Home Spin(8,C)-Higgs pairs over a compact Riemann surface
Article Open Access

Spin(8,C)-Higgs pairs over a compact Riemann surface

  • Álvaro Antón-Sancho EMAIL logo
Published/Copyright: November 16, 2023
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

Let X be a compact Riemann surface of genus g ≥ 2 , G be a semisimple complex Lie group and ρ : G → GL ( V ) be a complex representation of G . Given a principal G -bundle E over X , a vector bundle E ( V ) whose typical fiber is a copy of V is induced. A ( G , ρ ) -Higgs pair is a pair ( E , φ ) , where E is a principal G -bundle over X and φ is a holomorphic global section of E ( V ) ⊗ L , L being a fixed line bundle over X . In this work, Higgs pairs of this type are considered for G = Spin ( 8 , C ) and the three irreducible eight-dimensional complex representations which Spin ( 8 , C ) admits. In particular, the reduced notions of stability, semistability, and polystability for these specific Higgs pairs are given, and it is proved that the corresponding moduli spaces are isomorphic, and a precise expression for the stable and not simple Higgs pairs associated with one of the three announced representations of Spin ( 8 , C ) is described.

Keywords: Higgs pair; Higgs bundle; spin; Riemann surface; stability
MSC 2010: 14H10; 14H60; 57R57; 53C10

1 Introduction

Let X be a compact Riemann surface of genus g ≥ 2 and G be a semisimple complex Lie group that is equipped with a complex representation ρ : G → GL ( V ) . Every principal G -bundle E over X induces a vector bundle E ( V ) whose typical fiber is a copy of V and which is defined from the direct product E × V by identifying ( e , v ) ∼ ( e g , ρ ( g − 1 ) ( v ) ) for all e ∈ E , v ∈ V and g ∈ G . A ( G , ρ ) -Higgs pair (or simply G -Higgs pair, Higgs pair, or pair, when there is not possibility of doubt) over X is defined to be a pair ( E , φ ) , where E is a principal G -bundle over X and φ is a holomorphic global section of the vector bundle E ( V ) ⊗ L , the bundle L being a fixed line bundle over X (Definition 1). When the representation ρ is the adjoint one and V coincides with the Lie algebra g of G , this concept corresponds to that of G -Higgs bundle over X . Higgs bundles were introduced by Hitchin [1,2] for G = SL ( 2 , C ) and studied in the general case of semisimple (in fact, reductive) Lie groups by Simpson [3,4], who provided notions of stability and polystability aimed at constructing the moduli space of G -Higgs bundles over X . Since these foundational articles were published, moduli spaces of G -Higgs bundles over a compact Riemann surface have been intensely studied from different points of view, including automorphisms and subvarieties of the moduli space [5], stratifications [6,7], representations of the Riemann surface X [8], or Langlands program [9]. The concept of ( G , ρ ) -Higgs pair was first introduced in Banfield [10] as a natural generalization of that of G -Higgs bundle and has been studied in recent years because they appear in certain geometric contexts, for example as fixed points of certain automorphisms of moduli spaces of Higgs bundles [11]. The latter concept extends, in turn, that of the principal bundle over a curve X , whose geometry and topology are also intensively studied, in part along lines analogous to those followed with Higgs pairs and Higgs bundles, such as the study of automorphisms of the corresponding moduli space [12].

In this work, Higgs pairs over X are studied for the structure group Spin ( 8 , C ) , the universal cover of SO ( 8 , C ) and PSO ( 8 , C ) , and associated with the three eight-dimensional irreducible complex representations that Spin ( 8 , C ) admits, which are defined in (5), (6), and (7) and will be denoted by ρ , ρ + , and ρ − . The group Spin ( 8 , C ) is the only simple complex structure group that admits an order 3 outer automorphism, called triality. This singular fact makes Spin ( 8 , C ) a group with interesting geometric peculiarities, which has been the subject of great interest in the literature. For example, fixed point subvarieties have been specifically studied for automorphisms of the moduli space of principal Spin ( 8 , C ) -bundles induced by the action of outer automorphismsm of Spin ( 8 , C ) [13]. After that, it was proved that the fixed points for the action of the triality automorphism on the moduli space of Spin ( 8 , C ) -Higgs bundles can be described through certain Higgs pairs with structure group isomorphic to G 2 and PSL ( 3 , C ) [5]. In addition, further objects are related to Spin ( 8 , C ) -bundles, such as Galois Spin ( 8 , C ) -bundles [14], which are essentially fixed points of certain S 3 -action defined in the moduli space of principal Spin ( 8 , C ) -bundles over a curve and whose moduli space was constructed by Oxbury and Ramanan [14]. The interest in Spin ( 8 , C ) -Higgs pairs responds, therefore, to the interest in G -Higgs pairs in general (because they appear in many situations, such as in descriptions of fixed point of automorphisms) and in the interest in Spin ( 8 , C ) -bundles over curves in particular.

The three representations ρ , ρ + and ρ − of Spin ( 8 , C ) do not descend to induce representations of either SO ( 8 , C ) or PSO ( 8 , C ) , which is why this research is focused on the group Spin ( 8 , C ) . Following the general theory on ( G , ρ ) -Higgs pairs developed in the studies by Garcia-Prada et al. [15,16], in Propositions 4.1, 4.2, and 4.3, the reduced notions of stability, semistability, and polystability for that three types of Spin ( 8 , C ) -Higgs pairs are described. The triality automorphism of Spin ( 8 , C ) gives isomorphisms between the three considered representations of Spin ( 8 , C ) , which will be proved in Corollary 4.1 that induce isomorphisms between the moduli spaces of polystable ( Spin ( 8 , C ) , ρ ) , ( Spin ( 8 , C ) , ρ + ) , and ( Spin ( 8 , C ) , ρ − ) -Higgs pairs over X . After that, an application of this whole study is provided. Specifically, a description of the stable but not simple ( Spin ( 8 , C ) , ρ ) -Higgs pairs over X is provided, in the spirit of previous works devoted to principal or vector bundles [17]. Given any semisimple complex Lie group G , the deformation theory of G -Higgs bundles makes it possible to describe the tangent space of the moduli space of these objects at smooth elements in terms of certain hypercohomology groups. In particular, the smooth points of the moduli space of G -Higgs bundles over X can be identified as the stable and simple Higgs bundles. Taking advantage of this theory, in the study by Garcia-Prada et al. [16], a description of the singular points of the moduli space of G -Higgs bundles is given. That deformation theory is not directly adaptable, as far it has been studied, to the general case of Higgs pairs. However, in Theorem 5.1, it is proved the following description of the stable and not simple ( Spin ( 8 , C ) , ρ ) -Higgs pairs over X .

Theorem

Let ( E , φ ) be a stable and non-simple ( Spin ( 8 , C ) , ρ ) -Higgs pair over X, where ρ : Spin ( 8 , C ) → SO ( 8 , C ) ↪ GL ( 8 , C ) is given by the double cover Spin ( 8 , C ) → SO ( 8 , C ) . Let E SO be the principal SO ( 8 , C ) -bundle over X associated with E through ρ . Then the underlying vector bundle of E SO is isomorphic to one of the following vector bundles:

  1. L k ⊕ L 8 − k for k = 0 , 1 , 2 , 3 , 4 ;

  2. F 2 r ⊕ L k ⊕ L 8 − 2 r − k for r = 1 , 2 , 3 and k = 0 , 1 , … , 4 − r ;

  3. F 2 r ⊕ F 2 s ⊕ L k ⊕ L 8 − 2 r − 2 s − k for r = 1 , 2 , 3 , s = r , … , 4 − r , and k = 0 , … , 4 − r − s ;

  4. F 2 ⊕ F 2 ⊕ F 2 r ⊕ L k ⊕ L 4 − 2 r − k for r = 1 , 2 and k = 2 − r , … , 4 − 2 r ,

where F j is an SL ( j , C ) -bundle and L j is an SO ( j , C ) -bundle for all j ≥ 1 , F 0 = 0 , L 0 = 0 , and L 1 = O .

This article is organized as follows. In Section 2, the concept of ( G , ρ ) -Higgs pair over a compact Riemann surface X associated with a semisimple complex Lie group G and a complex representation ρ of it is defined, and the notions of stability, semistability, and polystability for Higgs pairs are presented to establish the precise formulation of the Hitchin-Kobayashi correspondence for Higgs pairs. Section 3 is devoted to presenting the main properties of the groups Spin ( 2 n , C ) for n ≥ 2 , focusing on Spin ( 8 , C ) . The interest in Spin groups with even rank other than 8 is that they will naturally appear in the description of stable and not simple ( Spin ( 8 , C ) , ρ ) -Higgs pairs made in Theorem 5.1. The reduced notions of stability and polystability for Higgs pairs with structure group Spin ( 8 , C ) and associated with the representations ρ , ρ + , and ρ − introduced earlier are stated and proved in Section 4, where it is also proved that the Higgs pairs corresponding to the three representations mentioned earlier are in bijective correspondence through a correspondence that preserves the polystability condition. Finally, in Section 5, a precise description of the stable and not simple ( Spin ( 8 , C ) , ρ ) -Higgs pairs over X is given.

2 Stability and polystability notions for Higgs pairs

Let X be a compact Riemann surface of genus g ≥ 2 , G be a semisimple complex Lie group, and ρ : G → GL ( V ) be a complex representation of G . In this section, the concept of ( G , ρ ) -Higgs pair over X is introduced and reduced notions of stability, semistability, and polystability are provided for such pairs. The survey material presented in this section has been adapted from the study by Garcia-Prada et al. [16].

Definition 1

Let G be a semisimple complex Lie group and ρ : G → GL ( V ) be a complex representation of G . A ( G , ρ ) -Higgs pair over X is a pair ( E , φ ) , where E is a principal G -bundle over X and φ ∈ H 0 ( X , E ( V ) ⊗ L ) , E ( V ) being the vector bundle obtained by making the quotient of E × V where the identification ( e , v ) ∼ ( e g , ρ ( g − 1 ) ( v ) ) is made for all g ∈ G and all ( e , v ) ∈ E × V , and L being a fixed line bundle over X .

Observe that the notion of ( G , ρ ) -Higgs pair extends that of G -Higgs bundle, for which the representation ρ is the adjoint one and V is the underlying vector bundle of the Lie algebra g of G .

Let G be a semisimple complex Lie group with Lie algebra g . Having fixed a maximal compact connected Lie subgroup H of G with Lie algebra h and such that h C = g , and denoting by Δ the set of simple roots of g , and in the stud by Garcia-Prada et al. [16, Section 2.5], it is proved that the proper subsets of Δ and the parabolic subalgebras of g (hence the parabolic subgroups of G ) are in bijective correspondence. Given any parabolic subgroup P of G and any antidominant character χ of G , which belongs to the dual c * of the Cartan subalgebra c of g , the Killing form induces an element s χ ∈ c , which is in fact an element of i h . Denote by P s χ the maximal parabolic subgroup induced by s χ and by L s χ a choice of a Levi subgroup of P s χ .

Let now ρ : G → GL ( V ) be a complex representation of G . Given a parabolic subgroup P of G and an antidominant character χ of P , the following subspaces of V are defined [11]:

(1) V χ − = { v ∈ V : ρ ( e t s χ ) v is bounded as t → ∞ } , V χ 0 = { v ∈ V : ρ ( e t s χ ) v = v ∀ t } .

The subspaces V χ − and V χ 0 thus defined are invariant under the action of P s χ and L s χ , respectively, on them.

Definition 2

Let G be a semisimple complex Lie group, ρ : G → GL ( V ) be a complex representation of G , and ( E , φ ) be a ( G , ρ ) -Higgs pair over X . Then ( E , φ ) is stable (resp. semistable) if for every parabolic subgroup P of G , every antidominant character χ of P , and every reduction of structure group E P of E to P such that φ takes values in E P ( V χ − ) ⊗ L , where V χ − is defined in (1), and it is satisfied that deg χ * E P > 0 (resp. deg χ * E P ≥ 0 ).

The ( G , ρ ) -Higgs pair ( E , φ ) is polystable if it is semistable, and for every parabolic subgroup P of G , every antidominant character χ of P , and every reduction of structure group E P of E to P such that φ takes values in E P ( V χ − ) ⊗ L , where V χ − is defined in (1), and such that deg χ * E P = 0 , there exists a reduction of structure group E L of E P to a Levi subgroup L of P such that φ takes values in E L ( V χ 0 ) ⊗ L , where V χ 0 is also defined in (1).

The precise notions of stability and polystability of Higgs pairs, which extend that of Higgs bundles, were given by García-Prada et al. [16] to obtain a bijective correspondence between polystable Higgs pairs and solutions to the Hermite-Einstein equations.

The condition of polystability of a ( G , ρ ) -Higgs pair ( E , φ ) when applied to a faithful representation ρ can be expressed in terms of filtrations of certain vector bundle associated with E through other fixed representation ρ G of G satisfying the hypothesis stated in the following result, which is derived from the study by Garcia-Prada et al. [16, Lemma 2.12]. The idea is applying this in the cases in which G is naturally embedded in some GL ( n , C ) , for example, SL ( n , C ) , where ρ G is the natural embedding.

Proposition 2.1

Let G be a semisimple complex Lie group, ρ : G → GL ( V ) be a faithful complex representation of G, and ( E , φ ) be a ( G , ρ ) -Higgs pair over X . Suppose that there exists a representation ρ G : G → GL ( W ) , with W ≅ C n for some n ∈ N , such that for any a , b ∈ ( Ker d ρ G ) ⊥ it is satisfied that ⟨ a , b ⟩ = Tr d ρ G ( a ) d ρ G ( b ) , where the product is the Euclidean product of W . Denote E = E ( W ) . Then

  1. The ( G , ρ ) -Higgs pair ( E , φ ) is semistable if for every parabolic subgroup P of G, any antidominant character χ of P , and any filtration E 0 = 0 ⊊ E 1 ⊊ ⋯ ⊊ E k = E induced by a reduction of structure group of E to P and such that φ takes values in the space V χ − defined in (1) in each fiber over X, it is satisfied that the degree of the filtration, defined by

    (2) λ k deg E + ∑ j = 1 k − 1 ( λ j − λ j + 1 ) deg E j ,

    is greater than or equal to 0, where λ 1 < ⋯ < λ k are the eigenvalues of d ρ ( s χ ) .

  2. The ( G , ρ ) -Higgs pair ( E , φ ) is polystable if it is semistable, and there exists a parabolic subgroup P of G and an antidominant character χ of P such that E admits a decomposition of the form E = ⊕ j = 1 k E j ⁄ E j − 1 into vector subbundles, where E 0 = 0 and E j ⁄ E j − 1 is the λ j -eigenspace of d ρ ( s χ ) for all j = 1 , … , k , the degree defined in (2) equals 0, and φ takes values, in each fiber over X, in the space V χ 0 defined in (1).

The Hitchin-Kobayashi correspondence for Higgs pairs will be now introduced. This was first formulated and proved by Hitchin [2] for the case of rank 2 Higgs bundles and was generalized by Simpson [3,4] for Higgs bundles whose structure group is any semisimple complex Lie group. The version presented in this work, developed in the study by Garcia-Prada et al. [16], covers the case of Higgs pairs, which are the objects of interest. Given a semisimple complex Lie group G and any complex representation ρ : G → GL ( V ) , and having fixed a maximal compact subgroup H of G , a Hermitian structure h on V , and a Hermitian metric h L on the line bundle L over X , whose curvature will be denoted by F L , let ρ H : H → U ( V ) be the unitary representation of H obtained by restriction of ρ to H . Given also a ( G , ρ ) -Higgs pair ( E , φ ) as in Definition 1, the vector bundle E ( V ) admits a Hermitian metric induced by that of V and the same is true for the vector bundle E H ( V ) , which is canonically isomorphic to E ( V ) , where E H is any reduction of structure group of E to H . Let F H be the curvature on E H ( V ) , which corresponds to the Chern connection. From the fact that H 0 ( ( X , End ( V ) ⊗ L ) * ) = H 0 ( X , End ( V ) * ) = H 0 ( X , E ( u ( V ) ) * ) , the existence of a skew-symmetric element φ ⊗ φ * h , h L of H 0 ( X , E ( u ( V ) ) * ) is deduced. Define

(3) μ ( φ ) = ρ H * − i 2 φ ⊗ φ * h , h L ,

which may be understood as an element of H 0 ( X , E H ( h ) ) , since h ≅ h * and d ρ H * induces an isomorphism E ( u ( V ) ) * ≅ E H ( h ) * . Notice that, throughout this explanation, the same symbols have been used to denote the C ∞ -objects and their holomorphic structures, by a slight abuse of notation.

Theorem 2.1

Let G be a semisimple complex Lie group, ρ : G → GL ( V ) be a complex representation of G, and ( E , φ ) be a ( G , ρ ) -Higgs pair over X. Then ( E , φ ) is polystable if and only if E admits a reduction of structure group E H to a maximal compact subgroup H of G such that

∧ ( F H + F L ) + μ ( φ ) = 0 ,

where μ ( φ ) is defined in (3) and ∧ denotes the adjoint wedging with the volume form on X .

The following result, which is where the interest in the Hitchin-Kobayashi correspondence lies, derives directly from Theorem 2.1.

Corollary 2.1

Let G be a semisimple complex Lie group, ρ : G → GL ( V ) be a complex representation of G, G ′ be a subgroup of G, and ρ G ′ be the restriction of ρ to G ′ . Let ( E , φ ) be a polystable ( G , ρ ) -Higgs pair over X and let E G ′ be a reduction of structure group of E to G ′ such that φ takes values in E G ′ ( V ) ⊗ L . Then the ( G ′ , ρ G ′ ) -Higgs pair ( E G ′ , φ ) over X is polystable.

3 The groups Spin ( 8 , C ) and Spin ( 2 n , C )

It will now be considered the simple complex Lie group Spin ( 8 , C ) , whose Lie algebra is so ( 8 , C ) , of type D 4 . The group Spin ( 8 , C ) is the simply connected complex group with Lie algebra so ( 8 , C ) , and it is a double cover of SO ( 8 , C ) , and a cover of order 4 of the projective group PSO ( 8 , C ) , the centerless group with Lie algebra so ( 8 , C ) . Let Z be the center of Spin ( 8 , C ) , which is isomorphic to Z 2 × Z 2 . The group Out ( Spin ( 8 , C ) ) of outer automorphisms of Spin ( 8 , C ) acts faithfully on Z , from which follows the existence of a nontrivial injective homomorphism Out ( Spin ( 8 , C ) ) → S ( Z \ { 1 } ) . Since the last group is isomorphic to the group S 3 of permutations of three elements and Out ( Spin ( 8 , C ) ) is isomorphic to the group of symmetries of the Dynkin diagram of D 4 , which is also a copy of S 3 , it is deduced that the homomorphism Out ( Spin ( 8 , C ) ) → S ( Z \ { 1 } ) is actually an isomorphism of groups [18, Section 1].

The three elements of Z \ { 1 } correspond, in the following sense, to the three irreducible complex representations of dimension 8 that Spin ( 8 , C ) admits: each one of these representations leaves invariant exactly one element of Z \ { 1 } and permutes the other two. The set of nontrivial outer involutions of Spin ( 8 , C ) , the set of eight-dimensional irreducible complex representations of G , and the set Z \ { 1 } are in bijective correspondence in a way that each z ∈ Z \ { 1 } admits exactly one outer involution σ of Spin ( 8 , C ) such that every representative of order 2 of σ in Aut ( Spin ( 8 , C ) ) leaves z invariant, and exactly one eight-dimensional irreducible complex representation ρ of Spin ( 8 , C ) such that ρ ( z ) = 1 . This representation ρ thus defined actually descends to a representation of Spin ( 8 , C ) ⁄ ⟨ 1 , z ⟩ ≅ SO ( 8 , C ) . The triality automorphism τ of Spin ( 8 , C ) is a choice of a nontrivial outer automorphism of Spin ( 8 , C ) of order 3 whose effect on Z turns out to be to permute the three nontrivial elements of Z without leaving fixed points. The triality automorphism τ interchanges then the three eight-dimensional irreducible complex representations of Spin ( 8 , C ) , in the sense that a choice of an order 3 representative T of τ in Aut ( Spin ( 8 , C ) ) acts as an order 3 permutation on the set of the aforementioned three representations [18].

Notice that the triality automorphism τ defines an outer automorphism of order 3 of PSO ( 8 , C ) , but it does not define an outer automorphism of SO ( 8 , C ) , because for that to happen, a representative of τ in Aut ( Spin ( 8 , C ) ) should leave invariant the center of SO ( 8 , C ) , which is not possible. On the other hand, the three announced eight-dimensional complex representations of Spin ( 8 , C ) do not descend to give rise to representations of PSO ( 8 , C ) (specifically, they descend to projective representations of PSO ( 8 , C ) ). For these reasons, throughout the article, only Higgs pairs over a compact Riemann surface whose structure group is Spin ( 8 , C ) are considered.

The construction of the three eight-dimensional irreducible complex representations of Spin ( 8 , C ) , which have been considered in the previous paragraphs will now be sketched following the study by Fulton and Harris[19, Chapter 20]. Let V be an eight-dimensional complex vector space equipped with a nondegenerate quadratic form q . Then SO ( 8 , C ) is isomorphic to the group SL ( V , q ) of determinant 1 complex automorphisms of the vector space V , which preserves the quadratic form q . Let

(4) π : Spin ( 8 , C ) → SO ( 8 , C )

be the double covering. The representation ρ of Spin ( 8 , C ) , which will be faithful, is then defined to be the representation

(5) ρ : Spin ( 8 , C ) → π SO ( 8 , C ) ≅ SL ( V , q ) ↪ GL ( V ) ,

where the last map SL ( V , q ) ↪ GL ( V ) is the natural inclusion of SL ( V , q ) in the general linear group associated with V . The different choices of the isomorphism SL ( V , q ) ≅ SO ( 8 , C ) induce of course equivalent representations of Spin ( 8 , C ) .

Consider now the Clifford algebra C ( V , q ) associated with V . It can be understood as a quotient of the tensor algebra of V where the identification v ⊗ v = q ( v ) ⋅ 1 is made for all v ∈ V . Let now W be a maximal isotropic complex subspace of V (recall that a subspace of V is isotropic if q ( v ) = 0 for all v in that subspace). In the study by Fulton and Harris [19, Lemma 20.9], it is proved that C ( V , q ) is isomorphic to gl ( ∧ W ) , where ∧ W = ⊕ k = 0 8 ∧ k W . Let ∧ W = ∧ + W ⊕ ∧ − W be the decomposition of ∧ W into the direct sum of even and odd exterior powers, respectively, and let C ( V , q ) + be the subalgebra of C ( V , q ) of even tensor powers. In [19, Lemma 20.7], it is also proved that the Lie algebra gl ( V , q ) is contained in C ( V , q ) + , which is, by [19, Lemma 20.9], isomorphic to gl ( ∧ + W ) ⊕ gl ( ∧ − W ) , so the Lie algebra so ( 8 , C ) comes with two representations: so ( 8 , C ) → gl ( ∧ + W ) and so ( 8 , C ) → gl ( ∧ − W ) , thus constructed. Since W has complex dimension 4, it is easy to check that dim ∧ + W = dim ∧ − W = 8 . This defines two eight-dimensional faithful complex representations of Spin ( 8 , C ) :

(6) ρ + : Spin ( 8 , C ) → GL ( ∧ + W ) ,

(7) ρ − : Spin ( 8 , C ) → GL ( ∧ − W ) .

These representations are irreducible [19, Proposition 20.15]. The triality automorphism τ interchanges the three representations ρ , ρ + , and ρ − of Spin ( 8 , C ) . Specifically, in the study by Fulton and Harris [19, Section 20.3], it is constructed a complex linear automorphism of vector spaces

(8) J : V ⊕ ∧ + W ⊕ ∧ − W → V ⊕ ∧ + W ⊕ ∧ − W ,

such that J ( V ) = ∧ + W , J ( ∧ + W ) = ∧ − W , and J ( ∧ − W ) = V , which satisfies

(9) J ∘ ρ = ( ρ + ∘ T ) ∘ J ,

(10) J ∘ ρ + = ( ρ − ∘ T ) ∘ J ,

(11) J ∘ ρ − = ( ρ ∘ T ) ∘ J ,

in the sense that J ( ρ ( g ) ( v ) ) = ρ + ( T ( g ) ) ( J ( v ) ) for all g ∈ Spin ( 8 , C ) and all v ∈ V , where T is some order 3 representative of τ in Aut ( Spin ( 8 , C ) ) (and analogous expressions for the other two identities).

For the study of ( Spin ( 8 , C ) , ρ ) -Higgs pairs made in Theorem 5.1, it will be necessary to consider Higgs pairs whose structure group is Spin ( 2 n , C ) for n = 2 , 3 . For any integer number n ≥ 2 , Spin ( 2 n , C ) is the simply connected complex Lie group with Lie algebra so ( 2 n , C ) , and it is the universal cover of the group SO ( 2 n , C ) through the double covering

(12) π 2 n : Spin ( 2 n , C ) → SO ( 2 n , C ) .

Let V 2 n be a complex vector space of dimension 2 n equipped with a holomorphic nondegenerate quadratic form q 2 n . Then the representation so ( 2 n , C ) → gl ( 2 n , C ) given by the natural inclusion lifts to a faithful complex irreducible representation

(13) ρ 2 n : Spin ( 2 n , C ) → GL ( 2 n , C ) ,

which factors through SO ( 2 n , C ) , so it induces a representation

(14) ρ 2 n SO : SO ( 2 n , C ) → GL ( 2 n , C )

given by the inclusion of groups. Observe that, with this notation, π 8 = π and ρ 8 = ρ , where π and ρ were defined in (4) and (5), respectively.

To conclude, it is useful to establish some facts about the parabolic subgroups of Spin ( 2 n , C ) for n ≥ 2 , which will be done following the study by Procesi [20, Chapters 10 and 11]. Parabolic subgroups of Spin ( 2 n , C ) are in bijective correspondence, through the covering map π 2 n defined in (12), with the parabolic subgroups of SO ( 2 n , C ) . For its part, a parabolic subgroup of SO ( 2 n , C ) corresponds to a filtration of V of the form

(15) 0 ⊂ U 1 ⊂ ⋯ ⊂ U k ⊆ U k ⊥ ⊂ ⋯ ⊂ U 1 ⊥ ⊂ V 2 n ,

where U 1 , … , U k are complex vector subspaces of V 2 n isotropic for q 2 n and ⊥ denotes the orthogonality with respect to the nondegenerate symmetric bilinear form induced by q 2 n in V 2 n . The conjugacy class of the parabolic subgroup is univocally determined by the number k and the ranks of the subbundles. The parabolic subgroup is maximal exactly when k = 1 in the preceding filtration, that is, when the induced filtration of V 2 n is of the form

(16) 0 ⊂ U 1 ⊆ U 1 ⊥ ⊂ V 2 n

for some isotropic subbundle U 1 of V 2 n .

4 Stability conditions for Higgs pairs with structure group Spin ( 8 , C )

Let X be a compact Riemann surface of genus g ≥ 2 . In this section, the reduced stability, semistability, and polystability conditions for Higgs pairs over X with structure group Spin ( 8 , C ) and associated with the representations ρ , ρ + , and ρ − of it defined, respectively, in (5), (6), and (7), will be given.

Given any principal Spin ( 8 , C ) -bundle E over X , the covering map π defined in (4) induces a principal SO ( 8 , C ) -bundle E SO given by the image of E by the map

(17) H 1 ( X , Spin ( 8 , C ) ) → H 1 ( X , SO ( 8 , C ) ) , E ↦ E SO ,

which comes from the exact sequence of groups

1 → Z 2 → Spin ( 8 , C ) → SO ( 8 , C ) → 1 .

The principal SO ( 8 , C ) -bundle E SO can be understood as a holomorphic complex vector bundle of rank 8 over X equipped with a global nondegenerate holomorphic symmetric bilinear form ω . For any n ≥ 2 , the covering map π 2 n defined in (12) also defines a map

(18) H 1 ( X , Spin ( 2 n , C ) ) → H 1 ( X , SO ( 2 n , C ) ) , E ↦ E SO ,

where E SO is a holomorphic complex vector bundle of rank 2 n over X equipped with a global nondegenerate holomorphic symmetric bilinear form ω 2 n (of course, ω 8 = ω ).

Let V be an eight-dimensional complex vector space equipped with a nondegenerate quadratic form q and let W be a maximal isotropic vector subspace of V . Let ρ , ρ + , and ρ − be the representations of Spin ( 8 , C ) defined in (5), (6), and (7), whose associated vector spaces are V , ∧ + W , and ∧ − W , respectively. The rank 8 holomorphic vector bundle E ( V ) is also a special orthogonal vector bundle, and the vector bundles E ( ∧ + W ) and E ( ∧ − W ) are also rank 8 holomorphic vector bundles, which are subbundles of ∧ + E SO and ∧ − E SO , the even and odd exterior powers, respectively, of E SO .

From the description in terms of filtrations of the parabolic subgroups of Spin ( 2 n , C ) given in (15) and (16), if follows that a reduction of structure group of a principal Spin ( 2 n , C ) -bundle E over X to a parabolic subgroup of Spin ( 2 n , C ) gives a filtration of E SO of the form

(19) 0 ⊂ F 1 ⊂ ⋯ ⊂ F k ⊆ F k ⊥ ⊂ ⋯ ⊂ F 1 ⊥ ⊂ E SO ,

where F 1 , … , F k are isotropic holomorphic vector subbundles of E SO for 1 ≤ k ≤ n (isotropy and orthogonality are taken with respect to the holomorphic symmetric bilinear form ω 2 n of E SO ). A reduction of structure group of E to a maximal parabolic subgroup of Spin ( 2 n , C ) gives a filtration of the form

(20) 0 ⊂ F ⊆ F ⊥ ⊂ E SO ,

where F is an isotropic subbundle of E SO .

Proposition 4.1

Let ( E , φ ) be a ( Spin ( 2 n , C ) , ρ 2 n ) -Higgs pair over X for the representation ρ 2 n of Spin ( 2 n , C ) defined in (13) for some n ≥ 2 . Let E SO be the principal SO ( 2 n , C ) -bundle defined in (18), and let ω 2 n be its associated global nondegenerate holomorphic symmetric bilinear form. The ( Spin ( 2 n , C ) , ρ 2 n ) -Higgs pair ( E , φ ) is stable (resp. semistable) if for every proper subbundle F of E SO , which is isotropic for ω 2 n and such that φ takes values in F ⊥ ⊗ L , it is satisfied that deg F < 0 (resp. deg F ≤ 0 ). In particular, if ( E , φ ) is a ( Spin ( 8 , C ) , ρ ) -Higgs pair over X for the representation ρ of Spin ( 8 , C ) defined in (5) and ω is the global nondegenerate holomorphic symmetric bilinear form of E SO , then ( E , φ ) is stable (resp. semistable) if for every proper subbundle F of E SO isotropic for ω and such that φ takes values in F ⊥ ⊗ L we have that deg F < 0 (resp. deg F ≤ 0 ).

The ( Spin ( 8 , C ) , ρ ) -Higgs pair ( E , φ ) is polystable if it is semistable and E SO admits a filtration of the form

0 ⊂ F 1 ⊂ ⋯ ⊂ F k ⊆ F k ⊥ ⊂ ⋯ ⊂ F 1 ⊥ ⊂ E SO

described in (19), where F 1 , … , F k are holomorphic vector subbundles of E SO isotropic for ω with deg F 1 = ⋯ = deg F k = 0 , such that φ takes values in F k ⊥ ⁄ F k ⊗ L , and E SO admits the following decomposition into a direct sum of subspaces:

E SO = F 1 ⊕ F 2 ⁄ F 1 ⊕ ⋯ ⊕ F k ⁄ F k − 1 ⊕ F k ⊥ ⁄ F k ⊕ F k − 1 ⊥ ⁄ F k ⊥ ⊕ ⋯ ⊕ E SO ⁄ F 1 ⊥ .

Proof

Let P be a parabolic subgroup of Spin ( 8 , C ) , χ be any antidominant character of P , and s χ be the associated element of i h . If the filtration of E SO is as described in (19), then the element s χ diagonalizes in the form

λ 1 I F 1 ⋱ λ k I F k ⁄ F k − 1 0 I F k ⊥ ⁄ F k − λ k I F k − 1 ⊥ ⁄ F k ⊥ ⋱ − λ 1 I E SO ⁄ F 1 ⊥ ,

where λ 1 , … , λ k ∈ R and λ 1 < ⋯ < λ k < 0 . The degree defined in (2) takes the value

∑ i = 1 k ( λ i − λ i + 1 ) ( deg F i + deg F i ⊥ ) ,

so it is greater than or equal to 0 exactly when deg F i ≤ 0 for all i and it is equal to 0 when deg F i = 0 for all i . On the other hand, the condition for φ of taking values in the space V χ − (resp. in V χ 0 ) defined in (1) clearly requires that φ takes values in F k ⊥ ⊗ L (resp. in F k ⊥ ⁄ F k ⊗ L ). Then the semistability condition requires that deg F i ≤ 0 for all i whenever φ takes values in F k ⊥ ⊗ L and for every filtration as the considered one. Since the satisfaction of this condition for filtrations induced by reductions to a maximal parabolic subgroup as in (20) gives the fulfilment of the condition for every filtration, the first part of the result is proved. For polystability, observe that a reduction of structure group of ( E , φ ) to a Levi subgroup of P gives a decomposition of E SO into a direct sum of vector subbundles of the form

E SO = F 1 ⊕ F 2 ⁄ F 1 ⊕ ⋯ ⊕ F k ⁄ F k − 1 ⊕ F k ⊥ ⁄ F k ⊕ F k − 1 ⊥ ⁄ F k ⊥ ⊕ ⋯ ⊕ E SO ⁄ F 1 ⊥ .

Therefore, ( E , φ ) is polystable if it admits a decomposition into direct sum of vector subbundles as those described in the statement such that deg F j = 0 for all j and φ takes values in F k ⊥ ⁄ F k ⊗ L .□

The reduced notions of stability and polystability for ( Spin ( 8 , C ) , ρ + ) and ( Spin ( 8 , C ) , ρ − ) -Higgs pairs over X , where ρ + and ρ − are the eight-dimensional complex representations of Spin ( 8 , C ) defined in (6) and (7), respectively, will be now described. The proofs of Propositions 4.2 and 4.3 keep many analogous elements with each other and with the proof of Proposition 4.1. However, it has been preferred to keep the details of the proofs, at the risk of being repetitive, to make explicit the differences that exist between the cases considered in the three results.

Proposition 4.2

Let ( E , φ ) be a ( Spin ( 8 , C ) , ρ + ) -Higgs pair over X for the representation ρ + of Spin ( 8 , C ) defined in (6). Let E SO be the principal SO ( 8 , C ) -bundle defined in (17), and let ω be its global nondegenerate holomorphic symmetric bilinear form. The ( Spin ( 8 , C ) , ρ + ) -Higgs pair ( E , φ ) is stable (resp. semistable) if for every proper isotropic subbundle F of E SO such that φ takes values in

( F ∧ E SO + F ⊥ ∧ F ⊥ ) ∐ ( F ∧ F ∧ E SO ∧ E SO + F ∧ F ⊥ ∧ F ⊥ ∧ E SO + F ⊥ ∧ F ⊥ ∧ F ⊥ ∧ F ⊥ ) ,

where the reference to the line bundle L is omitted for clarity, and it is satisfied that deg F < 0 (resp. deg F ≤ 0 ).

The ( Spin ( 8 , C ) , ρ + ) -Higgs pair ( E , φ ) is polystable if one of the following conditions holds (again, the reference to L has been omitted in the vector subbundles where, in each case, φ takes values, for clarity):

  1. There exists a proper isotropic subbundle F of E SO with rk F ≤ 3 such that E SO admits a decomposition of the form

    E SO = F ⊕ F ⊥ ⁄ F ⊕ E SO ⁄ F ⊥

    and φ takes values in

    ( F ∧ E SO ⁄ F ⊥ ⊕ F ⊥ ⁄ F ∧ F ⊥ ⁄ F ) ∐ ( F ∧ F ∧ E SO ⁄ F ⊥ ∧ E SO ⁄ F ⊥ ⊕ F ∧ F ⊥ ⁄ F ∧ F ⊥ ⁄ F ∧ E SO ⁄ F ⊥ ⊕ F ⊥ ⁄ F ∧ F ⊥ ⁄ F ∧ F ⊥ ⁄ F ∧ F ⊥ ⁄ F ) .

  2. There exists a rank 4 isotropic subbundle F of E SO such that E SO admits a decomposition of the form

    E SO = F ⊕ E SO ⁄ F

    and φ takes values in

    ( F ∧ E SO ⁄ F ⊥ ) ∐ ( F ∧ F ∧ E SO ⁄ F ⊥ ∧ E SO ⁄ F ⊥ ) .

  3. There exists a filtration 0 ⊊ F 1 ⊊ F 2 of E SO into isotropic subbundles with 2 ≤ rk F 2 ≤ 3 such that E SO admits a decomposition of the form

    E SO = F 1 ⊕ F 2 ⁄ F 1 ⊕ F 2 ⊥ ⁄ F 2 ⊕ F 1 ⊥ ⁄ F 2 ⊥ ⊕ E SO ⁄ F 1 ⊥

    and φ takes values in

    ( F 1 ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 2 ⊥ ⁄ F 2 ∧ F 2 ⊥ ⁄ F 2 ) ∐ ( F 1 ∧ F 1 ∧ E SO ⁄ F 1 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 2 ⊥ ⁄ F 2 ∧ F 2 ⊥ ⁄ F 2 ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⊥ ⁄ F 2 ∧ F 2 ⊥ ⁄ F 2 ∧ F 2 ⊥ ⁄ F 2 ∧ F 2 ⊥ ⁄ F 2 ⊕ F 2 ⁄ F 1 ∧ F 2 ⊥ ⁄ F 2 ∧ F 2 ⊥ ⁄ F 2 ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ F 1 ⊥ ⁄ F 2 ⊥ ) .

  4. There exists a filtration 0 ⊊ F 1 ⊊ F 2 of E SO into isotropic subbundles with rk F 2 = 4 such that E SO admits a decomposition of the form

    E = F 1 ⊕ F 2 ⁄ F 1 ⊕ F 1 ⊥ ⁄ F 2 ⊥ ⊕ E SO ⁄ F 1 ⊥

    and φ takes values in

    ( F 1 ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ) ∐ ( F 1 ∧ F 1 ∧ E SO ⁄ F 1 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ F 1 ⊥ ⁄ F 2 ⊥ ) .

  1. There exists a filtration 0 ⊊ F 1 ⊊ F 2 ⊊ F 3 of E SO into isotropic subbundles with rk F 3 = 3 such that E SO admits a decomposition of the form

    E SO = F 1 ⊕ F 2 ⁄ F 1 ⊕ F 3 ⁄ F 2 ⊕ F 3 ⊥ ⁄ F 3 ⊕ F 2 ⊥ ⁄ F 3 ⊥ ⊕ F 1 ⊥ ⁄ F 2 ⊥ ⊕ E SO ⁄ F 1 ⊥

    and φ takes values in

    ( F 1 ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ⊕ F 3 ⊥ ⁄ F 3 ∧ F 3 ⊥ ⁄ F 3 ) ∐ ( F 1 ∧ F 1 ∧ E SO ⁄ F 1 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 3 ⊥ ⁄ F 3 ∧ F 3 ⊥ ⁄ F 3 ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 3 ⊥ ⁄ F 3 ∧ F 3 ⊥ ⁄ F 3 ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 3 ⁄ F 2 ∧ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ F 2 ⊥ ⁄ F 3 ⊥ ⊕ F 3 ⁄ F 2 ∧ F 3 ⊥ ⁄ F 3 ∧ F 3 ⊥ ⁄ F 3 ∧ F 2 ⊥ ⁄ F 3 ⊥ ⊕ F 3 ⊥ ⁄ F 3 ∧ F 3 ⊥ ⁄ F 3 ∧ F 3 ⊥ ⁄ F 3 ∧ F 3 ⊥ ⁄ F 3 ) .

  1. There exists a filtration 0 ⊊ F 1 ⊊ F 2 ⊊ F 3 of E SO into isotropic subbundles with rk F 3 = 4 such that E SO admits a decomposition of the form

    E SO = F 1 ⊕ F 2 ⁄ F 1 ⊕ F 3 ⁄ F 2 ⊕ F 2 ⊥ ⁄ F 3 ⊥ ⊕ F 1 ⊥ ⁄ F 2 ⊥ ⊕ E SO ⁄ F 1 ⊥

    and φ takes values in

    ( F 1 ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ) ∐ ( F 1 ∧ F 1 ∧ E SO ⁄ F 1 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 3 ⁄ F 2 ∧ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ F 2 ⊥ ⁄ F 3 ⊥ ) .

  1. There exists a filtration 0 ⊊ F 1 ⊊ F 2 ⊊ F 3 ⊊ F 4 of E SO into isotropic subbundles such that E SO admits a decomposition of the form

    E SO = F 1 ⊕ F 2 ⁄ F 1 ⊕ F 3 ⁄ F 2 ⊕ ⊕ F 4 ⁄ F 3 ⊕ F 3 ⊥ ⁄ F 4 ⊥ ⊕ F 2 ⊥ ⁄ F 3 ⊥ ⊕ F 1 ⊥ ⁄ F 2 ⊥ ⊕ E SO ⁄ F 1 ⊥

    and φ takes values in

    ( F 1 ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ⊕ F 4 ⁄ F 3 ∧ F 3 ⊥ ⁄ F 4 ⊥ ) ∐ ( F 1 ∧ F 1 ∧ E SO ⁄ F 1 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 4 ⁄ F 3 ∧ F 3 ⊥ ⁄ F 4 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 2 ⁄ F 1 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 4 ⁄ F 3 ∧ F 3 ⊥ ⁄ F 4 ⊥ ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 3 ⁄ F 2 ∧ F 3 ⁄ F 2 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ F 2 ⊥ ⁄ F 3 ⊥ ⊕ F 3 ⁄ F 2 ∧ F 4 ⁄ F 3 ∧ F 3 ⊥ ⁄ F 4 ⊥ ∧ F 2 ⊥ ⁄ F 3 ⊥ ⊕ F 4 ⁄ F 3 ∧ F 4 ⁄ F 3 ∧ F 3 ⊥ ⁄ F 4 ⊥ ∧ F 3 ⊥ ⁄ F 4 ⊥ ) .

Proof

Let P be any parabolic subgroup of Spin ( 8 , C ) and χ be any antidominant character of P . With the notation of Section 2, the associated element s χ ∈ i h diagonalizes in the form

λ 1 I F 1 ⋱ λ k I F k ⁄ F k − 1 0 I F k ⊥ ⁄ F k − λ k I F k − 1 ⊥ ⁄ F k ⊥ ⋱ − λ 1 I E SO ⁄ F 1 ⊥ .

for the filtration

0 ⊂ F 1 ⊂ ⋯ ⊂ F k ⊆ F k ⊥ ⊂ ⋯ ⊂ F 1 ⊥ ⊂ E SO

of E SO induced by a restriction of structure group of E SO to P , where λ 1 < ⋯ < λ k < 0 . Since the space V χ defined in (1) is a subspace of the corresponding space induced by a reduction to a maximal parabolic subgroup (that is, when k = 1 ), it is enough to check the semistability condition on filtrations of the form 0 ⊂ F ⊆ F ⊥ ⊂ E SO . In this case, the space V χ − is clearly

( F ∧ E SO + F ⊥ ∧ F ⊥ ) ∐ ( F ∧ F ∧ E SO ∧ E SO + F ∧ F ⊥ ∧ F ⊥ ∧ E SO + F ⊥ ∧ F ⊥ ∧ F ⊥ ∧ F ⊥ ) ,

so the first part of the result is proved. It is easily checked that the kernel of the corresponding endomorphism of ∧ + E SO is exactly the space announced in each one of the seven cases described in the second part of the statement; the case depends on the value of k = 1 , 2 , 3 , 4 and the ranks of the isotropic subspaces involved:

  1. k = 1 and rk F ≤ 3 . In this case, F ⊊ F ⊥ .

  2. k = 1 and rk F = 4 . In this case, F = F ⊥ .

  3. k = 2 and rk F 2 ≤ 3 . In this case, F 2 ⊊ F 2 ⊥ .

  4. k = 2 and rk F 2 = 4 . In this case, F 2 = F 2 ⊥ .

  5. k = 3 and rk F 3 ≤ 3 . In this case, F 3 ⊊ F 3 ⊥ .

  6. k = 3 and rk F 3 = 4 . In this case, F 3 = F 3 ⊥ .

  7. k = 4 . In this case, necessarily F 4 = F 4 ⊥ .

This finally shows the result.□

Proposition 4.3

Let ( E , φ ) be a ( Spin ( 8 , C ) , ρ − ) -Higgs pair over X for the representation ρ − of Spin ( 8 , C ) defined in (7). Let E SO be the principal SO ( 8 , C ) -bundle defined in (17), and let ω be its global nondegenerate holomorphic symmetric bilinear form. The ( Spin ( 8 , C ) , ρ − ) -Higgs pair ( E , φ ) is stable (resp. semistable) if for every proper isotropic subbundle F of E SO such that φ takes values in

( F ⊥ ) ∐ ( F ∧ F ⊥ ∧ E SO ⊕ F ⊥ ∧ F ⊥ ∧ F ⊥ ) ∐ ( F ∧ F ∧ F ⊥ ∧ E SO ∧ E SO ⊕ F ∧ F ⊥ ∧ F ⊥ ∧ F ⊥ ∧ E SO ) ,

where the reference to the line bundle L is omitted for clarity, it is satisfied that deg F < 0 (resp. deg F ≤ 0 ).

The ( Spin ( 8 , C ) , ρ − ) -Higgs pair ( E , φ ) is polystable if one of the following conditions holds (again, the reference to L has been omitted in the vector subbundles where, in each case, φ takes values, for clarity):

  1. There exists a proper isotropic subbundle F of E SO with rk F ≤ 3 such that E SO admits a decomposition of the form

    E SO = F ⊕ F ⊥ ⁄ F ⊕ E SO ⁄ F ⊥

    and φ takes values in

    ( F ⊥ ⁄ F ) ∐ ( F ∧ F ⊥ ⁄ F ∧ E SO ⁄ F ⊥ ⊕ ⋀ 3 F ⊥ ⁄ F ) ∐ ( ⋀ 2 F ∧ F ⊥ ⁄ F ∧ ⋀ 2 E SO ⁄ F ⊥ ⊕ F ∧ ⋀ 3 F ⊥ ⁄ F ∧ E SO ⁄ F ⊥ ) .

  2. There exists a filtration 0 ⊊ F 1 ⊊ F 2 of E SO into isotropic subbundles of E SO with 2 ≤ rk F 2 ≤ 3 such that E SO admits a decomposition of the form

    E SO = F 1 ⊕ F 2 ⁄ F 1 ⊕ F 2 ⊥ ⁄ F 2 ⊕ F 1 ⊥ ⁄ F 2 ⊥ ⊕ E SO ⁄ F 1 ⊥

    and φ takes values in

    ( F 2 ⊥ ⁄ F 2 ) ∐ ( F 1 ∧ F 2 ⊥ ⁄ F 2 ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 2 ⊥ ⁄ F 2 ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ ⋀ 3 F 2 ⊥ ⁄ F 2 ) ∐ ( ⋀ 2 F 1 ∧ F 2 ⊥ ⁄ F 2 ∧ ⋀ 2 E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ ⋀ 3 F 2 ⊥ ⁄ F 2 ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 2 ⁄ F 1 ∧ F 2 ⊥ ⁄ F 2 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ ⋀ 5 F 2 ⊥ ⁄ F 2 ⊕ F 2 ⁄ F 1 ∧ ⋀ 3 F 2 ⊥ ⁄ F 2 ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ ⋀ 2 F 2 ⁄ F 1 ∧ F 2 ⊥ ⁄ F 2 ∧ ⋀ 2 F 1 ⊥ ⁄ F 2 ⊥ ) .

  3. There exists a filtration 0 ⊊ F 1 ⊊ F 2 ⊊ F 3 of E SO into isotropic subbundles of E SO with rk F 3 = 3 such that E SO admits a decomposition of the form

    E SO = F 1 ⊕ F 2 ⁄ F 1 ⊕ F 3 ⁄ F 2 ⊕ F 3 ⊥ ⁄ F 3 ⊕ F 2 ⊥ ⁄ F 3 ⊥ ⊕ F 1 ⊥ ⁄ F 2 ⊥ ⊕ E SO ⁄ F 1 ⊥

    and φ takes values in

    ( F 3 ⊥ ⁄ F 3 ) ∐ ( F 1 ∧ F 3 ⊥ ⁄ F 3 ∧ E SO ⁄ F 1 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 3 ⊥ ⁄ F 3 ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 3 ⁄ F 2 ∧ F 3 ⊥ ⁄ F 3 ∧ F 2 ⊥ ⁄ F 3 ⊥ ⊕ ⋀ 3 F 3 ⊥ ⁄ F 3 ) ∐ ( ⋀ 2 F 1 ∧ F 3 ⊥ ⁄ F 3 ∧ ⋀ 2 E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ F 2 ⁄ F 1 ∧ F 3 ⊥ ⁄ F 3 ∧ F 1 ⊥ ⁄ F 2 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ F 1 ∧ ⋀ 3 F 2 ⊥ ⁄ F 2 ∧ E SO ⁄ F 1 ⊥ ⊕ ⋀ 2 F 2 ⁄ F 1 ∧ F 3 ⊥ ⁄ F 3 ∧ ⋀ 2 F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 2 ⁄ F 1 ∧ F 3 ⁄ F 2 ∧ F 3 ⊥ ⁄ F 3 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 2 ⁄ F 1 ∧ ⋀ 3 F 3 ⊥ ⁄ F 3 ∧ F 1 ⊥ ⁄ F 2 ⊥ ⊕ F 1 ∧ F 3 ⁄ F 2 ∧ F 3 ⊥ ⁄ F 3 ∧ F 2 ⊥ ⁄ F 3 ⊥ ∧ E SO ⁄ F 1 ⊥ ⊕ ⋀ 5 F 3 ⊥ ⁄ F 3 ⊕ F 3 ⁄ F 2 ∧ ⋀ 3 F 3 ⊥ ⁄ F 3 ∧ F 2 ⊥ ⁄ F 3 ⊥ ⊕ ⋀ 2 F 3 ⁄ F 2 ∧ F 3 ⊥ ⁄ F 3 ∧ ⋀ 2 F 2 ⊥ ⁄ F 3 ⊥ ) .

Proof

The same proof of Proposition 4.2, with the necessary formal differences, works here. A few words to the reduced notion of polystability. If P is any parabolic subgroup of Spin ( 8 , C ) , χ is any antidominant character of P and

0 ⊂ F 1 ⊂ ⋯ ⊂ F k ⊆ F k ⊥ ⊂ ⋯ ⊂ F 1 ⊥ ⊂ E SO

is the filtration of E SO induced by a restriction of structure group of E to P , then the kernel of the endomorphism of ∧ − E SO induced by s χ is 0 when rk F k = 4 , since F k ⊥ ⁄ F k = 0 in this case and each wedge product has an odd number of factors, so a factor of type F k ⊥ ⁄ F k must appear in every element of the considered kernel. Then it must be k ≤ 3 and rk F k ≤ 3 . Therefore, the possible cases are the following:

  1. k = 1 and rk F ≤ 3 .

  2. k = 2 and rk F ≤ 3 .

  3. k = 3 and rk F = 3 .

These three cases turn to the cases described in the statement.□

Example

Let E be a rank 8 and trivial determinant vector bundle over X , which admits a globally defined nondegenerate symmetric bilinear form, and whose second Stiefel-Whitney class is 0. Suppose that the maximal isotropic subbundle of E is F . Take L to be the trivial line bundle O over X . Then E can be understood as a principal SO ( 8 , C ) -bundle over X and it lifts to a principal Spin ( 8 , C ) -bundle over X (because the second Stiefel-Whitney class is 0). A Higgs pair associated with the representation ρ defined in (5) is, in this situation, given by E together with a holomorphic global section of E . Similarly, a Higgs pair for the representation ρ + defined in (6) (resp, ρ − defined in (7)) is given by E together with a holomorphic global section of ∧ k F for some even number k with k ≤ 8 (resp. odd number k with k ≤ 7 ). In the case of the representation ρ , the stability of E as a Higgs pair depends on the degree of F and where the global section takes values. If deg F ≥ 0 , then it should be required that the global section does not take values in F ⊥ for the Higgs pair to be stable, by Proposition 4.1.

In Proposition 4.4, the action of the group Out ( G ) of outer automorphisms of G on the set of principal G -bundles over X , introduced in [21, Section 5] for a semisimple complex Lie group G , is considered. Specifically, if σ ∈ Out ( G ) and E is a principal G -bundle over X , σ ( E ) is defined to be the principal G -bundle over X whose total space coincides with that of E and such that the action of G on it derives from that of G in E in the following way: if g ∈ G and e ∈ E , then

(21) e ⋅ g = e S − 1 ( g ) ,

where S is an automorphism of G that represents σ .

Proposition 4.4

Let G be a semisimple complex Lie group, ρ : G → GL ( V ) and ρ ′ : G → GL ( W ) be complex representations of G, σ be an outer automorphism of G, S be a representative of σ in Aut ( G ) , and F : V → W be an isomorphism of vector spaces such that F ∘ ρ = ( ρ ′ ∘ S ) ∘ F . Then F induces a bijective correspondence between polystable ( G , ρ ) -Higgs pairs and ( G , ρ ′ ) -Higgs pairs over X that preserves polystability.

Proof

Under the conditions and the notation of the statement, the map f is defined in the following way: if ( E , φ ) is a polystable ( G , ρ ) -Higgs pair over X , then f ( E , φ ) = ( σ ( E ) , F ( φ ) ) , where σ ( E ) is defined in (21) and F ( φ ) acts on each fiber by taking the image by F of the image of φ , that is, if x ∈ X and φ ( x ) = [ e , v ] ⊗ l , where e ∈ E , v ∈ V and l ∈ L , then F ( φ ( x ) ) = [ e , F ( v ) ] ⊗ L . Notice that this F ( φ ) is well defined, and gives a global section of σ ( E ) ( W ) ⊗ L . To prove this, take any g ∈ G . Since [ e , v ] = [ e g − 1 , ρ ( g ) ( v ) ] and F ∘ ρ = ( ρ ′ ∘ S ) ∘ F , it follows that

[ e g − 1 , F ( ρ ( g ) ( v ) ) ] = [ e g − 1 , ρ ′ ( S ( g ) ) ( F ( v ) ) ] = [ e σ − 1 ( S ( g ) ) − 1 , ρ ′ ( S ( g ) ) ( F ( v ) ) ] ,

so F ( φ ) is well defined, and it takes values in W . In fact, F ( φ ) ∈ H 0 ( X , σ ( E ) ( W ) ) .

If ( E , φ ) is a semistable ( G , ρ ) -Higgs pair over X then F ( E , φ ) is also semistable. To show this, take any parabolic subgroup P of G , any antidominant character χ of P , and a representative S of σ in Aut ( G ) . Then S ( P ) and χ ∘ S − 1 are generic parabolic subgroup of G and antidominant character of S ( P ) , so it suffices to check the semistability condition stated in Definition 2, applied to F ( E , φ ) = ( σ ( E ) , F ( φ ) ) , for S ( P ) and χ ∘ S − 1 . Let σ ( E ) S ( P ) be a reduction of structure group of σ ( E ) to S ( P ) such that F ( φ ) takes values in W χ ∘ S − 1 − . It is then clear that σ − 1 ( σ ( E ) S ( P ) ) defines a reduction of structure group E P of E to P , and this reduction satisfies that φ takes values in V χ − because S ( V χ − ) = W χ ∘ S − 1 − , as an immediate consequence of the expression F ∘ ρ = ( ρ ′ ∘ S ) ∘ F of the hypotheses, the definition of the spaces V χ − and W χ ∘ σ − 1 − given in (1), and the fact that F is linear. Then the reduction E P satisfies that φ takes values in V χ − . Since ( E , φ ) is semistable, then deg χ * E P ≥ 0 . This together with the observation that the line bundles ( χ ∘ S − 1 ) * σ ( E ) S ( P ) and χ * E P over X are isomorphic, concludes that F ( E , φ ) is semistable.

Let now ( E , φ ) be a polystable ( G , ρ ) -Higgs pair over X . Then F ( E , φ ) is itself semistable. Let P and χ be a parabolic subgroup of G and an antidominant character of P such that F ( E , φ ) = ( σ ( E ) , F ( φ ) ) admits a reduction of structure group σ ( E ) S ( P ) to S ( P ) with

F ( φ ) ∈ H 0 ( X , σ ( E ) S ( P ) ( W χ ∘ S − 1 − ) ⊗ L )

and deg ( χ ∘ S − 1 ) * σ ( E ) S ( P ) = 0 . Then, as mentioned earlier, E P = σ − 1 ( σ ( E ) S ( P ) ) is a reduction of structure group of E to P such that φ takes values in V χ − and

deg χ * E P = deg ( χ ∘ S − 1 ) * σ ( E ) S ( P ) = 0 .

Since ( E , φ ) is polystable, there exists a reduction of structure group E L of E P to a Levi subgroup L of P such that φ takes values in the space V χ 0 defined in (1). Then it is easily checked that σ ( E L ) defines a reduction of structure group of σ ( E ) S ( P ) to S ( L ) , which is a Levi subgroup of S ( P ) , and F ( φ ) takes values in W χ ∘ S − 1 0 since it is satisfied that S ( V χ 0 ) = W χ ∘ S − 1 0 , due to the hypothesis relation F ∘ ρ = ( ρ ′ ∘ S ) ∘ F . This proves that F ( E , φ ) is polystable.

All this proves that the correspondence f is well defined. Observe that the isomorphism of vector spaces F − 1 defines also a correspondence, which is clearly inverse to f , what proves that f is bijective.□

Corollary 4.1

The automorphism J defined in (8) induces bijective correspondences between polystable ( Spin ( 8 , C ) , ρ ) -Higgs pairs, ( Spin ( 8 , C ) , ρ + ) -Higgs pairs, and ( Spin ( 8 , C ) , ρ − ) -Higgs pairs over X that preserves the polystability, where the representations ρ , ρ + , and ρ − are defined in (5), (6), and (7), respectively.

Proof

Let V be the eight-dimensional vector space on which Spin ( 8 , C ) acts through the representation ρ defined in (5), and let q be its nondegenerate quadratic form. Let W be the maximal isotropic subspace of V such that the representations ρ + and ρ − of Spin ( 8 , C ) defined in (6) and (7), respectively, define actions of Spin ( 8 , C ) on ∧ + W and ∧ − W . Consider the automorphism

J : V ⊕ ∧ + W ⊕ ∧ − W → V ⊕ ∧ + W ⊕ ∧ − W

defined in (8). It takes values in ∧ + W when restricted to V , so it defines an isomorphism J : V → ∧ + W . From the relation expressed in (9), it is satisfied that J ∘ ρ = ( ρ + ∘ T ) ∘ J for a representative T in Aut ( Spin ( 8 , C ) ) of the triality automorphism τ of Spin ( 8 , C ) . The hypotheses of Proposition 4.4 are satisfied, so the correspondence defined by j ( E , φ ) = ( τ ( E ) , J ( φ ) ) given in Proposition 4.4 is bijective.□

5 Stability and simplicity of ( Spin ( 8 , C ) , ρ ) -Higgs pairs

Let X be a compact Riemann surface of genus g ≥ 2 . Let ( E , φ ) be a ( Spin ( 2 n , C ) , ρ 2 n ) -Higgs pair over X . Denote by Aut ( E , φ ) the group of automorphisms of ( E , φ ) , that is,

(22) Aut ( E , φ ) = { f ∈ Aut ( E ) : f ρ 2 n ( φ ) = φ } ,

where f ρ 2 n denotes the automorphism of E ( V 2 n ) induced by f . The space of infinitesimal automorphisms of ( E , φ ) is also defined to be the space

(23) aut ( E , φ ) = { f ∈ End ( E ) : f ρ 2 n ( φ ) = 0 } ,

Let Z ( Spin ( 2 n , C ) ) be the center of Spin ( 2 n , C ) , which satisfies Z ( Spin ( 2 n , C ) ) ≅ Z 2 × Z 2 . Let z ∈ Z ( Spin ( 2 n , C ) ) be a choice of a central element. This choice induces the definition of an automorphism f z : E → E of E given by multiplication by z . The corresponding automorphism f ρ 2 n z of E ( V 2 n ) is defined by f ρ 2 n z ( [ e , v ] ) = [ e z , ρ 2 n ( z ) ( v ) ] , where e ∈ E and v ∈ V 2 n . This is a good definition since, for any g ∈ Spin ( 2 n , C ) ,

[ e g z , ρ 2 n ( z ) ρ 2 n ( g − 1 ) ( v ) ] = [ e z g , ρ 2 n ( g − 1 ) ρ 2 n ( z ) ( v ) ] = [ e z , ρ 2 n ( z ) ( v ) ] .

This fact, together with the additional observation that every central element of Spin ( 2 n , C ) has order 2, proves that Z ( Spin ( 2 n , C ) ) can be understood as a subgroup of Aut ( E , φ ) .

Definition 3

A ( Spin ( 2 n , C ) , ρ 2 n ) -Higgs pair ( E , φ ) over X is said to be simple if the group Aut ( E , φ ) coincides with Z ( Spin ( 2 n , C ) ) .

In previous studies [13,16,22], it is proved that stable and simple Higgs bundles represent smooth points of the moduli space of Higgs bundles for any reductive complex structure group. They use arguments that involve deformation theory and that, as far as it has been studied, are not easily adaptable to the situation in which pairs associated with a representation different from the adjoint one are considered.

Proposition 5.1

Let ( E , φ ) be a stable ( Spin ( 2 n , C ) , ρ 2 n ) -Higgs pair over X . Then every element of Aut ( E , φ ) is semisimple, where Aut ( E , φ ) is defined in (22).

Proof

Under the conditions of the statement, the identity component Aut ( E , φ ) 0 of Aut ( E , φ ) is semisimple, since every element of its Lie algebra, the space aut ( E , φ ) of infinitesimal automorphisms of ( E , φ ) , is semisimple as a consequence of [16, Proposition 2.14]. Let g ∈ Aut ( E , φ ) , and let g u be the unipotent part of g . Let p : Aut ( E , φ ) → π 0 ( Aut ( E , φ ) ) be the projection. Since p is a morphism that preserves the unipotent parts and π 0 ( Aut ( E , φ ) ) is a finite group, then every element of π 0 ( Aut ( E , φ ) ) is also semisimple, so p ( g u ) = 0 and g u ∈ Aut ( E , φ ) 0 ; hence, g u is itself 0 because the elements of Aut ( E , φ ) 0 are all of them semisimple.□

Lemma 5.1

Let G be any semisimple complex Lie group, and let g ∈ G . Then the centralizer Z G ( g ) of g in G is defined up to conjugation.

Proof

Let g ∈ V , x ∈ Z G ( g ) , and h ∈ G . It is clear that h x h − 1 commutes with h g h − 1 (because g and x commute). This proves that the inner automorphism i h : G → G defined by i h ( y ) = h y h − 1 restricts to an isomorphism between Z G and Z G ( h g h − 1 ) .□

Lemma 5.2

Let n be any integer number with n ≥ 2 , π 2 n : Spin ( 2 n , C ) → SO ( 2 n , C ) be the covering map defined in (12), and let g ∈ Spin ( 2 n , C ) . Then π 2 n ( Z Spin ( 2 n , C ) ( g ) ) = Z SO ( 2 n , C ) ( π 2 n ( g ) ) .

Proof

Since π 2 n is a homomorphism of groups, it is obvious that

π 2 n ( Z Spin ( 2 n , C ) ( g ) ) ⊆ Z SO ( 2 n , C ) ( π 2 n ( g ) ) .

To prove the other contention, take any x ∈ Z SO ( 2 n , C ) ( π 2 n ( g ) ) and let h ∈ Spin ( 2 n , C ) be such that π 2 n ( h ) = x . Then h g h − 1 g − 1 ∈ ker π 2 n . It may be supposed that h g h − 1 g − 1 = 1 (if this is not the case, then the other element in the fiber π 2 n − 1 ( g ) should satisfy this relation), so h ∈ Z Spin ( 2 n , C ) ( g ) , hence x ∈ π 2 n ( Z Spin ( 2 n , C ) ) .□

Lemma 5.3

Let n be any integer number with n ≥ 2 , and let g be any element of SO ( 2 n , C ) . Then Z SO ( 2 n , C ) ( g ) is isomorphic to one of the following groups:

  1. SO ( 2 n , C ) (if and only if g is a central element).

  2. SL ( 2 r 1 , C ) × ⋯ × SL ( 2 r d , C ) × SO ( k 1 , C ) × SO ( k 2 , C ) , where d ≥ 0 , k 1 , k 2 ≥ 0 , and 2 r 1 + ⋯ + 2 r d + k 1 + k 2 = 2 n .

Proof

Let T = SO ( 2 , C ) × ⋯ n × SO ( 2 , C ) be a maximal torus of SO ( 2 n , C ) . Every element in SO ( 2 n , C ) can be conjugated into T , since SO ( 2 n , C ) is connected. Then every element in SO ( 2 n , C ) is conjugate to an element of the form

M = M 1 ⋱ d M d I k 1 − I k 2 ,

where

M i = A i ⋱ r i A i ,

each A i being an element in SO ( 2 , C ) different from ± I and such that A i ≠ A j for i ≠ j . An element in SO ( 2 n , C ) that commutes with M should preserve the blocks. Since the centralizer of each M i is isomorphic to SL ( 2 r i , C ) , the result comes.□

Lemma 5.4

Let n be any integer number with n ≥ 2 , let ( E , φ ) be a polystable ( Spin ( 2 n , C ) , ρ 2 n ) -Higgs pair over X, and let ( E SO , φ ) the associated ( SO ( 2 n , C ) , ρ 2 n SO ) -Higgs pair over X, where the representations ρ 2 n and ρ 2 n SO are defined in (13) and (14), respectively. Then ( E , φ ) is simple if and only if ( E SO , φ ) is simple.

Proof

Observe first that every automorphism of ( E , φ ) descends to give an automorphism of ( E SO , φ ) in such a way that two automorphisms of ( E , φ ) that descend to the same automorphism of ( E SO , φ ) differ in one central element of Spin ( 2 n , C ) . This proves that if ( E SO , φ ) is simple then ( E , φ ) is also simple. Reciprocally, suppose that ( E , φ ) is simple, and take any automorphism f of ( E SO , φ ) . The automorphism f defines an element of the adjoint bundle Ad ( E SO ) , which is isomorphic to Ad ( E ) , so f defines an endomorphism of ( E , φ ) , say f ¯ , such that f ¯ descends to f . For the same reason, there exists an endomorphism f − 1 ¯ of ( E , φ ) that descends to f − 1 . The endomorphisms f ¯ and f − 1 ¯ clearly differ in one central element of Spin ( 2 n , C ) , so they are isomorphisms. Since ( E , φ ) is simple, f ¯ must consist of multiplication by a central element of Spin ( 2 n , C ) , so f consists also in multiplication by a central element of SO ( 2 n , C ) , what proves that ( E SO , φ ) is itself simple.□

Theorem 5.1

Let ( E , φ ) be a stable and non-simple ( Spin ( 8 , C ) , ρ ) -Higgs pair over X, where the representation ρ : Spin ( 8 , C ) → GL ( 8 , C ) is defined in (5). Let E SO be the principal SO ( 8 , C ) -bundle over X defined in (17) associated with E. Then the underlying vector bundle of E SO is isomorphic to one of the following vector bundles:

  1. L k ⊕ L 8 − k for k = 0 , 1 , 2 , 3 , 4 ;

  2. F 2 r ⊕ L k ⊕ L 8 − 2 r − k for r = 1 , 2 , 3 and k = 0 , 1 , … , 4 − r ;

  3. F 2 r ⊕ F 2 s ⊕ L k ⊕ L 8 − 2 r − 2 s − k for r = 1 , 2 , 3 , s = r , … , 4 − r , and k = 0 , … , 4 − r − s ;

  4. F 2 ⊕ F 2 ⊕ F 2 r ⊕ L k ⊕ L 4 − 2 r − k for r = 1 , 2 and k = 2 − r , … , 4 − 2 r ,

where F j is an SL ( j , C ) -bundle and L j is an SO ( j , C ) -bundle for all j ≥ 1 , F 0 = 0 , L 0 = 0 , and L 1 = O .

Remark

All the Higgs pairs described in the four cases stated in Theorem 5.1 are polystable, as a consequence of Corollary 2.1.

Proof of Theorem 5.1

Let ( E , φ ) be a stable and non-simple ( Spin ( 8 , C ) , ρ ) -Higgs pair over X . Then there exists an automorphism f of ( E , φ ) that does not belong to the center Z of Spin ( 8 , C ) . Since the group Spin ( 8 , C ) is semisimple and ( E , φ ) is stable, it is ensured [16, Proposition 2.14] that the space aut ( E , φ ) defined in (23) is 0. This space is the Lie algebra of the group Aut ( E , φ ) at the identity, so Aut ( E , φ ) 0 = Z 0 = { 1 } and, moreover, since π 0 ( Aut ( E , φ ) ) is finite because it is an algebraic group, and Z is a normal subgroup of Aut ( E , φ ) , the quotient Aut ( E , φ ) ⁄ Z is a finite group. Let f 1 , … , f k be a family of automorphisms of ( E , φ ) not coming from the center of Spin ( 8 , C ) such that the nontrivial elements of Aut ( E , φ ) ⁄ Z are exactly the set of their classes modulo Z , { [ f 1 ] , … , [ f k ] } . Each f i corresponds to an element g i ∈ Spin ( 8 , C ) . In the study by Garcia-Prada and Oliveira [23, Theorem 3.17], it is shown that, in this situation, E admits a reduction of structure group to the centralizer Z Spin ( 8 , C ) ( g i ) of g i for every i = 1 , … , k . Of course, φ takes values in that reduction, since f i is an automorphism of ( E , φ ) . Notice that the choice of representatives is well defined except for one element of the center Z , but this does not change the centralizers that are being considered. Let ( E g 1 , φ g 1 ) be the reduction of structure group of ( E SO , φ ) to Z Spin ( 8 , C ) ( π ( g 1 ) ) , which exists by Lemma 5.2. Then, since g 1 does not belong to Z , from Lemma 5.3 applied to n = 4 , we deduce the result.□

Remark

The same proof made in Theorem 5.1 works to give a similar description of stable and non-simple ( SO ( 8 , C ) , ι ) -Higgs pairs, where ι is the representation of SO ( 8 , C ) induced by the natural inclusion of groups SO ( 8 , C ) → GL ( 8 , C ) . Specifically, if ( E , φ ) is a stable and non-simple ( SO ( 8 , C ) , ι ) -Higgs pair over X , then it admits exactly one of the three forms described in Theorem 5.1.

Example

Take a rank 8 and trivial determinant vector bundle E over X , which admits a globally defined nondegenerate symmetric bilinear form, and whose second Stiefel-Whitney class is 0. Then this vector bundle can be understood as a principal SO ( 8 , C ) -bundle over X . The bundle E lifts to a principal Spin ( 8 , C ) -bundle over X . Suppose that this principal bundle is stable and not simple. Suppose, in addition, that E admits a nonzero holomorphic global section, whose induced line subbundle of E is not isotropic. The pair consisting of E together with this global section is a stable Higgs pair for the representation ρ (5), where the fixed line bundle L considered is the trivial line bundle O over X . In this situation, E satisfies the conditions of Theorem 5.1. Then it is deduced that E admits a decomposition into 2, 3, 4, or 5 vector subbundles. Moreover, if there are more than 2 subbundles, all but perhaps two of them must be of even rank.

6 Conclusion

The group Spin ( 8 , C ) is the only simple complex Lie group that admits an outer automorphism of order 3, called triality automorphism. It also admits three non-isomorphic irreducible eight-dimensional complex representations, so that the triality automorphism acts as an order 3 permutation on the set of these representations. One of them is the representation ρ induced by the double covering Spin ( 8 , C ) → SO ( 8 , C ) with which Spin ( 8 , C ) is equipped. If α : Spin ( 8 , C ) → GL ( V ) is a complex representation of Spin ( 8 , C ) and E is a principal Spin ( 8 , C ) -bundle over a compact Riemann surface X , then a complex rank 8 vector bundle E ( V ) is induced by E and α . A ( Spin ( 8 , C ) , α ) -Higgs pair over X is a pair ( E , φ ) , where E is a principal Spin ( 8 , C ) -bundle over X and φ ∈ H 0 ( X , E ( V ) ⊗ L ) , L being a fixed line bundle over X . In this work, reduced notions of stability and polystability for Higgs pairs over X with structure group Spin ( 8 , C ) and associated with the representations cited above are given, and it is proved that the three moduli spaces of Higgs pairs considered are isomorphic. It is also given an explicit expression of the vector bundles associated with the stable and not simple ( Spin ( 8 , C ) , ρ ) -Higgs pairs over X through the representation ρ of Spin ( 8 , C ) . Specifically, it has been shown that, if ( E , φ ) is a stable and non-simple ( Spin ( 8 , C ) , ρ ) -Higgs pair over X , then the vector bundle induced by E and ρ is isomorphic to one of the following:

  1. L k ⊕ L 8 − k for k = 0 , 1 , 2 , 3 , 4 ;

  2. F 2 r ⊕ L k ⊕ L 8 − 2 r − k for r = 1 , 2 , 3 and k = 0 , 1 , … , 4 − r ;

  3. F 2 r ⊕ F 2 s ⊕ L k ⊕ L 8 − 2 r − 2 s − k for r = 1 , 2 , 3 , s = r , … , 4 − r and k = 0 , … , 4 − r − s ;

  4. F 2 ⊕ F 2 ⊕ F 2 r ⊕ L k ⊕ L 4 − 2 r − k for r = 1 , 2 and k = 2 − r , … , 4 − 2 r ,

where F j is an SL ( j , C ) -bundle and L j is an SO ( j , C ) -bundle for all j ≥ 1 , F 0 = 0 , L 0 = 0 , and L 1 = O .

  1. Funding information: The author states no funding involved.

  2. Author contributions: The author has conducted the whole research process, written the manuscript, and revised the text.

  3. Conflict of interest: The author states no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

[1] N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114, DOI: https://doi.org/10.1215/S0012-7094-87-05408-1. Search in Google Scholar

[2] N. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. (3) 55 (1987), no. 1, 59–126, DOI: https://doi.org/10.1112/plms/s3-55.1.59. Search in Google Scholar

[3] C. T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867–918, DOI: https://doi.org/10.1090/S0894-0347-1988-0944577-9. Search in Google Scholar

[4] C. T. Simpson, Higgs bundles and local systems, Publ. Math. Inst. 75 (1992), 5–95, DOI: https://doi.org/10.1007/BF02699491. Search in Google Scholar

[5] A. Antón-Sancho, Automorphisms of order three of the moduli space of Spin-Higgs bundles, Hokkaido Math. J. 47 (2018), no. 2, 387–426, DOI: https://doi.org/10.14492/hokmj/1529308825. Search in Google Scholar

[6] A. Antón-Sancho, Shatz and Bialynicki-Birula stratifications of the moduli space of Higgs bundles, Hokkaido Math. J. 51 (2022), no. 1, 25–56, DOI: https://doi.org/10.14492/hokmj/2019-202. Search in Google Scholar

[7] P. Gothen and R. Zúñiga-Rojas, Stratifications on the moduli space of Higgs bundles, Port. Math. 74 (2017), no. 2, 127–148, DOI: https://doi.org/10.4171/pm/1996. Search in Google Scholar

[8] S. B. Bradlow, O. Garcia-Prada, and P. B. Gothen, Surface group representations and U(p,q)-Higgs bundles, J. Differential Geom. 64 (2003), 111–170, DOI: https://doi.org/10.4310/jdg/1090426889. Search in Google Scholar

[9] R. Donagi and T. Pantev, Langlands duality for Hitchin systems, Invent. Math. 189 (2012), 653–735, DOI: https://doi.org/10.1007/s00222-012-0373-8. Search in Google Scholar

[10] D. Banfield, Stable pairs and principal bundles, Quart. J. Math. 51 (2000), 417–436, DOI: https://doi.org/10.1093/qjmath/51.4.417. Search in Google Scholar

[11] A. Antón-Sancho, F4 and PSp(8,C)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface, Open Math. 20 (2022), no. 1, 1723–1733, DOI: https://doi.org/10.1515/math-2022-0543. Search in Google Scholar

[12] R. Fringuelli, Automorphisms of moduli spaces of principal bundles over a smooth curve, arXiv:2112.08750v3, 2022, https://doi.org/10.48550/arXiv.2112.08750. Search in Google Scholar

[13] O. Garcia-Prada and S. Ramanan, Involutions and higher order automorphisms of Higgs bundle moduli spaces, arXiv:1605.05143v4, 2019, https://doi.org/10.48550/arXiv.1605.05143. Search in Google Scholar

[14] W. Oxbury and S. Ramanan, Trigonal curves and Galois Spin(8)-bundles, arXiv:math/9907114v1, 1999, https://doi.org/10.48550/arXiv.math/9907114. Search in Google Scholar

[15] S. B. Bradlow, O. Garcia-Prada, and I. Mundet, Relative Hitchin-Kobayashi correspondences for principal pairs, Quart. J. Math. 54 (2003), 171–208, DOI: https://doi.org/10.1093/qmath/hag013. Search in Google Scholar

[16] O. Garcia-Prada, P. Gothen, and I. Mundet, The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations, arXiv:0909.4487v3, 2012, https://doi.org/10.48550/arXiv.0909.4487. Search in Google Scholar

[17] G. Reyes-Ahumada, L. Roa-Leguizamón, and H. Torres-López, A note on rank two stable bundles over surfaces, Glasg. Math. J. 64 (2022), no. 2, 397–410, DOI: https://doi.org/10.1017/S0017089521000185. Search in Google Scholar

[18] A. Antón-Sancho, The moduli space of principal Spin bundles, Rev. Un. Mat. Argentina 57 (2016), no. 2, 25–51. Search in Google Scholar

[19] W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, Vol. 129, Springer, New York, NY, 2004. Search in Google Scholar

[20] C. Procesi, Lie Groups: An Approach through Invariants and Representations, Springer, New York, 2007. Search in Google Scholar

[21] A. Antón-Sancho, Principal Spin-bundles and triality, Rev. Colombiana Mat. 49 (2015), 235–259, DOI: https://doi.org/10.15446/recolma.v49n2.60442. Search in Google Scholar

[22] S. B. Bradlow, O. Garcia-Prada, and P. B. Gothen, Higgs bundles for the non-compact dual of the special orthogonal group, Geom. Dedicata 175 (2015), 1–48, DOI: https://doi.org/10.1007/s10711-014-0026-8. Search in Google Scholar

[23] O. Garcia-Prada and A. Oliveira, Connectedness of Higgs bundle moduli for complex reductive Lie groups, Asian J. Math. 21 (2017), no. 5, 791–810, DOI: https://doi.org/10.4310/AJM.2017.v21.n5.a1. Search in Google Scholar
Received: 2023-04-05
Revised: 2023-10-17
Accepted: 2023-10-25
Published Online: 2023-11-16

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
https://doi.org/10.1515/math-2023-0153
Keywords for this article
Higgs pair; Higgs bundle; spin; Riemann surface; stability
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 9.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2023-0153/html
Scroll to top button