⊗a})
(g, R) ⟼ g · R

of G on the Graßmannian variety Gr_{s}(Q^{⊗a}) of s-dimensional sub vector spaces of Q^{⊗a}. Here, G acts on Q^{⊗a} by the a-fold tensor power of the initial “tautological” action of G on Q.

The task would be to describe the set of G-orbits in the respective space. For the original action, there are two orbits, namely {0} and Q \ {0}. In Example (3), there is only one orbit. For Example (1), there is, e.g., the theory of the Jordan normal form which exhibits in each orbit a “nice” representative. However, Case (2) has not been solved, so far, and is, in fact, considered a wild problem which one does not expect to solve completely. The same goes for Case (4). In order to say something meaningful in those cases, we need to endow the set of orbits with an additional structure. The easiest structure is that of a topological space — simply endow the set of orbits with the quotient topology. As algebraic geometers, we prefer the structure of an algebraic variety. If the set of G-orbits carries the structure of an algebraic variety, the topological space described before needs to be a Hausdorff space. If there are non-closed orbits, as in Example (1) for r ≥ 2, this is not true. So, we may hope at best to equip the set of closed orbits with the structure of an algebraic variety. In the case that G acts on an affine variety, such as in Example (1) and (2), Hilbert’s seminal work [10] and [11], as reinterpreted by Mumford [22], shows that the set of closed orbits carries a natural structure of an affine algebraic variety. That variety is called the categorical quotient. If G acts on a projective variety as in Example (3) and (4), then one needs an additional datum, called linearization, in order to find a G-invariant open subset U, such that the set of closed orbits inside U carries the structure of a projective variety and yields the categorical quotient (see [22]).

Before we proceed, let us mention some related actions which occur in applications. First, we look at the diagram

1 ⟶ 2 ⟶ ⋯ ⟶ n − 1 ⟶ n

and pick positive integers r_{1}, ..., r_{n}. Then, there is the action

_{i=1}^{n} GL_{r_{i}}(ℂ) × ⨁_{i=1}^{n−1} Mat_{r_{i+1},r_{i}}(ℂ) ⟶ ⨁_{i=1}^{n−1} Mat_{r_{i+1},r_{i}}(ℂ)
(g_{1}, ..., g_{n}, m_{1}, ..., m_{n−1}) ⟼ (g_{2} · m_{1} · g_{1}^{−1}, ..., g_{n} · m_{n−1} · g_{n−1}^{−1}).

In this case, there are only finitely many orbits, and representatives are direct sums of the basic diagrams

0 ⟶ ⋯ ⟶ 0 ⟶ ℂ ⟶^{id} ⋯ ⟶^{id} ℂ ⟶ 0 ⟶ ⋯ ⟶ 0.

This allows to classify orbits by bar codes and plays an important rôle in persistent homology and big data analysis [24]. More generally, for each directed graph and choice of a positive integer for each vertex, one has a similar action. There are either finitely many orbits, or the classification problem is as complex as the one for Example (1), or the problem is as wild as Example (2). (We refer to [24], Appendix A, for a guide to the literature.) Some of the latter diagrams are considered in control theory. The choice of a linearization is important for obtaining compactifications of moduli spaces of linear systems. We refer to [9] and [1] for overviews and general results. The paper [12] explains the rôle of Example (1) in control theory more explicitly.

Now, we work relative to a smooth projective algebraic curve X over the complex numbers or a compact Riemann surface. Given an action

σ: G × V ⟶ V

of G = GL_{r}(ℂ) on a finite dimensional complex vector space V, corresponding to the group homomorphism

ϱ: G ⟶ GL(V)
g ⟼ (v ⟼ σ(g, v)),

called a representation, it is possible to associate with every algebraic or holomorphic vector bundle E of rank r on X an algebraic or holomorphic vector bundle E_{ϱ} on X with typical fiber V. In addition, we fix an algebraic or holomorphic line bundle L on X. The task, here, is to classify pairs (E, s) which consist of an algebraic or holomorphic vector bundle E on X and a section s: O_{X} ⟶ E_{ϱ} ⊗ L. (In fact, we may consider this problem over any algebraic or complex analytic variety X. The case that X is a point is the one that we discussed at the beginning.) For the first action which we considered and L = O_{X}, we look at vector bundles E together with a global section s ∈ H^{0}(X, E). These are the Bradlow pairs introduced in [3]. For Example (1) and L = ω_{X}, the cotangent bundle of X, we have vector bundles E together with a twisted endomorphism s: E ⟶ E ⊗ ω_{X}. These are known as Higgs bundles (see [13]). Building on these and many more examples, the author developed a procedure for constructing moduli spaces for these objects (see [32]). These moduli spaces are the analogs to the categorical quotients discussed above. The differential geometric counterparts of these objects occur in mathematical physics in the context of gauge theory ([35], Chapter V, Section 4-6). There is a so-called Kobayashi–Hitchin correspondence between related objects in gauge theory and semistable decorated vector bundles in the above sense ([21], [20]). The theory of decorated vector bundles that we have just outlined is closely related to the theory of stable gauged maps which has been used in quantum cohomology. We refer to [7] for an introduction.

The example of the action of G on the Graßmannian has been generalized to the moduli problem of Brill–Noether pairs or coherent systems. These are pairs (E, Γ) in which E is a vector bundle of rank r and Γ ⊂ H^{0}(X, E) is a subspace of prescribed dimension. Raghavendra and Vishwanath [26] formulated a notion of semistability for coherent systems which depends on a parameter δ ∈ ℚ_{>0} and constructed the moduli spaces for some values of δ. Constructions of moduli spaces which work in general were given by King and Newstead [15] and Le Potier [19]. In [34], the author considered the corresponding analog for the action of G on Gr_{s}(Q^{⊗a}). To this end, we fix again a line bundle L on X. Then, a coherent system is a pair (E, Γ) which consists of a vector bundle E of rank r on X and a vector space Γ ⊂ H^{0}(X, E^{⊗a} ⊗ L). The paper [34] introduces a notion of stability for these coherent systems which generalizes the notion of Raghavendra and Vishwanath and seems quite natural. This notion also depends on a stability parameter δ ∈ ℚ_{>0}. The paper also contains a construction of moduli spaces. However, the construction works only for stability parameters below a certain threshold. In the explicit study of moduli spaces, the variation of the moduli spaces with the stability parameter δ plays an important rôle and the moduli spaces for large values of δ are typically easier to understand. We refer to [39] and [4] for two characteristic examples. Motivated by this, we look in this note at another possible notion of semistability which also depends on a positive rational number δ. It is more complicated to state, but the construction of moduli spaces we propose does not impose an upper bound on the stability parameter. This approach follows the natural idea to construct the moduli spaces for coherent systems as GL_{s}(ℂ)-quotients of the moduli spaces for decorated vector bundles associated with ϱ^{⊕s}, ϱ: GL_{r}(ℂ) ⟶ GL(ℂ^{r}^{⊗a}) being the afold tensor power of the standard representation and s := dim_{ℂ}(Γ). In addition, it seems that, for some of the moduli spaces, there exists a Hitchin map. Besides the definition of semistability and a sketch of the construction of moduli spaces, this note explores some examples which should help to understand the general picture, in particular, the Hitchin map. In future work, we hope to address the relation between the constructions of [34] and this note in order to fully understand the variation of moduli spaces. We would also like to point out that twisted versions of Brill–Noether pairs appear, e.g., in [38] and [14]. One fixes a vector bundle V on X and looks at pairs (E, Γ) where E is a vector bundle on X and Γ ⊂ H^{0}(X, E ⊗ V) is a linear subspace. I was informed by Peter Newstead that the case when both E and V vary is also of interest for Brill–Noether theory. This motivates to develop the results of [34] and this note further to a theory of coherent systems for decorated principal bundles. The case of Brill– Noether pairs with varying E and V would correspond to the structure group GL_{r}(ℂ) × GL_{s}(ℂ) and the natural representation of this group on ℂ^{r} ⊗ ℂ^{s}.

Notation

We will work on a connected smooth projective curve X of genus g at least two which is defined over the field ℂ of complex numbers, and we will fix a point x_{0} ∈ X. We write O_{X}(k) for O_{X}(k · x_{0}), and, given a coherent O_{X}-module F, the symbol F(k) stands for the O_{X}-module F ⊗_{O_{X}} O_{X}(k), k ∈ ℤ.

Given a scheme S and a vector bundle A on S, we write P (A) for the projective bundle of lines in the fibers of A, i.e., for \underline{Proj}(\underline{Sym}^{⋆}(A^{∨})). Occasionally, we will also use ℙ(A) := P (A^{∨}), i.e., Grothendieck’s convention for the projectivization.

For a cartesian product A × B in a category, we let π_{A}: A × B ⟶ A and π_{B}: A × B ⟶ B be the natural projections.

2 Review of some moduli spaces of decorated bundles

In [6], a general theory for tuples of vector bundles on a smooth projective curve X decorated with the help of a multihomogeneous representation was developed. It includes the classification of triples (F, E, φ), consisting of a vector bundle F of rank s, a vector bundle E of rank r, and a homomorphism φ: F ⟶ E^{⊗a}⊗L, for a fixed line bundle L on X. The case F ≅ O_{X}^{⊕s} corresponds to coherent systems. We will first review the relevant results from the theory of decorated vector bundles from [30] and then present a moduli problem for decorated pairs of vector bundles. It will be important to understand some features of that moduli problem in order to understand in which respects the moduli problem for coherent systems is different.

2.1 Decorated vector bundles

Let a, s, and r be positive integers. Suppose that ϱ: GL_{r}(ℂ) ⟶ GL(V) is a representation which is homogeneous of degree a, e.g., the a-th tensor power of the standard representation with V = (ℂ^{r})^{⊗a}.

Then, given a vector bundle E of rank r on X, we may use ϱ to assign to E a vector bundle E_{ϱ} of rank dim_{ℂ}(V). Fix also a line bundle L on X. A ϱ-pair is a pair (E, φ) which consists of a vector bundle E of rank r on X and a non-zero homomorphism φ: O_{X}^{⊕s} ⟶ E_{ϱ} ⊗ L.

Recall that a weighted filtration of a vector bundle E on X is a pair (E_{•}, α_{•}) in which

0 ⊊ E_{1} ⊊ ⋯ ⊊ E_{a} ⊊ E

is a filtration of E by subbundles and α_{•} = (α_{1}, ..., α_{a}) is a vector of positive rational numbers. For such a weighted filtration (E_{•}, α_{•}), we set

M(E_{•}, α_{•}) = ∑_{i=1}^{a} α_{i} · (deg(E) · rk(E_{i}) − deg(E_{i}) · rk(E)).

Given a homomorphism φ: O_{X}^{⊕s} ⟶ E_{ϱ} ⊗ L and a weighted filtration (E_{•}, α_{•}) of E, the quantity μ(E_{•}, α_{•}; φ) is defined in [30], p. 175f, [32], p. 139.

Let δ be a positive rational number. A ϱ-pair (E, φ) is δ-(semi)stable, if the inequality

M(E_{•}, α_{•}) + δ · μ(E_{•}, α_{•}; φ)(≥)0

is satisfied, for every weighted filtration (E_{•}, α_{•}) of E.

For the rest of this section, we fix a line bundle M on X and look only at ϱ-pairs (E, φ) with det(E) ≅ M. Recall from [30], Theorem 2.3.7.1, that there are critical values 0 =: c_{0} < c_{1} < ⋯ < c_{t} < c_{t+1} := ∞, such that the notions of δ-semistability and δ-stability for ϱ-pairs (E, φ) with det(E) ≅ M remain constant within each of the intervals (c_{i}, c_{i+1}) ∩ ℚ, i = 0, ..., t. A ϱ-pair (E, φ) is asymptotically (semi)stable, if it is δ-(semi)stable with respect to a stability parameter δ ∈ (c_{t}, ∞) ∩ ℚ.

Lemma 2.1 A ϱ-pair (E, φ) is asymptotically (semi)stable if and only if it satisfies the following two conditions.

a) μ(E_{•}, α_{•}; φ) ≥ 0, for every weighted filtration (E_{•}, α_{•}) of E,

b) M(E_{•}, α_{•})(≥)0, for every weighted filtration (E_{•}, α_{•}) of E with μ(E_{•}, α_{•}; φ) = 0.

Remark 2.2 Let η be the generic point of X, 𝕂 := ℂ(X) the function field of the curve X, 𝔼 the restriction of E to η, and 𝕍 the restriction of E_{ϱ} to η. Note that 𝔼 and 𝕍 are vector spaces over the field 𝕂. The restriction of φ to η yields a point φ_{η} ∈ Hom(𝕂^{s}, 𝕍). Since we are working in characteristic zero, the semistability of φ_{η} may be tested with the Hilbert–Mumford criterion (see [32], Section 1.7.1, for a brief discussion). Therefore, φ_{η} is semistable if and only if Condition a) in Lemma 2.1 is satisfied. In this case, we say that φ is generically semistable.

For a positive rational number δ, we let P_{X/M/L/ϱ}^{δ} be the moduli space of δsemistable ϱ-pairs (E, φ) in which det(E) ≅ M. (We will briefly review the construction in Section 4.1.) For δ ∈ (c_{t}, ∞), the above moduli space is equipped with a special gadget, the so-called Hitchin map. To this end, we set ℍ := Hom(ℂ^{s}, V). The coordinate algebra of ℍ is isomorphic to the symmetric algebra of ℍ^{∨},

ℂ[ℍ] ≅ Sym^{⋆}(ℍ^{∨}) = ⨁_{d≥0} Sym^{d}(ℍ^{∨}),

and the action of SL_{r}(ℂ) preserves this grading, so that, in particular,

ℂ[ℍ]^{SL_{r}(ℂ)} = ⨁_{d≥0} Sym^{d}(ℍ^{∨})^{SL_{r}(ℂ)}.

Note that the GIT quotient of P (ℍ) by the action of SL_{r}(ℂ) is given as

P (ℍ)// SL_{r}(ℂ) := Proj(ℂ[ℍ]^{SL_{r}(ℂ)}).

For any k > 0, we set

sym_{k}^{⋆} := ⨁_{d≥0} sym_{k}^{d}, sym_{k}^{d} := Sym^{k·d}(ℍ^{∨})^{SL_{r}(ℂ)}, d ≥ 0.

Then,

P (ℍ)// SL_{r}(ℂ) ≅ Proj(sym_{k}^{⋆}), k > 1.

We may choose k > 0 in such a way that sym_{k}^{⋆} is generated by sym_{k}^{1}, i.e,

P (ℍ)// SL_{r}(ℂ) ↪ ℙ(sym_{k}^{1}).

Remark 2.3 A function I ∈ ℂ[ℍ] is a ϱ-semiinvariant, if there exists an integer w, such that

∀g ∈ GL_{r}(ℂ)∀h ∈ ℍ : I(g · h) = det(g)^{w} · I(h).

The ring ℂ[ℍ]^{SL_{r}(ℂ)} agrees with the ring of ϱ-semiinvariants. For a homogeneous element I ∈ ℂ[ℍ]^{SL_{r}(ℂ)}, the number a · deg(I) is a multiple of r. In particular, we may write a · k = r · l.

Set

h_{1} := ℙ(sym_{k}^{1} ⊗H^{0}(X, M^{⊗l} ⊗ L^{⊗k})).  (2.3.1)

Let I_{1}, ..., I_{u} be a basis for sym_{k}^{1} and φ: O_{X}^{⊕s} ⟶ E_{ϱ} ⊗ L a homomorphism. We pick a suitable open covering (U_{i})_{i∈I} and trivializations E_{ϱ|U_{i}} ≅ V ⊗ O_{U_{i}}, L_{|U_{i}} ≅ O_{U_{i}}, i ∈ I. So, φ defines morphisms f_{i}: U_{i} ⟶ ℍ and, thus, functions I_{j} ∘ f_{i}, i ∈ I. For j ∈ { 1, ..., u }, the functions I_{j} ∘ f_{i}, i ∈ I, glue to a section

I_{j}(φ) ∈ H^{0}(X, M^{⊗l} ⊗ L^{⊗k}).

If φ is generically semistable, then there exists an index j ∈ { 1, ..., u } with I_{j}(φ) ≠ 0, so that we may define

\overline{I}(φ) := [I_{1}(φ) : ⋯: I_{u}(φ)] ∈ h_{1}.

This construction leads to the morphism

χ_{1} : P_{X/M/L/ϱ}^{δ} ⟶ h_{1}
[E, φ] ⟼ \overline{I}(φ).

which we call the Hitchin map (see Page 24 for more details). Since the moduli space P_{X/M/L/ϱ}^{δ} is projective, the Hitchin map is automatically projective.

Remark 2.4 The above construction works also for negative a and the value a = 0. In the latter case, the line bundle M does not enter the definition of the Hitchin space, and we do not have to fix it, i.e., it suffices to fix just the degree of the participating vector bundles. The equivalence relation has to be formulated as for the example of the adjoint representation discussed in the next section.

2.2 Example: Coherent Higgs systems

In the special case s = 1, the moduli space P_{X/M/L/ϱ}^{δ} solves the classification problem for pairs (E, Γ) which consist of a vector bundle E and a onedimensional subspace Γ ⊂ H^{0}(X, E_{ϱ} ⊗ L), i.e., for certain coherent systems. As pointed out in Remark 2.4, we may apply the results also in the case a = 0, and we do not need to fix the determinant in that case. In particular, we may apply them to the adjoint representation of GL_{r}(ℂ) on Mat_{r}(ℂ). Since this is the starting point of the present work, let us discuss this example in some detail.

The classification problem

We consider pairs (E, φ) that consist of a vector bundle E on X and a non-zero twisted endomorphism φ: E ⟶ E ⊗ L. We say that (E_{1}, φ_{1}) is isomorphic to (E_{2}, φ_{2}), if there exist an isomorphism ψ: E_{1} ⟶ E_{2} and a non-zero complex number λ ∈ ℂ^{⋆} with

φ_{2} = λ · ((ψ ⊗ id_{L}) ∘ φ_{1} ∘ ψ^{−1}).

The set of isomorphism classes of such pairs clearly agrees with the set of isomorphism classes of coherent Higgs systems (E, Γ) with dim_{ℂ}(Γ) = 1 presented in the introduction.

Asymptotic semistability

We let H_{X}^{δ}(r, d) be the projective moduli space of δ-semistable pairs (E, φ) with rk(E) = r and deg(E) = d. As in Section 2.1, we let 0 =: c_{0} < c_{1} < ⋯ < c_{t} < c_{t+1} := ∞ be the critical values for the concept of δ-semistability for pairs (E, φ) with deg(E) = d. The two extremal cases have special properties.

Lemma 2.5 Suppose that δ ∈ (c_{0}, c_{1})∩ℚ. Then, (E, φ) is δ-(semi)stable if and only if the following two conditions are satisfied.

a) The vector bundle E is semistable.

b) For every weighted filtration (E_{•}, α_{•}) of E with μ(E_{i}) = d/r, i = 1, ..., a, one has μ(E_{•}, α_{•}; φ)(≥)0.

Remark 2.6 Note that the lemma shows that (E, φ) is stable for δ ∈ (c_{0}, c_{1}) ∩ ℚ, if E is a stable vector bundle. This shows that

P (T_{M_{X}^{s}(r,d)}^{⋆}) ⊂ H_{X}^{δ}(r, d).

Here, M_{X}^{s}(r, d) is the moduli space of stable vector bundles of rank r and degree d on X, and T_{M_{X}^{s}(r,d)}^{⋆} is its cotangent bundle.

Lemma 2.7 Suppose that δ ∈ (c_{t}, ∞)∩ℚ. Then, (E, φ) is δ-(semi)stable if and only if the following two conditions are satisfied.

a) The twisted endomorphism φ is not nilpotent, i.e., (φ ⊗ id_{L^{⊗(r−1)}}) ∘ ⋯ ∘ φ: E ⟶ E ⊗ L^{⊗r} ≠ 0.

b) For every subbundle 0 ⊊ F ⊊ E with φ(F) ⊂ F ⊗L, one has μ(F)(≥)0.

Remark 2.8 i) Part b) implies that (E, φ) is a semistable Higgs bundle. For δ ∈ (c_{t}, ∞), the moduli space H_{X}^{δ}(r, d) is the divisor that compactifies Hitchin’s moduli space ℌ_{X}(r, d) in the compactification constructed in [28]. It can be interpreted as a ℂ^{⋆}-quotient of the latter, i.e.,

H_{X}^{δ}(r, d) = ℌ_{X}(r, d)//ℂ^{⋆}.

ii) Define h := ⨁_{i=1}^{r} H^{0}(X, L^{⊗i}) and let ℂ^{⋆} act on H^{0}(X, L^{⊗i}) with weight

i, i = 1, ..., r, Here, we may define h_{1} := h//ℂ^{⋆} as the resulting weighted projective space. Condition b) grants that there is a Hitchin map

χ_{1} : H_{X}^{δ}(r, d) ⟶ h_{1}.

It is automatically projective, because H_{X}^{δ}(r, d) is projective.

The rank two case

A problem for applications might be that the Hitchin map χ_{1} is not defined on the whole projectivized cotangent bundle P (T_{M_{X}^{s}(r,d)}^{⋆}).

Example 2.9 Let X be a curve of genus g ≥ 2. We look at extensions

0 ⟶ O_{X} ⟶^{ι} E ⟶^{π} ω_{X} ⟶ 0.

These extensions are parameterized by Ext^{1}(ω_{X}, O_{X}) ≅ H^{1}(X, ω_{X}^{∨}) ≅ H^{0}(X, ω_{X}^{⊗2})^{∨}. This is a vector space of dimension 3g − 3. Proposition 2.4 in [17] and [16], Lemma 1, imply that there exist extensions for which E is stable. Pick an extension for which E is stable and define

φ := (ι ⊗ id_{ω_{X}}) ∘ π: E ⟶ E ⊗ ω_{X}.

Then, (E, φ) defines a point in the projectivized cotangent bundle, but φ is clearly nilpotent, so that χ_{1} is not defined at (E, φ).

There is the one parameter subgroup

λ: ℂ^{⋆} ⟶ SL_{2}(ℂ)
z ⟼ ( \begin{matrix}z^{−1} & 0 \\ 0 & z \end{matrix} ) .

Note

∀m = ( \begin{matrix}a & b \\ c & d \end{matrix} ) , ∀z ∈ ℂ^{⋆} : λ(z) · m · λ(z)^{−1} = ( \begin{matrix}a & z^{−2} · b \\ z^{2} · c & d \end{matrix} ) .

This shows that, for a vector bundle E of rank two, a non-zero homomorphism φ: E ⟶ E ⊗ L, and a sub line bundle F ⊂ E, one has

μ(0 ⊊ F ⊊ E, (1); φ) = \{ \begin{matrix}−2, & if F ⊂ Ker(φ) \\ 0, & if 0 ≠ φ(F) ⊂ F ⊗ L \\ 2, & if φ(F) ⊄ F ⊗ L \end{matrix} .  (2.9.1)

Example 2.10 We return to the setting of Example 2.9. A sub line bundle F ⊂ E distinct from O_{X} will not be invariant under φ. So, it follows from (2.9.1) that (E, φ) will be δ-(semi)stable if and only if the condition of δ-(semi)stability is satisfied for the sub line bundle O_{X} of E. This shows that (E, φ) is δ-stable for δ < g − 1, properly (g − 1)-semistable, and not δ-semistable for δ > g − 1.

2.3 Decorated pairs of vector bundles

We fix the same data as at the beginning of Section 2.1. A holomorphic ϱ-triple is a triple (F, E, φ) which consists of a vector bundle F of rank s on X, a vector bundle E of rank r on X, and a non-zero homomorphism φ: F ⟶ E_{ϱ} ⊗ L.

Remark 2.11 We may write F as F′ ⊗ L for some other vector bundle F′. Then, φ corresponds to a homomorphism φ′: F′ ⟶ E_{ϱ}. So, one might assume without loss of generality that L = O_{X}.

Semistability

The representation ϱ̃: GL_{s}(ℂ) × GL_{r}(ℂ) ⟶ GL(ℍ) is bihomogeneous of bidegree (−1, a), i.e., for all z, w ∈ ℂ^{⋆}, ϱ̃(z · E_{s}, w · E_{r}) = z^{−1} · w^{a} · id_{ℍ}. Therefore, the results of Section 3 in [6] may be applied.

Let (F, E, φ) be a holomorphic ϱ-triple. For weighted filtrations (F_{•}, β_{•}) and (E_{•}, α_{•}) of F and E, respectively, Formula (20) in [6] may be used to define

μ((F_{•}, β_{•}), (E_{•}, α_{•}); φ).

Fix a positive rational number δ. We say that a holomorphic ϱ-triple is δ- (semi)stable, if the inequality

M(F_{•}, β_{•}) + M(E_{•}, α_{•}) + δ · μ((F_{•}, β_{•}), (E_{•}, α_{•}); φ)(≥)0

is satisfied, for every pair ((F_{•}, β_{•}), (E_{•}, α_{•})) of weighted filtrations of F and E.

For the following, we fix line bundles N and M on X. As before, there are critical values 0 =: c_{0} < c_{1} < ⋯ < c_{t} < c_{t+1} := ∞, such that the notions of δsemistability and δ-stability for holomorphic ϱ-triples (F, E, φ) with det(F) ≅ N and det(E) ≅ M remain constant within each of the intervals (c_{i}, c_{i+1}) ∩ ℚ, i = 0, ..., t. We say that (F, E, φ) is asymptotically (semi)stable, if it is δ- (semi)stable with respect to a stability parameter δ ∈ (c_{t}, ∞) ∩ ℚ.

Lemma 2.12 A holomorphic ϱ-triple (F, E, φ) is asymptotically (semi)stable if and only if it satisfies the following two conditions.

a) μ((F_{•}, β_{•}), (E_{•}, α_{•}); φ) ≥ 0, for all pairs ((F_{•}, β_{•}), (E_{•}, α_{•})) of weighted filtrations,

b) M(F_{•}, β_{•})+M(E_{•}, α_{•})(≥)0, for all pairs ((F_{•}, β_{•}), (E_{•}, α_{•})) of weighted filtrations with μ((F_{•}, β_{•}), (E_{•}, α_{•}); φ) = 0.

Remark 2.13 We use the notation from Remark 2.2. In addition, we define 𝔽 as the restriction of F to η. So, the restriction of φ to η yields a point φ_{η} ∈ Hom(𝔽, 𝕍). The group SL_{s}(𝕂) × SL_{r}(𝕂) acts on that vector space. The point φ_{η} will be semistable with respect to that group action if and only if Condition a) in Lemma 2.12 is satisfied. In that case, we call φ generically semistable.

For δ ∈ ℚ_{>0}, we let T_{X/N/M/L/ϱ}^{δ} be the moduli space of δ-semistable holomorphic ϱ-triples (F, E, φ) in which det(F) ≅ N and det(E) ≅ M. For δ ∈ (c_{t}, ∞) ∩ ℚ, there will be again a Hitchin map. Set 𝕊 := SL_{s}(ℂ) × SL_{r}(ℂ). This group acts on ℍ = Hom(ℂ^{s}, V). This time, we will be interested in the invariant ring

ℂ[ℍ]^{𝕊} = ⨁_{d≥0} Sym^{d}(ℍ^{∨})^{𝕊}.

Fix k > 0 and define

{ \atop sym}_{k}^{⋆} := ⨁_{d≥0} { \atop sym}_{k}^{d}, { \atop sym}_{k}^{d} := Sym^{k·d}(ℍ^{∨})^{𝕊}, d ≥ 0.

As before,

P (ℍ)//𝕊 ≅ Proj({ \atop sym}_{k}^{⋆}), k > 0.

We choose k > 0 in such a way that { \atop sym}_{k}^{⋆} is generated by { \atop sym}_{k}^{1}. In particular,

P (ℍ)//𝕊 ↪ ℙ({ \atop sym}_{k}^{1}).

Write k = s · n and a · k = r · m. The Hitchin space is now

h_{2} := ℙ({ \atop sym}_{k}^{1} ⊗ H^{0}(X, N^{⊗−n} ⊗ M^{⊗m} ⊗ L^{⊗k})).  (2.13.1)

For δ ∈ (c_{t}, ∞) ∩ ℚ, we will have a Hitchin map

χ_{2} : T_{X/N/M/L/ϱ}^{δ} ⟶ h_{2}.

In the next section, we will discuss examples of GIT problems which will help to understand Hitchin spaces and maps (see Section 5).

3 The underlying problem from geometric invariant theory

The understanding of the notion of semistability for coherent systems and the Hitchin map rests on the analysis of the underlying model given by the representation of the reductive affine algebraic group SL_{s}(ℂ) × SL_{r}(ℂ) on the vector space Hom(ℂ^{s}, V) or rather its projectivization. As an illustration, we look at two specific examples.

Remark 3.1 i) We may view the GIT problem just introduced as the problem of classifying coherent systems when the base variety is a point rather than a smooth projective curve X.

ii) The GIT quotient can be formed in two steps, e.g., by first dividing by the action of the group SL_{s}(ℂ) and then by the action of the group SL_{r}(ℂ). This means that we may alternatively study the action of SL_{r}(ℂ) on the Graßmannian 𝔊 of s-dimensional subspaces of V. We will see in Section 4.2 how these two viewpoints lead to different approaches for constructing moduli spaces for coherent systems on a smooth projective curve X which seem to work for different ranges of the stability parameter.

3.1 Symmetric tensors

We choose a = s = r = 2 and look at the subspace Sym^{2}(ℂ^{2}) ⊂ (ℂ^{2})^{⊗2}. Let us set Q := ℂ^{2}. Denote by (e_{0}, e_{1}) the standard basis of Q and by (x_{0}, x_{1}) the dual basis of Q^{∨}. The space Sym^{2}(Q^{∨}) is the space of quadratic forms on Q. We write a quadratic form q ∈ Sym^{2}(Q^{∨}) as

q = q(x_{0}, x_{1}) = a · x_{0}^{2} + 2b · x_{0}x_{1} + c · x_{1}^{2}.

We may represent q also by the symmetric (2 × 2)-matrix

m_{q} = ( \begin{matrix}a & b \\ b & c \end{matrix} ) .

Let

ω: SL_{2}(ℂ) ⟶ SL_{2}(ℂ)
g ⟼ (g^{−1})^{t}

be the outer automorphism. If τ: SL_{2}(ℂ) ⟶ SL(Sym^{2}(Q^{∨})) is the second symmetric power of the dual of the standard representation, the representation κ := τ ∘ ω is isomorphic to the second symmetric power of the standard representation of SL_{2}(ℂ) on Q. So, we may write the action associated with the representation of SL_{2}(ℂ) on Sym^{2}(Q) as

SL_{2}(ℂ) × Sym^{2}(Q^{∨}) ⟶ Sym^{2}(Q^{∨})
(g, q) ⟼ (v ⟼ q(g^{t} · v))
(g, m_{q}) ⟼ g · m_{q} · g^{t}.

Next, note that there is the non-degenerate anti-symmetric pairing

Q × Q ⟶ ℂ
(v, w) ⟼ det(v|w).

It is also invariant under the action of SL_{2}(ℂ). So, it induces an isomorphism Q ≅ Q^{∨} of representations of SL_{2}(ℂ). We write ℍ := Hom(Q^{∨}, Sym^{2}(Q^{∨})) and 𝕊 := SL_{2}(ℂ) × SL_{2}(ℂ). By our conventions, the 𝕊-action A on ℍ has the form

((( \begin{matrix}α & β \\ γ & δ \end{matrix} ) , g) , (m_{q_{1}}, m_{q_{2}})) ⟼  (3.1.1)
⟼ (g · (α · m_{q_{1}} + γ · m_{q_{2}}) · g^{t}, g · (β · m_{q_{1}} + δ · m_{q_{2}}) · g^{t}) .

As recalled in Remark 3.1, ii), we may first form the quotient by the action of the second factor. For this, we need to determine the invariant ring

ℂ[ℍ]^{SL_{2}(ℂ)}.

It appears from time to time in the literature, but we haven’t found a recent reference where it is explicitly determined. So, let us do it, here. We look at a point (q_{1}, q_{2}) ∈ ℍ with

(m_{q_{1}}, m_{q_{2}}) = (( \begin{matrix}a_{1} & a_{2} \\ a_{2} & a_{3} \end{matrix} ) , ( \begin{matrix}b_{1} & b_{2} \\ b_{2} & b_{3} \end{matrix} )) .

The first invariant is the determinant of m_{q_{1}}, i.e., we set

Δ_{1}(q_{1}, q_{2}) := a_{1} · a_{3} − a_{2}^{2}.

Likewise, the second invariant is given by the determinant of m_{q_{2}}, that is,

Δ_{2}(q_{1}, q_{2}) := b_{1} · b_{3} − b_{2}^{2}.

The third invariant is the so-called codiscriminant. It is defined by the formula

Γ(q_{1}, q_{2}) := \frac{1}{2} · a_{1} · b_{3} + \frac{1}{2} · a_{3} · b_{1} − a_{2} · b_{2}.

Remark 3.2 i) Suppose that I ∈ ℂ[ℍ]^{SL_{2}(ℂ)} is an invariant. Denoting by E_{2} the identity matrix, we define, for h ∈ SL_{2}(ℂ), the map

h^{⋆}(I): ℍ ⟶ ℍ
(q_{1}, q_{2}) ⟼ I(A((h, E_{2}), (q_{1}, q_{2}))).

Since the two actions of SL_{2}(ℂ) on ℍ commute with each other, it follows that h^{⋆}(I) is also an element of ℂ[ℍ]^{SL_{2}(ℂ)}. Now, let

h = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ SL_{2}(ℂ).

Then, one computes

h^{⋆}(Δ_{1}) = α^{2} · Δ_{1} + γ^{2} · Δ_{2} + 2(αγ) · Γ,
h^{⋆}(Δ_{2}) = β^{2} · Δ_{1} + δ^{2} · Δ_{2} + 2(βδ) · Γ,  (3.2.1)
h^{⋆}(Γ) = (αβ) · Δ_{1} + (γδ) · Δ_{2} + (αδ + βγ) · Γ.

We can choose h, such that αγ ≠ 0. Since Δ_{1}, Δ_{2}, and h^{⋆}(Δ_{1}) are invariants, the first formula shows that Γ is an invariant, too.

ii) The above computations are a special case of a procedure for computing the invariant rings for direct sums of representations (see [36]).

iii) We now explain what it means that all three invariants Δ_{1}, Δ_{2}, and Γ vanish at a point (q_{1}, q_{2}) ∈ ℍ. As is well known, Δ_{i} vanishes if and only if m_{q_{i}} has rank at most one, i = 1, 2. Recall that a (2 × 2)-matrix m defines the linear map

D_{m}: Q ⟶ Q^{∨}
v ⟼ (w ⟼ v^{t} · m · w).

The radical Rad(m) of m is the kernel of D_{m}. We infer that Δ_{1}(q_{1}, q_{2}) = Δ_{2}(q_{1}, q_{2}) = Γ(q_{1}, q_{2}) = 0 is equivalent to the following: a) rk(m_{q_{i}}) ≤ 1, i = 1, 2, and,

b) if q_{1} ≠ 0 and q_{2} ≠ 0, then Rad(m_{q_{1}}) = Rad(m_{q_{2}}).

Proposition 3.3 The invariants Δ_{1}, Δ_{2}, and Γ are algebraically independent and generate the invariant ring, i.e.,

ℂ[ℍ]^{SL_{2}(ℂ)} = ℂ[Δ_{1}, Δ_{2}, Γ].

Proof. We will use Hilbert’s “algorithm” for computing the invariant ring (see [11], [37], Section 4.6). We leave it to the reader to check that these invariants are algebraically independent. This will also be a consequence of the following discussion.

Abbreviate R := ℂ[Δ_{1}, Δ_{2}, Γ] and S := ℂ[ℍ]^{SL_{2}(ℂ)}. The first thing to show is that Δ_{1}, Δ_{2}, and Γ cut out the nullforms. Below, we will determine the nullforms with respect to the action of the second factor of 𝕊 with the Hilbert– Mumford criterion. It turns out that the nullforms are exactly those pairs satisfying the conditions stated in Remark 3.2, iii), i.e., exactly those pairs in which Δ_{1}, Δ_{2}, and Γ vanish. Then, according to [37], Theorem 4.6.1, R ⊂ S is a finite ring extension. Since R is a normal ring, it suffices to show that the field extension Q(R) ⊂ Q(S) has degree one. This amounts to showing that, for the morphism

π: Spec(S) ⟶ Spec(R),

there is a non-empty open subset U ⊂ Spec(S), such that π^{−1}(U) ⟶ U is bijective. We look at pairs (q_{1}, q_{2}), such that Δ_{1}(q_{1}, q_{2}) ≠ 0, Δ_{2}(q_{1}, q_{2}) ≠ 0, and q_{1} and q_{2} have no common linear factor.

Since SL_{2}(ℂ) acts triply transitively on ℙ^{1}, we may assume that there exist complex numbers λ, μ ∈ ℂ^{⋆}, and a ∈ ℂ \ { 0, 1 } with

q_{1} = λ · x_{0}x_{1} and q_{2} = μ · (x_{0} − x_{1}) · (x_{0} − a · x_{1}).

The matrix

g := ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) ∈ SL_{2}(ℂ)

transforms q_{1} into −λ · x_{0}x_{1} (and q_{2} into −a · μ · (x_{0} − x_{1}) · (x_{0} − (1/a) · x_{1})). So, we may assume

λ ∈ { z = u + v · i | (v > 0) ∨ (v = 0 ∧ u > 0) }.

Under this assumption Δ_{1}(q_{1}, q_{2}) = −λ^{2} allows to reconstruct λ. Next,

Δ_{2}(q_{1}, q_{2}) = −\frac{1}{4} · μ^{2} · (a − 1)^{2}, Γ(q_{1}, q_{2}) = λ · μ · (a + 1).

Since we know λ, we may determine the ratio

(\frac{a − 1}{a + 1})^{2} .

Now, suppose that b is another complex number with

(\frac{a − 1}{a + 1})^{2} = (\frac{b − 1}{b + 1})^{2} .

If

\frac{a − 1}{a + 1} = \frac{b − 1}{b + 1},

then a = b. Moreover, if μ · (a + 1) = ν · (b + 1) and a = b, then also μ = ν. If, on the other hand,

\frac{a − 1}{a + 1} = \frac{1 − b}{b + 1},

then b = 1/a. The equality μ · (a + 1) = ν · (b + 1) then yields ν = a · μ. Pick a complex number c with c^{2} = a. Then, the matrix

g := ( \begin{matrix}c & 0 \\ 0 & \frac{1}{c} \end{matrix} ) ∈ SL_{2}(ℂ)

leaves q_{1} = λ · x_{0}x_{1} invariant and transforms q_{2} = μ · (x_{0} − x_{1}) · (x_{0} − a · x_{1}) into a · μ · (x_{0} − x_{1}) · (x_{0} − (1/a) · x_{1}). So, under the assumptions on q_{1} and q_{2} made above, we may reconstruct the SL_{2}(ℂ)-orbit of (q_{1}, q_{2}) from Δ_{1}(q_{1}, q_{2}), Δ_{2}(q_{1}, q_{2}), and Γ(q_{1}, q_{2}).

Proposition 3.4 Set Θ := Δ_{1} · Δ_{2} − Γ^{2}. Then,

ℂ[ℍ]^{𝕊} = ℂ[Θ].

Proof. The formulas in (3.2.1) show that the action of the first factor of 𝕊 on the vector space ⟨ Δ_{1}, Δ_{2}, Γ ⟩ is isomorphic to the representation of SL_{2}(ℂ) on Sym^{2}(ℂ^{2}). The implies the assertion.

Remark 3.5 As the first (or second) formula in (3.2.1) shows, the zeros of Θ correspond to pairs (q_{1}, q_{2}) ∈ ℍ for which P (⟨ m_{q_{1}}, m_{q_{2}} ⟩) has exactly one point that corresponds to a matrix of rank one.

Let us determine the canonical nullforms with the Hilbert–Mumford criterion. We define, for μ, ν ≥ 0 and μ > 0 or ν > 0, the one parameter subgroup

λ_{μ,ν}: ℂ^{⋆} ⟶ 𝕊
z ⟼ (( \begin{matrix}z^{−μ} & 0 \\ 0 & z^{μ} \end{matrix} ) , ( \begin{matrix}z^{−ν} & 0 \\ 0 & z^{ν} \end{matrix} )) .

Let us record the weights of the different entries of a pair of matrices associated with an element of ℍ:

(( \begin{matrix}−μ − ν^{2} & −μ \\ −μ & −μ + ν^{2} \end{matrix} ) , ( \begin{matrix}μ − ν^{2} & μ \\ μ & μ + ν^{2} \end{matrix} )) .

Remark 3.6 i) We are looking for non-zero pairs of (2 × 2)-matrices in ℍ for which all weights are negative. If the second matrix is zero, this happens for ν = 0.

ii) If μ = 0, then all weights will be negative if and only if the only non-zero entry in both m_{q_{1}} and m_{q_{2}} is the top left one. One readily checks that (q_{1}, q_{2}) is a nullform with respect to the action of the second factor of 𝕊 if and only if the conditions stated in Remark 3.2, iii), are satisfied.

iii) Let us assume that both q_{1} ≠ 0 and q_{2} ≠ 0 and that both μ ≠ 0 and ν ≠ 0. Then, in m_{q_{2}}, only the top left entry is non-zero and μ < ν^{2}. This forces the bottom right entry of m_{q_{1}} to be zero. In other words, Rad(m_{q_{2}}) is an isotropic subspace for m_{q_{1}}.

iv) If m_{q_{1}} and m_{q_{2}} do not have a common isotropic subspace, then (q_{1}, q_{2}) is stable with respect to the action of the second factor of 𝕊.

Proposition 3.7 A pair (q_{1}, q_{2}) ∈ ℍ is semistable with respect to the action of 𝕊 if and only if a) ⟨ q_{1}, q_{2} ⟩ is a two-dimensional space and b) m_{q_{1}} and m_{q_{2}} have no common isotropic subspace.

Proof. If (q_{1}, q_{2}) satisfies Condition a) and b), then it follows readily from the previous discussions that (q_{1}, q_{2}) is semistable. Now, suppose that (q_{1}, q_{2}) fails to meet Condition b). Since (α, β) ⟼ det(α·m_{q_{1}} +β·m_{q_{2}}) has to admit a zero, the vector space ⟨ m_{q_{1}}, m_{q_{2}} ⟩ has a non-zero element of rank one. By applying the action of the first factor of 𝕊, we may assume without loss of generality that m_{q_{2}} has rank one. Then, the previous computations immediately show that (q_{1}, q_{2}) is a nullform.

The vector space ℍ has dimension six, the group 𝕊 has dimension six, as well, but the categorical quotient ℍ//𝕊 has dimension one and not zero. This implies that there are no points in ℍ which are stable with respect to the action of the group 𝕊. Still, we have the following interesting observation.

Proposition 3.8 Every point (q_{1}, q_{2}) ∈ ℍ which is semistable with respect to the action of 𝕊 is also polystable with respect to the action of 𝕊. In particular, the quotient

ℍ ⟶ ℍ//𝕊

is a geometric quotient Proof. A point (q_{1}, q_{2}) ∈ ℍ is semistable (polystable) with respect to the action of 𝕊 if and only if it is semistable (polystable) with respect to the action of the second factor 𝕊 and the image of (q_{1}, q_{2}) in ℍ// SL_{2}(ℂ) is semistable (polystable) with respect to the action of the first factor of 𝕊 ([23], Proposition 1.3.1 and 1.3.2). Now, for the action of SL_{2}(ℂ) on ℍ// SL_{2}(ℂ) ≅ Sym^{2}(Q^{∨}), every point which is semistable is also polystable, and the latter happens, according to Proposition 3.7, if and only if ⟨ m_{q_{1}}, m_{q_{2}} ⟩ is a two-dimensional space and m_{q_{1}} and m_{q_{2}} have no common isotropic subspace. It follows from Remark 3.6, iv), that (q_{1}, q_{2}) is stable and, thus, polystable with respect to the action of the second factor of 𝕊.

Remark 3.9 Suppose that (q_{1}, q_{2}) is semistable in the above sense. We also need to know when the maximal weight is zero. If μ were zero, the lower right entry of both m_{q_{1}} and m_{q_{2}} would have to be zero. This is ruled out by semistability. So, the only possibility is that μ = ν^{2}, and, in the matrix m_{q_{2}}, only the top left entry is non-zero (and the lower right entry of the first matrix is non-zero).

Example 3.10 a) The point (q_{1}, q_{2}) ∈ ℍ corresponding to the matrices

m_{q_{1}} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) and m_{q_{2}} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )

is semistable with respect to the action of the second factor of 𝕊 but not with respect to the action of 𝕊. In fact, m_{q_{2}} is - up to a scalar multiple - the only non-zero matrix in ⟨ m_{q_{1}}, m_{q_{2}} ⟩ of rank one.

b) The 𝕊 orbit of a point (q_{1}, q_{2}) ∈ ℍ which is semistable with respect to the action of 𝕊 contains an element (r_{1}, r_{2}) with

m_{r_{1}} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) and m_{r_{2}} = ( \begin{matrix}0 & 0 \\ 0 & a \end{matrix} ) .

Here,

a^{2} = −4 · Θ(r_{1}, r_{2}) = −4 · Θ(q_{1}, q_{2}).

By requiring Im(a) > 0 or Im(a) = 0 and a > 0, we may make a unique.

3.2 Homomorphisms

For classical coherent systems, the model in geometric invariant theory would be the action of GL_{s}(ℂ) × GL_{r}(ℂ) on the vector space Hom(ℂ^{s}, ℂ^{r}) and the quest for invariants with respect to the action of the group SL_{s}(ℂ) × SL_{r}(ℂ). This problem is, unfortunately, rather uninteresting. Invariants exist only in the case r = s and are given as polynomials in the determinant, in this case. So, let us first look at the action of GL_{r}(ℂ) on the vector space Hom(ℂ^{s}, ℂ^{r}). The invariants for the action of the special linear group SL_{r}(ℂ) are given by the first fundamental theorem of invariant theory for the special linear group.

Theorem 3.11 For s < r, one has ℂ[Hom(ℂ^{s}, ℂ^{r})]^{SL_{r}(ℂ)} = ℂ, and, for s ≥ r, the invariant ring is generated by the (r × r)-minors.

Proof. [5], Theorem 2.1.

Let us look at the special case s = r + 1. Then, ℂ[Hom(ℂ^{s}, ℂ^{r})]^{SL_{r}(ℂ)} is

the symmetric algebra of the vector space ⋀^{r} ℂ^{r+1}. Now, the wedge product

ℂ^{r+1} ⊗ ⋀^{r} ℂ^{r+1} ⟶ ⋀^{r+1} ℂ^{r+1}

shows that, as a representation of SL_{r+1}(ℂ), ⋀^{r} ℂ^{r+1} is isomorphic to (ℂ^{r+1})^{∨}.

In the construction of the Hitchin space h_{3} (4.6.2), we need to study the action of SL_{r+1}(ℂ) on ((ℂ^{r+1})^{∨})^{⊕h}, for some h > 0. So, we may use Theorem 3.11 again, in order to do this.

Remark 3.12 In connection with Hitchin spaces (see (2.3.1) and (4.6.2)), we need to study invariants in S^{⊕h} for an SL_{s}(ℂ)-module S and and h > 0. There are generalizations of the first fundamental theorem of invariant theory to this situation (see [36]). The module S itself is obtained from an invariant ring ℂ[T]^{SL_{r}(ℂ)} by picking a system of generators and homogenizing. So, in order to find nice examples, one could start with situations where ℂ[T]^{SL_{r}(ℂ)} is already a polynomial algebra. We refer to [25], §8, for an introduction to this topic. Interesting recent applications of such GIT problems are given in [2]. Some of the representations occurring there are of interest in the setting of coherent systems.

4 Moduli spaces

We fix a homogeneous representation ϱ: GL_{r}(ℂ) ⟶ GL(V), a positive integer s, line bundles L, M on X, and a stability parameter δ ∈ ℚ_{>0}. The aim is the definition of an appropriate notion of δ-semistability for coherent ϱ-systems and the sketch of a construction of moduli spaces. Particular care will be spent on the construction of the Hitchin map. An alternative approach has already been worked out in [34].

4.1 Review of the construction of moduli spaces of pairs

In this part, we will discuss some elements of the construction of moduli spaces of δ-semistable ϱ-pairs which are useful or even necessary for the understanding of the corresponding construction for moduli spaces of coherent systems. The definitions and constructions for holomorphic ϱ-triples are completely analogous.

The moduli functors

Let S be a scheme of finite type over ℂ. A family of ϱ-pairs parameterized by S is a tuple (E_{S}, L_{S}, φ_{S}) which consists of

• a vector bundle E_{S} of rank r on S ×X, such that there exists a line bundle M_{S} on S with det(E_{S}) ≅ π_{S}^{⋆}(M_{S}) ⊗ π_{X}^{⋆}(M),

• a line bundle L_{S} on S,

• and a homomorphism φ_{S}: O_{S×X}^{⊕s} ⟶ E_{S,ϱ} ⊗ π_{S}^{⋆}(L_{S}) ⊗ π_{X}^{⋆}(L).

We say that two families (E_{S}^{1}, L_{S}^{1}, φ_{S}^{1}) and (E_{S}^{2}, L_{S}^{2}, φ_{S}^{2}) of ϱ-pairs parameterized by S are isomorphic, if there exist isomorphisms

ψ_{S}: E_{S}^{1} ⟶ E_{S}^{2} and λ_{S}: L_{S}^{1} ⟶ L_{S}^{2},

such that

φ_{S}^{2} = (ψ_{S,ϱ} ⊗ π_{S}^{⋆}(λ_{S}) ⊗ id_{π_{X}^{⋆}(L)}) ∘ φ_{S}^{1}.

Given a scheme S of finite type over ℂ, a family (E_{S}, L_{S}, φ_{S}) of ϱ-pairs parameterized by S, and a closed point s ∈ S, the ϱ-pair (E_{S,s}, φ_{S,s}) is obtained by restricting (E_{S}, L_{S}, φ_{S}) to {s}×X. We say that (E_{S}, L_{S}, φ_{S}) is δ-(semi)stable, if (E_{S,s}, φ_{S,s}) is δ-(semi)stable, for every closed point s ∈ S. (This might involve defining δ-(semi)stability in the case that the residue field of s is not algebraically closed.) In this way, we obtain the moduli functors P_{X/M/L/ϱ}^{δ-(s)s} which assign to a scheme S of finite type over ℂ the set of isomorphy classes of δ- (semi)stable families of ϱ-pairs parameterized by S. There is an obvious notion of the pullback of a family of ϱ-pairs via a morphism f: T ⟶ S which enables us to define the map P_{X/M/L/ϱ}^{δ-(s)s}(f): P_{X/M/L/ϱ}^{δ-(s)s}(S) ⟶ P_{X/M/L/ϱ}^{δ-(s)s}(T).

The parameter space and the group action

Fix n ∈ ℕ, set p := d + r · (n + 1 − g), d := deg(M), and choose a complex vector space W of dimension p. There exist a quasi-projective scheme

𝔓 := 𝔓_{X/M/L/p/ϱ},

line bundles L_{𝔓}, M_{𝔓} on 𝔓, as well as families

q_{𝔓}: W ⊗ π_{X}^{⋆}(O_{X}(−n)) ⟶ E_{𝔓}

and

φ_{𝔓}: O_{𝔓×X}^{⊕s} ⟶ E_{𝔓,ϱ} ⊗ π_{𝔓}^{⋆}(L_{𝔓}) ⊗ π_{X}^{⋆}(L)

on 𝔓 × X, such that

• E_{𝔓} is a vector bundle of rank r on 𝔓 × X with det(E_{𝔓}) ≅ π_{𝔓}^{⋆}(M_{𝔓}) ⊗ π_{X}^{⋆}(M),

• q_{𝔓} is a surjective homomorphism,

• π_{𝔓⋆}(q_{𝔓} ⊗ id_{π_{X}^{⋆}(O_{X}(n))}): W ⊗ O_{𝔓} ⟶ π_{𝔓⋆}(E_{𝔓} ⊗ π_{X}^{⋆}(O_{X}(n)) is an isomorphism.

We will write points of 𝔓 in the form (q, φ). The (set-theoretic) map

A: GL_{s}(ℂ) × GL(W) × 𝔓 ⟶ 𝔓  (4.0.1)

(g, h, q, φ) ⟼ (q ∘ (g^{−1} ⊗ id_{π_{X}^{⋆}(O_{X}(n))}), φ ∘ (h^{−1} ⊗ id_{O_{𝔓×X}}))

underlies a scheme theoretic group action. We let

B: GL(W) × 𝔓 ⟶ 𝔓

be the restriction of A to GL(W) × 𝔓.

If we choose n large enough, there exist GL(W)-invariant open subsets 𝔓^{δ-ss} and 𝔓^{δ-s} of 𝔓, such that,

• for every point s = (q: W ⊗ O_{X}(−n) ⟶ E, φ) ∈ 𝔓^{δ-(s)s}, the ϱ-pair (E, φ) is δ-(semi)stable,

• for every scheme S, every δ-(semi)stable family (E_{S}, L_{S}, φ_{S}) of ϱ-pairs parameterized by S, and every point s ∈ S, there exist an open neighborhood s ∈ U ⊂ S and a morphism κ_{U}: U ⟶ 𝔓^{δ-(s)s}, such that the restriction of (E_{S}, L_{S}, φ_{S}) to U × X is isomorphic to the pullback of the family (E_{𝔓}, L_{𝔓}, φ_{𝔓}) via κ_{U} × id_{X},

• for every scheme S, and every pair κ_{S}^{i}: S ⟶ 𝔓, i = 1, 2, of morphisms, such that the pullback (E_{S}^{1}, L_{S}^{1}, φ_{S}^{1}) of (E_{𝔓}, L_{𝔓}, φ_{𝔓}) via κ_{S}^{1} × id_{X} is isomorphic to the pullback (E_{S}^{2}, L_{S}^{2}, φ_{S}^{2}) of (E_{𝔓}, L_{𝔓}, φ_{𝔓}) via κ_{S}^{2} × id_{X}, there exists a morphism g_{S}: S ⟶ GL(W), such that κ_{S}^{2} = B∘(g_{S}×κ_{S}^{1}).

These properties imply that the (coarse) moduli spaces for the functors P_{X/M/L/ϱ}^{δ-ss} and P_{X/M/L/ϱ}^{δ-s} are given by the categorical quotients

P_{X/M/L/ϱ}^{δ} := 𝔓^{δ-ss}// GL(W) and P_{X/M/L/ϱ}^{δ-s} := 𝔓^{δ-s}// GL(W).

Remark 4.1 Since the center ℂ^{⋆} · id_{W} ⊂ GL(W) acts trivially on 𝔓^{δ-(s)s}, it suffices to prove the existence of the categorical quotients 𝔓^{δ-ss}// SL(W) and 𝔓^{δ-s}// SL(W).

Construction of the Hitchin map

Now, assume that δ > c_{t}, so that the notions of δ-semistability and asymptotic semistability are characterized by Lemma 2.1. For every invariant I ∈ sym_{k}^{1}, we use the construction described after (2.3.1) to define a global section

I(φ_{S}): O_{𝔓×X} ⟶ π_{𝔓}^{⋆}(M_{𝔓}^{⊗l} ⊗ L_{𝔓}^{⊗k}) ⊗ π_{X}^{⋆}(M^{⊗l} ⊗ L^{⊗k}).

(Of course, we need to fix a suitable number k as described before.) We push this forward to 𝔓 and get

O_{𝔓} ⟶ H^{0}(X, M^{⊗l} ⊗ L^{⊗k}) ⊗ M_{𝔓}^{⊗l} ⊗ L_{𝔓}^{⊗k}.

Choosing a basis I_{1}, ..., I_{u} for sym_{k}^{1}, the associated sections may be used to define a morphism

χ̃_{𝔓}: 𝔓^{δ-ss} ⟶ h_{1}

with

χ_{𝔓}^{⋆}(O_{h_{1}}(1)) ≅ M_{𝔓}^{⊗l} ⊗ L_{𝔓}^{⊗k}.

This map is invariant under the action of GL(W). For this reason, it descends to a morphism

χ_{1} : P_{X/M/L/ϱ}^{δ} ⟶ h_{1}.

This is the Hitchin map.

4.2 The moduli problem for coherent systems

The moduli problem for coherent systems has already been discussed in our previous paper [34]. The parameter space for coherent ϱ-systems on which the GIT construction was performed in that source is itself a GIT quotient of the parameter space 𝔓 for ϱ-pairs from Section 4.1. This suggests that a construction of the moduli space for coherent ϱ-systems which is based on GIT on the parameter space 𝔓 should be possible. We will now try to explain this alternative construction.

Semistability

Let Γ be a finite dimensional complex vector space. A weighted flag in Γ is a pair (Γ_{•}, α_{•}) which consists of a (not necessarily complete) flag

0 ⊊ Γ_{1} ⊊ ⋯ ⊊ Γ_{b} ⊊ Γ

in Γ and a vector β_{•} = (β_{1}, ..., β_{b}) of positive rational numbers.

Remark 4.2 A weighted flag (Γ_{•}, β_{•}) determines a weighted filtration (F_{•}, β_{•}) of F := Γ ⊗ O_{X} with F_{i} := Γ_{i} ⊗ O_{X}, i = 1, ..., b. Clearly, not every weighted filtration of F comes from a weighted flag of Γ.

Now, let (E, Γ) be a coherent ϱ-system. We define

φ_{Γ} : F = Γ ⊗ O_{X} ⟶ H^{0}(X, E_{ϱ} ⊗ L) ⊗ O_{X} ⟶^{ev} E_{ϱ} ⊗ L.

For a coherent ϱ-system (E, Γ), a weighted flag (Γ_{•}, β_{•}) in Γ, and a weighted filtration (E_{•}, α_{•}) of E, we let (F_{•}, β_{•}) be the associated weighted filtration of F = Γ ⊗ O_{X} and set, using the definitions in Section 2.3,

μ((Γ_{•}, β_{•}), (E_{•}, α_{•}); Γ) := μ((F_{•}, β_{•}), (E_{•}, α_{•}); φ_{Γ}).

Let δ ∈ ℚ_{>0} be a stability parameter. We call a coherent ϱ-system (E, Γ) δ- (semi)stable, if the inequality

M(E_{•}, α_{•}) + δ · μ((Γ_{•}, β_{•}), (E_{•}, α_{•}); Γ)(≥)0

holds true, for every pair ((Γ_{•}, β_{•}), (E_{•}, α_{•})), consisting of a weighted flag (Γ_{•}, β_{•}) in Γ and a weighted filtration (E_{•}, α_{•}) of E.

Remark 4.3 We may choose Γ_{•} = 0 ⊊ Γ and β_{•} = (). In this case, the identity

μ((F_{•}, β_{•}), (E_{•}, α_{•}); Γ) = μ((E_{•}, α_{•}); φ_{Γ})

is satisfied, for every weighted filtration (E_{•}, α_{•}) of E. So, given a stability parameter δ ∈ ℚ_{>0} and a δ-semistable coherent ϱ-system (E, Γ), it follows that the ϱ-pair (E, φ_{Γ}) is δ-semistable, as well.

Now, fix the rank r, the degree d, and the dimension s. The main theorem in [31] or Theorem 2.3.4.3 in [32] now implies that the family of isomorphy classes of vector bundles E of rank r and degree d for which there exists an sdimensional subspace Γ ⊂ H^{0}(X, E_{ϱ} ⊗ L) and a stability parameter δ ∈ ℚ_{>0}, such that (E, Γ) is δ-semistable, is bounded.

In view of the strong boundedness result that we have just mentioned, we expect that the notion of δ-semistability for coherent ϱ-systems (E, Γ) with rk(E) = r, deg(E) = d, and dim_{ℂ}(Γ) = s stabilizes for large values of δ. We say that a coherent ϱ-system (E, Γ) is asymptotically (semi)stable, if

a) μ((F_{•}, β_{•}), (E_{•}, α_{•}); φ_{Γ}) ≥ 0 holds, for every pair ((Γ_{•}, β_{•}), (E_{•}, α_{•})), consisting of a weighted flag (Γ_{•}, β_{•}) in Γ and a weighted filtration (E_{•}, α_{•}) of E,

b) for every pair ((Γ_{•}, β_{•}), (E_{•}, α_{•})), consisting of a weighted flag (Γ_{•}, β_{•}) in Γ and a weighted filtration (E_{•}, α_{•}) of E with μ((F_{•}, β_{•}), (E_{•}, α_{•}); φ_{Γ}) = 0, one has M(E_{•}, α_{•})(≥)0.

Problem 4.4 Given r, d, and s as before, is it true that there is a positive rational number δ_{∞}, such that, for δ ≥ δ_{∞}, a coherent ϱ-system (E, Γ) with rk(E) = r, deg(E) = d, and dim_{ℂ}(Γ) = s is δ-(semi)stable if and only if it is asymptotically (semi)stable?

In view of the above boundedness result, the proof should proceed along the lines of the proof of Proposition 2.3.6.5 in [32].

Remark 4.5 As before, a coherent ϱ-system (E, Γ) defines a holomorphic ϱtriple (F, E, φ_{Γ}), and, in the notation of Remark 2.2 and 2.13, a point φ_{Γ,η} ∈ Hom(𝕂^{s}, V_{𝕂}). According to Remark 2.13, the first condition in the notion of asymptotic semistability for the holomorphic ϱ-triple (F, E, φ_{Γ}) requires that φ_{Γ,η} is semistable with respect to the action of SL_{s}(𝕂) × SL_{r}(𝕂). In contrast, Condition a) in the definition of asymptotic semistability for coherent systems requires that φ_{Γ,η} satisfies the Hilbert–Mumford criterion for pairs (λ′, λ′′) where λ′: 𝔾_{m}(ℂ) ⟶ SL_{s}(ℂ) ⊂ SL_{s}(𝕂) and λ′′: 𝔾_{m}(𝕂) ⟶ SL_{s}(𝕂) are one parameter subgroups. In order to understand this notion with the help of geometric invariant theory, it will probably be helpful to restrict φ_{Γ} to a suitable open subset U ⊂ X and choose appropriate trivializations in order to view φ_{Γ|U} as a morphism U ⟶ Hom(ℂ^{s}, V) as in [30], Lemma 1.4.

The moduli functors

Let S be a scheme of finite type over ℂ. A family of coherent ϱ-systems parameterized by S is a family (E_{S}, L_{S}, φ_{S}) of ϱ-pairs, such that

π_{S⋆}(φ_{S}): Γ ⊗ O_{S} ⟶ π_{S⋆}(E_{S,ϱ} ⊗ π_{X}^{⋆}(L)) ⊗ L_{S}

is universally injective, i.e., for every base change diagram

(4.5.1)

the homomorphism

π_{T ⋆}((f ×id_{X})^{⋆}(φ_{S})): Γ⊗O_{T} ⟶ π_{T ⋆}((f ×id_{X})^{⋆}(E_{S,ϱ})⊗π_{X}^{⋆}(L))⊗f^{⋆}(L_{S})

is injective.

Remark 4.6 Let us explain the notion of universal injectivity from a different viewpoint. First, we note that we may find an ample line bundle N on X, such that π_{S⋆}(E_{S,ϱ}⊗π_{X}^{⋆}(N))⊗L_{S} is locally free and commutes with base change. We may also fix an injective homomorphism L ⟶ N. For a base change diagram as (4.5.1), we get the commutative diagram ([8], Chapter III, Remark 9.3.1):

Here, we have set f_{X} := f × id_{X}. We point out the following:

• Since the projection T × X ⟶ X is flat, the homomorphism π_{T}^{⋆} (L) ⟶ π_{T}^{⋆} (N) is injective. The sheaf E_{S,ϱ} is locally free, so that the homomorphism f_{X}^{⋆}(E_{S,ϱ}) ⊗ π_{X}^{⋆}(L) ⟶ f_{X}^{⋆}(E_{S,ϱ}) ⊗ π_{X}^{⋆}(N) is injective. It follows that the second homomorphism in the bottom row of the above diagram is always injective. This means that the homomorphism π_{S⋆}(φ_{S}) is universally injective if and only if the corresponding homomorphism

ψ_{S}: Γ ⊗ O_{S} ⟶ π_{S⋆}(E_{S,ϱ} ⊗ π_{X}^{⋆}(N)) ⊗ L_{S}

is universally injective.

• Assume that S is reduced. Using the outer square in the above diagram and the fact that the leftmost and the rightmost vertical homomorphism are isomorphisms, it follows that ψ_{S} is universally injective if and only if Γ ⊗ O_{S} is a subbundle of π_{S⋆}(E_{S,ϱ} ⊗ π_{X}^{⋆}(N)) ⊗ L_{S}.

• Suppose that S is reduced and (E_{S}, L_{S}, φ_{S}) is an arbitrary family of ϱpairs paramterized by S. As before, let ψ_{S}: Γ ⊗ O_{S} ⟶ π_{S⋆}(E_{S,ϱ} ⊗ π_{X}^{⋆}(N)) ⊗ L_{S} be the induced homomorphism. Then, there is the open subscheme U of points s ∈ S, for which ψ_{S|{s}×X} has maximal rank. The foregoing discussion shows that, for the family (E_{U}, L_{U}, φ_{U}) of ϱ-pairs that is obtained by restricting (E_{S}, L_{S}, φ_{S}) to U × X, the homomorphism π_{U⋆}(φ_{U}) is universally injective.

Two families (E_{S}^{1}, L_{S}^{1}, φ_{S}^{1}) and (E_{S}^{2}, L_{S}^{2}, φ_{S}^{2}) parameterized by the scheme S are isomorphic, if there exist an isomorphism ψ_{S}: E_{S}^{1} ⟶ E_{S}^{2} and an isomorphism λ_{S} ∈ Γ(S, GL_{s}(O_{S})), such that

φ_{S}^{2} = (ψ_{S,ϱ} ⊗ id_{π_{X}^{⋆}(L)}) ∘ φ_{S}^{1} ∘ λ_{S}^{−1}.

Fix δ ∈ ℚ_{>0}. It is now clear how to define the moduli functors C_{X/M/L/ϱ}^{δ-(s)s} of δ-(semi)stable coherent ϱ-systems.

The construction of the moduli spaces

We start with the parameter scheme 𝔓 from Section 4.1. It is reduced. According to Remark 4.6, there is an open subscheme ℭ ⊂ 𝔓 that is characterized by the property that, for the universal family (E_{ℭ}, L_{ℭ}, φ_{ℭ}) that is obtained by restricting the universal family (E_{𝔓}, L_{𝔓}, φ_{𝔓}) to ℭ × X, the associated homomorphism π_{ℭ⋆}(φ_{ℭ}) is universally injective. The action defined in (4.0.1) induces an action

GL_{s}(ℂ) × GL(W) × ℭ ⟶ ℭ.

Using the techniques from the paper [6], it is now possible to carry out the program outlined for ϱ-pairs in Section 4.1also for coherent ϱ-systems, i.e., we find the (GL_{s}(ℂ) × GL(W))-invariant open subsets ℭ^{δ-ss} and ℭ^{δ-s} that parameterize the coherent ϱ-systems that are δ-semistable and δ-stable, and the categorical quotients

C_{X/M/L/ϱ}^{δ} := ℭ^{δ-ss}//(GL_{s}(C) × GL(W))

and

C_{X/M/L/ϱ}^{δ-s} := ℭ^{δ-s}//(GL_{s}(ℂ) × GL(W))

do exist and constitute the moduli spaces for the functors C^{δ-ss} and C^{δ-s}, respectively.

The Hitchin map

According to [23], Proposition 1.3.1, the moduli space C_{X/M/L/ϱ}^{δ} may be constructed as a GL_{s}(ℂ)-quotient of

P_{X/M/L/ϱ}^{δ} = 𝔓^{δ-ss}// GL(W).

By the same result, GL_{s}(ℂ) acts on sym_{k}^{1} and, thus, on h_{1} (2.3.1). The Hitchin map

χ_{1} : P_{X/M/L/ϱ}^{δ} ⟶ h_{1}  (4.6.1)

is GL_{s}(ℂ)-equivariant.

Now, we define

h_{3} := h_{1}// GL_{s}(ℂ).  (4.6.2)

Let us be a bit more precise. The center of GL_{s}(ℂ) acts trivially on h_{1}. So, as before,

h_{1}// GL_{s}(ℂ) = h_{1}// SL_{s}(ℂ).

The GL_{s}(ℂ)-action and, so, also the SL_{s}(ℂ)-action on h_{1} is naturally linearized in the line bundle O_{h_{1}}(1). The above quotient is then to be understood as the quotient constructed by Mumford for the linearized SL_{s}(ℂ)-action. Note that, if the center of GL_{s}(ℂ) acts with non-zero weight on

s_{1} := sym_{k}^{1} ⊗H^{0}(X, M^{⊗l} ⊗ L^{⊗k}),

then there are no points which are GL_{s}(ℂ)-semistable in the sense of Mumford. This is why we are working with the action of SL_{s}(ℂ).

Remark 4.7 As a representation of GL_{s}(ℂ), s_{1} is isomorphic to the direct sum of dim_{ℂ}(H^{0}(X, M^{⊗l} ⊗ L^{⊗k})) copies of sym_{k}^{1}. So, if one knows sym_{k}^{1} as a GL_{s}(ℂ)-module, then one may use the techniques described, e.g., in [36] in order to determine the Hitchin space h_{3}. Examples were presented in Section 3.

We expect that (4.6.1) induces a Hitchin map on C_{X/M/L/ϱ}^{δ} for suitable values of δ. So, we formulate the following.

Problem 4.8 Let h_{1}^{ss} ⊂ h_{1} be the SL_{s}(ℂ)-invariant open subset of points that are SL_{s}(ℂ)-semistable in the sense of Mumford, and let

H: 𝔓^{δ-ss} ⟶ P_{X/M/L/ϱ} ⟶^{h_{1}} h_{1}

be the quotient morphism followed by the Hitchin map. Is it true that, for δ ≫ 0,

ℭ^{δ} ⊂ H^{−1}(h_{1}^{ss})?

Even if the answer to the above problem were negative, it would make sense to study the Hitchin map and the Hitchin space. Indeed, in order to construct the moduli space C^{δ} as a GIT quotient of 𝔓^{δ-ss}, we have used a linearization of the group action in some ample line bundle 𝔏_{1}. Furthermore, the pullback 𝔏_{2} of O_{h_{1}}(1) via the equivariant morphism H is also linearized. Relative geometric invariant theory ([27], Theorem 2.1, [33], Proposition 1.2) shows that, for large values of c ∈ ℕ, the set C^{∘} of points that are semistable with respect to the linearization in 𝔏_{1} ⊗ 𝔏_{2}^{⊗c} is contained in H^{−1}(h_{1}^{ss}). The quotient C^{∘} := ℭ^{∘}//(GL_{s}(ℂ) × GL(W)) will then be endowed with a projective Hitchin map

χ_{3} : C^{∘} ⟶ h_{3}.

5 Examples

We will discuss some very basic examples which mainly serve to illustrate properties of the Hitchin map. They pick up the topics of Section 2.3. We will compare them with the corresponding examples of holomorphic triples.

5.1 Symmetric tensors

We let X be a smooth projective curve and L a line bundle on X. Consider a vector bundle E of rank two and a subspace Γ ⊂ H^{0}(X, Sym^{2}(E^{∨}) ⊗ L) of dimension two. The test objects are pairs, consisting of a one dimensional subspace Λ ⊂ Γ together with a positive rational number ν and a sub line bundle N ⊂ E together with a positive rational number μ, corresponding to the weighted flag (0 ⊊ Λ ⊊ Γ, (ν)) and the weighted filtration (0 ⊊ N ⊊ E, (μ)), respectively. We use the computations from Page 18f for the homomorphism φ_{Γ} = (β_{1}, β_{2}): O_{X}^{⊕2} ⟶ Sym^{2}(E^{∨}) ⊗ L. For Condition a) in the definition of asymptotic (semi)stability, we use the discussion leading to Proposition 3.7.

Remark 5.1 There is an important point to observe when drawing the conclusions from the computations. A homomorphism β: Sym^{2}(E) ⟶ L is given by a symmetric (2 × 2)-matrix

m_{β} = ( \begin{matrix}s_{1} & s_{2} \\ s_{2} & s_{3} \end{matrix} )

with s_{1}, s_{2}, s_{3} ∈ H^{0}(X, L). Suppose that s_{1}, s_{2} ∈ H^{0}(X, L) are linearly independent sections and choose

m_{β_{1}} = ( \begin{matrix}s_{1} & 0 \\ 0 & s_{1} \end{matrix} ) and m_{β_{2}} = ( \begin{matrix}0 & s_{2} \\ s_{2} & 0 \end{matrix} ) .

For λ_{1}, λ_{2} ∈ ℂ, not both zero, the discriminant of λ_{1} · m_{β_{1}} + λ_{2} · m_{β_{2}} is

λ_{1}^{2} · s_{1}^{2} − λ_{2}^{2} · s_{2}^{2} ≢ 0.

Therefore, every element in Γ := ⟨ β_{1}, β_{2} ⟩ has generic rank two, in contrast to the observation in the proof of Proposition 3.7.

The first property that we need is that β_{1} and β_{2} are linearly independent over ℂ. This is ensured by the fact that Γ is a two-dimensional subspace. The second condition requires that there is no sub line bundle N ⊂ E which is isotropic for both β_{1} and β_{2}, i.e., isotropic for all β ∈ Γ, and occurs as the radical of an element β ∈ Γ \ {0}. Finally, we point out that the one parameter subgroups with weight zero described in Remark 3.9 are associated with an element q ∈ ⟨ q_{1}, q_{2} ⟩ \ {0} and the radical Λ ⊂ ℂ^{2} of q. In addition, the case μ = 0 may occur here and corresponds to a sub line bundle N ⊂ E which is isotropic for all β ∈ Γ. Our discussion yields the following criterion.

Lemma 5.2 The coherent system (E, Γ) is asymptotically (semi)stable if and only if

a) there is no sub line bundle N ⊂ E which is isotropic for all β ∈ Γ and agrees with the radical of an element β ∈ Γ \ {0}, and

b) for every sub line bundle N ⊂ E which is either isotropic for all β ∈ Γ or occurs as the radical of one element β ∈ Γ \ {0}, the inequality

deg(N)(≤)μ(E) = \frac{deg(E)}{2}

is satisfied.

Example 5.3 In the example presented in Remark 5.1, the radical of every element in Γ is trivial and there is no sub line bundle which is isotropic for all β ∈ Γ. This means that (E, Γ) is asymptotically stable.

Remark 5.4 i) The Hitchin space h_{1} for pairs(E, φ: O_{X}^{⊕2} → Sym^{2}(E^{∨}) ⊗ L) is

P (H^{0}(X, (M^{∨})^{⊗4} ⊗ L^{⊗2})^{⊕3}).

The three summands correspond to the invariants Δ_{1}, Δ_{2}, and Γ, respectively. ii) Let (E, Γ) be a coherent system which fails to meet Condition a) in Lemma 5.2. Assume that the sub line bundle N ⊂ E is the radical of β_{1} and isotropic for β_{2}. As usual, let 𝕂 = ℂ(X) be the function field of X. We choose a trivialization E_{|{η}} ≅ 𝕂^{2} of E at the generic point of X and apply a suitable transformation from SL_{2}(𝕂), such that, up to a factor in 𝕂^{⋆}, we get

m_{β_{1}} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) and m_{β_{2}} = ( \begin{matrix}1 & ⋆ \\ ⋆ & 0 \end{matrix} ) .

If ⋆ is zero, then the pair (E, φ_{Γ} : O_{X}^{⊕2} ⟶ Sym^{2}(E^{∨}) ⊗ L) is not generically semistable (see Remark 2.2), and the invariants Δ_{1}, Δ_{2}, and Γ all vanish at (E, φ_{Γ}), so that the Hitchin map χ_{1} is not defined at (E, φ_{Γ}). If ⋆ is different from zero, then the Hitchin map χ_{1} is defined at (E, φ_{Γ}) and lies in the summand corresponding to Δ_{1}. Using the arguments from Section 3, it is readily checked that these elements are nullforms for the action of SL_{2}(ℂ). So, the Hitchin map χ_{3} is not defined at (E, Γ).

When looking at holomorphic triples φ: F ⟶ Sym^{2}(E^{∨}) ⊗ L, the results of the section on symmetric tensors in Section 3 give the notion of generic semistability. By Proposition 3.4, the invariant ring is generated by a single invariant of degree four. So, according to (2.13.1), the base for the Hitchin map for holomorphic triples (F, E, φ) with rk(F) = rk(E) = 2, det(E) ≅ M, and det(F) ≅ O_{X} is

h_{2} = P (H^{0}(X, (M^{∨})^{⊗8} ⊗ L^{⊗4})).

In contrast, the Hitchin space for coherent systems (E, Γ) is the SL_{2}(ℂ)-quotient of

h_{1} = P (Sym^{2}(ℂ^{2}) ⊗ H^{0}(X, (M^{∨})^{⊗4} ⊗ L^{⊗2})).

The following example illustrates the difference between the notions of asymptotic semistability for the two species of objects.

Example 5.5 Let us choose X = ℙ^{1}, L = O_{X}(1), and E = O_{X}^{⊕2}. We let (s_{1}, s_{2}) be the usual basis of H^{0}(X, L). Pick

m_{β_{1}} = ( \begin{matrix}s_{1} & s_{1} \\ s_{1} & 0 \end{matrix} ) and m_{β_{2}} = ( \begin{matrix}s_{2} & s_{2} \\ s_{2} & 0 \end{matrix} ) .

The holomorphic triple given by these data is not generically semistable, but the coherent system (E, Γ) is asymptotically semistable. We compute

Δ_{1}(β_{1}, β_{2}) = −s_{1}^{2}, Δ_{2}(β_{1}, β_{2}) = −s_{2}^{2}, and Γ(β_{1}, β_{2}) = −s_{1} · s_{2}.

Using (s_{1}^{2}, s_{2}^{2}, s_{1} ·s_{2}) as a basis of H^{0}(X, L^{⊗2}) and identifying (Δ_{1}, Δ_{2}, Γ) with the basis (e_{1}^{2}, e_{2}^{2}, e_{1} · e_{2}) for Sym^{2}(ℂ^{2}), as in the proof of Proposition 3.4, we see that we have

χ_{1}([E, φ_{Γ}]) = [(e_{1}^{2}, 0, 0), (0, e_{2}^{2}, 0), (0, 0, e_{1} · e_{2})] ∈ h_{1}.

Since [e_{1} · e_{2}] ∈ P (Sym^{2}(ℂ^{2})) is semistable for the action of SL_{2}(ℂ), the point χ_{1}([E, φ_{Γ}]) is semistable for the action of SL_{2}(ℂ), and the Hitchin map χ_{3} is defined at [E, Γ].

5.2 Classical coherent systems

Let X be a smooth projective curve and L a line bundle on X. Here, we look at pairs (E, Γ) which consist of a holomorphic vector bundle E on X and a subspace Γ ⊂ H^{0}(X, E ⊗ L).

Remark 5.6 Replacing E by E ⊗ L, we may actually assume L = O_{X}. The reader may check that the condition of δ-(semi)stability is not affected by this manipulation. In this way, we may comply with the usual conventions in the literature. We will not make use of this in the example below to keep the notation closer to the general setting.

The condition of (semi)stability introduced by Raghavendra and Vishwanath [26] says that, for δ ∈ ℚ_{>0}, a coherent system (E, Γ) is δ-(semi)stable, if, for every subbundle 0 ⊊ G ⊊ E and every subspace Γ′ ⊂ Γ ∩ H^{0}(X, G ⊗ L), the inequality

\frac{deg(G) + δ · dim_{ℂ}(Γ′)}{rk(G)}(≤)\frac{deg(E) + δ · dim_{ℂ}(Γ)}{rk(E)}

is satisfied. We checked that the general notion of δ-(semi)stability for coherent ϱ-pairs that we defined in [34], Section 1.4, specializes to this one, for ϱ = id: GL_{r}(ℂ) ⟶ GL(ℂ^{r}) (see [34], Example 1.4.1). We also note that, in this special case, there is no restriction on the stability parameter in the construction from [34].

Problem 5.7 Does the notion of δ-semistability for coherent ϱ-systems defined in Section 4.2 specialize to the above condition for ϱ = id?

This will probably follow with the help of the arguments used in [29], Section 3.

In the sequel, we will work relative to X = ℙ^{1} and choose L := O_{X}(1). Pick a basis (s_{1}, s_{2}) for H^{0}(X, L). We look at the surjection φ: O_{X}^{⊕3} ⟶ E, E := O_{X}^{⊕2}, given by the matrix

m := ( \begin{matrix}s_{1} & 0 & s_{1} + s_{2} \\ 0 & s_{2} & s_{1} + s_{2} \end{matrix} ) .

The three minors are s_{1} · s_{2}, s_{1}^{2} + s_{1} · s_{2}, and s_{1} · s_{2} + s_{2}^{2}. They form a basis for H^{0}(X, L^{⊗2}). (The fact that they don’t have a common zero shows that φ is surjective.) Define Γ ⊂ H^{0}(X, E ⊗ L) as the subspace that is spanned by the columns of the matrix m. We claim that (E, Γ) is δ-stable, for all δ ∈ ℚ_{>0}. This amounts to the fact that there is no sub line bundle N ⊂ E with dim_{ℂ}(Γ_{N}) = 2, Γ_{N} := Γ ∩ H^{0}(X, N ⊗ L). For such a line bundle, we would find the commutative diagram

The snake lemma implies that (E/N) ⊗ L is globally generated by a single section, i.e., (E/N) ⊗ L ≅ O_{X}. The fact that (E/N) ⊗ L is a quotient of E ⊗ L implies that (E/N) ⊗ L ≅ O_{X}(k) with k ≥ 1. This is a contradiction.

Remark 5.8 This example and more general versions can be found in [4], Section 5.4, and [18], Example 6.6.

The Hitchin space is the SL_{3}(ℂ)-quotient of

P (V^{∨} ⊗ H^{0}(X, L)) ≅ P ((V^{∨})^{⊕3}), V := ℂ^{3},

i.e.,

h_{3} ≅ ⋆.

The Hitchin map χ_{3} is defined at the isomorphy class of (E, Γ). According to [18], Example 6.6, the moduli space contains just [E, Γ]. So, the Hitchin map is, in fact, an isomorphism.

Remark 5.9 i) If we replace O_{X}(1) by O_{X}(k) with k ≥ 2, we get positive dimensional Hitchin spaces, by Theorem 3.11. Using an embedding O_{X}(1) ⟶ O_{X}(k), we can extend the above example. We will still get a stable coherent system. In that case, Γ ⊗ O_{X} ⟶ E ⊗ O_{X}(k) is only generically surjective.

ii) Let 𝕂 ≅ ℂ(t) be the function field of X. There are no points in the projective space P (Hom(𝕂^{3}, 𝕂^{2})) which are semistable with respect to the action of SL_{3}(𝕂) × SL_{2}(𝕂). This means that for fixed degrees e and d and δ ≫ 0, there are no δ-semistable holomorphic triples φ: F ⟶ E with rk(F) = 3, deg(F) = e, rk(E) = 2, and deg(E) = 2 (compare Lemma 2.12 and Remark 2.13).

Acknowledgements

Leticia Brambila-Paz brought the problem of studying coherent Higgs systems to my attention. It was formulated by Bradlow, Brambila-Paz, García-Prada, and Gothen and served as starting point for [34] and this note. Bradlow, Brambila-Paz, García-Prada, and Gothen and Brambila-Paz and Edgar I. Castañeda-Gon-zález are carrying out independent research on the case of coherent Higgs systems. Their results will appear in forthcoming papers. I thank Peter Newstead for the references [14] and [38] and Ángel Muñoz Castañeda for the translation of the title and the abstract into Spanish. I am also grateful to Luisa González Campos and Ronald Zuñiga Rojas for their hospitality during my visit to Costa Rica for the XXII SIMMAC.

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