Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study

We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techn...

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2022
Main Author: Meneses, Claudio
Format: Article
Language:English
Published: Інститут математики НАН України 2022
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/211725
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Cite this:Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study. Claudio Meneses. SIGMA 18 (2022), 062, 41 pages

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author Meneses, Claudio
author_facet Meneses, Claudio
citation_txt Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study. Claudio Meneses. SIGMA 18 (2022), 062, 41 pages
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container_title Symmetry, Integrability and Geometry: Methods and Applications
description We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces of semi-stable parabolic Higgs bundles in genus 0, which is encoded into the combinatorics of weight polytopes. We also present a comprehensive analysis of the geometric models' behavior under variation of parabolic weights and wall-crossing, which is concentrated on their nilpotent cones.
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 062, 41 pages Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study Claudio MENESES Mathematisches Seminar, Christian-Albrechts Universität zu Kiel, Heinrich-Hecht-Platz 6, 24118 Kiel, Germany E-mail: meneses@math.uni-kiel.de URL: http://www.math.uni-kiel.de/geometrie/de/claudio-meneses/ Received September 21, 2021, in final form July 28, 2022; Published online August 13, 2022 https://doi.org/10.3842/SIGMA.2022.062 Abstract. We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of or- bit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces of semi-stable parabolic Higgs bundles in genus 0, which is encoded into the combinatorics of weight poly- topes. We also present a comprehensive analysis of the geometric models’ behavior under variation of parabolic weights and wall-crossing, which is concentrated on their nilpotent cones. Key words: parabolic Higgs bundle; Hitchin fibration; nilpotent cone 2020 Mathematics Subject Classification: 14H60; 14D22; 32G13; 22E25 To Professor Leon A. Takhtajan on his 70th birthday, with profound respect and admiration. 1 Introduction The moduli spaces of parabolic Higgs bundles on a Riemann surface of lowest possible dimen- sion are elliptic surfaces in virtue of their Hitchin fibrations [16]. These examples occur in low genus and encode at once, in its simplest possible form, the characteristic features of the rich and intricate geometry occuring in arbitrary dimensions. From the general theory of elliptic surfaces [25], they are necessarily biholomorphic to one of the fibrations in Kodaira’s list. Fol- lowing the seminal work of Gaiotto–Moore–Neitzke [8], their hyperkähler geometry has recently attracted renewed interest in relation to the study of gravitational instantons of ALG type [7]. The toy model [13] for the Hitchin fibration and C∗-action of the moduli spaces of rank 2 parabolic Higgs bundles over CP1 with four marked points, is the elliptic surface Mtoy associated to a choice (z1, z2, z3, z4) ∈ Conf4(CP1) as follows. Let ΣD → CP1 be the elliptic curve of D = z1+ z2+ z3+ z4, with involution τ and ramification points {w1, w2, w3, w4}. The extension of τ to an action in ΣD × C in terms of the character generated by ρ(τ) = −1 determines a 2-dimensional orbifold (τ × ρ)\(ΣD ×C) which is smooth away of the points [(wi, 0)]. Mtoy is This paper is a contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quan- tum in honor of Leon Takhtajan. The full collection is available at https://www.emis.de/journals/SIGMA/Takhtajan.html mailto:meneses@math.uni-kiel.de http://www.math.uni-kiel.de/geometrie/de/claudio-meneses/ https://doi.org/10.3842/SIGMA.2022.062 https://www.emis.de/journals/SIGMA/Takhtajan.html 2 C. Meneses defined as the desingularization of (τ × ρ)\(ΣD × C) by blow-up at those points. The elliptic fibration π2 : Mtoy → C is given by extending the map [(w, z)] 7→ z2 trivially along exceptional divisors, while its C∗-action is induced by the standard C∗-action on C. The loci where the (holomorphic) involution extending τ × ρ acts freely is a Zariski open subset U ∼= T ∗CP1\CP1, equipping Mtoy with an additional projection π1 : U → CP1. Intuitively, a biholomorphism between Mtoy and the aforementioned moduli spaces of semi-stable parabolic Higgs bundles could be constructed by relating π1 to the forgetful map to parabolic structures, and π2 to the determinant map on parabolic Higgs fields. There are some caveats when trying to carry out the previous biholomorphisms, though. First of all, they are not expected to hold for other ranks and number of marked points. A nontrivial feature of parabolic Higgs bundles is the dependence of their moduli problem on a choice of parabolic weights. While several classical results on this dependence [27, 32] are linked to the study of the birational geometry of the moduli problem, not only they are trivial when the moduli space is a complex surface, but also they shed no light on other subtle phenomena. For instance, it is unclear how holomorphic invariants such as Harder–Narasimhan stratifications may depend on parabolic weights, or what happens to the moduli space when semi-stable parabolic bundles don’t exist [1, 5].1 Moreover, the natural hyperkähler structures on these moduli spaces depend nontrivially on parabolic weights [17, 24], and the hyperkähler moduli problem motivates the search for suitable complex-analytic structures capturing this dependence. The will of attacking these problems is the main motivation to carry out this work. More specifically, this work sprung as a natural advance towards the parabolic Higgs bundle generalization of the results in [24], dealing with the study of Kähler forms on moduli spaces of semi-stable parabolic bundles, where the understanding of refined moduli space stratifications is crucial. We propose a construction of moduli spaces of rank 2 stable parabolic Higgs bundles on CP1 with four marked points, on which the dependence on parabolic weights and its wall- crossing phenomena becomes as transparent as possible. The construction is comprised of two steps. The first step is based on the observation that the peculiarities of genus 0 (namely, the infinitesimal rigidity of holomorphic bundles dictated by the Birkhoff–Grothendieck theorem) lead to the construction of Harder–Narasimhan strata for underlying holomorphic bundles E ∼= O(m1) ⊕ O(m2) as semi-stable orbit spaces under the action of bundle automorphisms, since any isomorphism class of quasi-parabolic Higgs bundles on E can be modeled as an orbit in an explicit complex manifold QPH(E) with respect to an action of the group P(Aut(E)) of projective automorphisms of E.2 The mechanism of construction is entirely self-contained, and follows the general localization philosophy to study invariants of bundle automorphism actions proposed in [22]. Once this is done, the remaining problem is to understand the way these strata can be subsequently glued into a smooth complex manifold modeling the moduli space. An important property of our construction is that it admits a generalization to arbitrary rank and number of marked points. This way, this is the first installment (or “toy model”) of a project focused on the study of complex-analytic structures of moduli spaces of parabolic Higgs bundles (which could be understood as an instance of the so-called Gelfand principle), where we exhaustively explore the simplest possible nontrivial example, and we rigorously exhibit the independence of the biholomorphism type of these moduli spaces under variation of parabolic weights and wall-crossing. Since the latter is exclusive of the case of four marked points, this is yet another reason to present it separately. 1Incidentally, in [28] Rayan introduced the notion of “quiver at the bottom of the nilpotent cone”, in the context of twisted Higgs bundles on CP1. The idea that for parabolic Higgs bundles, the subspace parametrized by these objects may not correspond to a moduli space of parabolic bundles, seems to be standard folklore [6], and resurfaces in our work in an explicit way. 2We remark that bundle automorphism groups are in general non-reductive. It should be possible to reinterpret our constructions as explicit examples of variations of non-reductive GIT quotients [3, 4] (cf. [12]). Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 3 1.1 Structure of the paper Even though the techniques employed in this work are all elementary, in the end the total number of steps involved in the construction of geometric models is substantial. This requires an adequate signposting strategy to guide the reader through it, which we now provide. The starting point are the model spaces QPH(E) of quasi-parabolic Higgs bundles on E, together with a pair of projections QPH(E) par ww det (( QP(E) := 4∏ i=1 P(E|zi) H0 ( CP1,K2 CP1 ( 4∑ i=1 zi )) par is the P(Aut(E))-equivariant forgetful map to quasi-parabolic structures, while det is the P(Aut(E))-invariant determinant map on parabolic Higgs fields. The construction and classification of all different Harder–Narasimhan strata as spaces of semi-stable P(Aut(E))-orbits in the complex manifolds QPH(E) follows from a systematic break down of the semi-stability condition for parabolic Higgs bundles into structural units, recollected in Sections 3–6. This process consists of four parts which are built progressively: Section 3: QP(E) is stratified in terms of interpolation loci for line sub-bundles of E. The resulting stratification is then related to the stratification by pointwise P(Aut(E))-stabilizer subgroups. Section 4: det stratifies QPH(E) into a (possibly empty) bulk QPHC∗(E) and its nilpotent locus QPH0(E). Potentially unstable points can only occur in QPH0(E) (Lemma 4.2). This leads to a stratification of QPH0(E) in terms of the interpolation properties of potentially destabilizing line sub-bundles, which arise as kernels of parabolic Higgs fields on E when the latter are nonzero. The stratifications of QPH(E) and QP(E) are then related by the projection par (Proposition 4.6). Section 5: A combinatorial classification of semi-stability walls H and open chambers C in the even and odd weight polytopes W• is explained. Open chambers are defined as the connected components in the complement of the union of all semi-stability walls in W•, and parametrize the different stable loci QPHs C(E) in QPH(E). Open chambers are split into interior and exterior (denoted as CI and CI respectively; this notation is carefully explained), according to whether stable parabolic bundles exist or not. Section 6: In order to determine stable loci QPHs C(E), the stratifications of QPH0(E) and the combinatorics of W• are combined to classify conditionally stable loci (Definition 6.3) as well as their (in general non-Hausdorff) orbit spaces. The latter are organized in terms of the combinatorics of some classical projective plane configurations, and then dissected into basic building blocks (Definition 6.6). Finally, the structure of basic building blocks is classified (they are biholomorphic to CP1 or C), and a nilpotent cone assembly kit (Definition 6.8) is conformed for every choice of open chamber C ⊂ W•. Once all Harder–Narasimhan strata are constructed, their glueing is brought in, and the interpretation of semi-stability via the geometry of P(Aut(E))-actions leads to an explicit classi- fication of wall-crossing as nilpotent cone transformations, in terms of the combinatorics of W•, as well as an additional characterization of parabolic S-equivalence along semi-stability walls. We list our main results, whose proof is presented in Section 7 together with a series of corollaries. Theorem 1.1 (orbit space models for Harder–Narasimhan strata). For any open chamber C ⊂ W• and vector bundle E, if QPHs C(E) ̸= ∅, then P(Aut(E)) acts freely and properly on it. 4 C. Meneses There is an isomorphism MC(E) ∼= QPHs C(E)/P(Aut(E)), where MC(E) is the Harder–Narasimhan stratum associated to E. MC(E) is smooth if it is 2-dimensional, and has at most two nodal singularities otherwise. Theorem 1.2 (glueing of Harder–Narasimhan strata). (i) For d even, the two orbit spaces QPHC∗(E)/P(Aut(E)), m1 −m2 = 0, 2, glue into a 2 : 1-branched covering over CP1 × C∗, ramified over {z1, 0, 1,∞}× C∗. (ii) For every choice of open chamber C ⊂ W•, the components in the nilpotent cone assembly kit are glued into a D4-configuration (Figure 1). Moreover, there is an isomorphism MC ∼= Mtoy, where MC is the smooth complex surface resulting from the glueing of geometric models of Harder–Narasimhan strata of corresponding degree. z1 0 1 ∞ CP1 Figure 1. The D4-configuration. Theorem 1.3 (wall-crossing classification; see Sections 5 and 6 for notations). For each bound- ary wall HI,|I|−3 between the exterior and interior chambers CI and CI(I), there is an isomorphism MCI ∼= MCI(I) exchanging the nilpotent cone’s central spheres SI and SI(I), and equal to the identity otherwise. For any pair of neighboring interior chambers CI(I) and CI(I′), there is an isomorphism MCI(I) ∼= MCI(I′) exchanging the compactified I- and I ′-basic building blocks, and equal to the identity otherwise. After this, the Hitchin elliptic fibrations, their degeneration into their nilpotent cones, as well as a C∗-action and a collection of Hitchin sections, are explicitly identified as by-products of the method of construction. In Appendix A we also introduce the alternative residue models M ev C (E) for Harder–Narasimhan strata.3 The generalization of this work to an arbitrary number of punctures requires a more careful analysis of automorphism group actions, including a suitable axiomatization of the intrinsic structures on display here, and will be treated elsewhere [23]. 3Cf. Komyo–Saito [18], who provide explicit descriptions of Zariski open sets on moduli spaces of logarithmic connections and parabolic Higgs bundles in odd degree, by apparent singularities and their dual parameters as coordinates. Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 5 1.2 Relation to other work The literature on moduli spaces emerging from parabolic structures on CP1 is vast and comprises different perspectives, strategies, and objectives. We compiled works with ideas overlapping substantially with ours. Further details can be found in those works and references therein. Related models were used by Loray–Saito–Simpson [20] in their proof of the non-abelian Hodge foliation conjecture for moduli of rank 2 logarithmic connections, in odd degree and n = 4 marked points, in terms of transversal foliations arising as Lagrangian fibrations. There, a bundle splitting type is necessarily generic, and their analysis of moduli dependence on parabolic weights leads to an analogous characterization of exterior chambers. These fibrations are studied for arbitrary n in [19] (cf. [18]), and in [15] the study of moduli of holomorphic connections in genus 2 is reduced to the genus 0 case and n = 6. The overall framework of these works is that of birational geometry. While similar in spirit, our proposal is considerably different in its methods and results. Godinho–Mandini [9, 10] defined an isomorphism between hyperpolygon spaces and the Zariski open subset of rank 2 parabolic Higgs bundles of degree 0 that are holomorphically trivial in terms of the residue data of the latter (a detailed description of the correspondence in the case n = 4, in terms of the Nakajima quiver variety for the affine Dynkin diagram D̃4, is discussed by Rayan–Schaposnik in [29, Section 4.3]). Similarly, Blaavand [6] provides a geomet- ric construction of the nilpotent cone in degree 1 and n = 4 under a choice of parabolic weights constrained to lie inside a specific open chamber (see Remark 7.7). Heller–Heller [14] applied the abelianization of logarithmic connections in degree 0 when n = 4 to compute symplectic volumes for moduli spaces of stable parabolic bundles. 2 Conventions and index of notation We will fix a point z1 ∈ CP1\{z2, z3, z4}, where z2 = [0 : 1] = 0, z3 = [1 : 1] = 1, z4 = [1 : 0] = ∞. When necessary, we will consider the choice of cross-ratio such that (z1, z2; z3, z4) = z1. Similarly, we will denote D = ∑4 i=1 zi. A parabolic structure supported on D on a rank 2 holomorphic vector bundle E → CP1 of degree d consists of flags Fi ⊂ E|zi , i = 1, 2, 3, 4 weighted by real numbers 0 ≤ βi < 1 such that Fi = E|zi if and only if βi = 0. A quasi-parabolic structure omits the choice of weights. Hereafter E∗ will denote a rank 2 parabolic bundle, i.e., a choice of parabolic structure over D on E, and E· the underlying quasi-parabolic bundle. Given E∗, to any subset I ⊂ {1, 2, 3, 4} we associate the weighted sum βI = ∑ i∈I βi − ∑ j∈Ic βj . For any E·, a line sub-bundle L ⊂ E determines a subset IL,E· ⊂ {1, 2, 3, 4} defined as IL,E· = { i ∈ {1, 2, 3, 4} : L|zi ⊂ Fi } . For any z ∈ CP1, n(E|z) will denote the cone of nilpotent endomorphisms of E|z, i.e., the singular quadric in sl(E|z) defined as the zero locus of its determinant map, and for any F ∈ P(E|z), n(F ) is the line of nilpotent endomorphisms ϕ such that ker(ϕ) ⊃ F . Let End(E) := E∨ ⊗ E. An element Φ ∈ H0(CP1,End(E) ⊗ KCP1(D)) is a (strongly) parabolic Higgs field of E∗ if for each i = 1, 2, 3, 4, the residue Reszi Φ satisfies Reszi Φ ∈ n(Fi). 6 C. Meneses A parabolic Higgs bundle is a pair of the form (E∗,Φ). The pair (E∗,Φ) is called stable (resp. semi-stable) if for every Φ-invariant line sub-bundle L ⊂ E we have that βIL,E· < d− 2 deg(L) (resp. ≤). (2.1) The (semi-)stability of a parabolic bundle E∗ is defined by letting Φ = 0. (E∗,Φ) is called strictly semi-stable if it semi-stable but not stable. Given a vector bundle E, a line bundle L, and their tensor product E′ = E ⊗ L, the bundle isomorphism End(E′) ∼= End(E) induces a bijection between the sets of quasi-parabolic Higgs bundles on E and E′. Since L′ is a line sub-bundle of E′ if and only if L−1 ⊗ L′ is a line sub- bundle of E, it follows that the stability of a parabolic Higgs bundle is preserved under tensor product by a line sub-bundle. Remark 2.1. In the case of rank 2 holomorphic vector bundles, the standard definition of parabolic stability [21, 30] involves parabolic weights 0 ≤ αi1 ≤ αi2 < 1, i = 1, 2, 3, 4. In that case, the parabolic degree and slope of E∗ are defined as par deg(E∗) := d+ 4∑ i=1 (αi1 + αi2), parµ(E∗) = par deg(E∗)/2. In turn, a line sub-bundle L ⊂ E acquires a set of parabolic weights α′ i defined as αi2 whenever i ∈ I and αi1 otherwise. The induced (semi-)stability condition then reads parµ(L∗) := deg(L) + 4∑ i=1 α′ i < parµ(E∗) (resp. ≤). It can be verified that the last inequality only depends on the effective parameters βi := αi2 − αi1 ∈ [0, 1), i = 1, 2, 3, 4, and reduces to (2.1). A set of parabolic weights will be called admissible whenever the equation par deg(E∗) = 0 is satisfied. The admissibility of parabolic weights, together with the stability condition for parabolic Higgs bundles, are necessary for the non-abelian Hodge correspondence to hold (see [30]), even though the moduli problem is meaningful without the former; e.g., in [26] Mukai considers parabolic weights corresponding to the choices αi1 = 0 and αi2 = βi, for n marked points on CP1. On the other hand, Bauer [1] considers the SU(2)-constraints par deg(E∗) = 0, 0 < αi1 = αi < 1/2, αi2 = 1− αi, i = 1, . . . , n, fixing the value of d. When d ≡ n (mod 2), up to the tensor product of E by a suitable line bundle, every set {βi} can be lifted to a unique set of parabolic weights satisfying the SU(2)- constraints. When d ̸≡ n (mod 2), a lift to a set of admissible parabolic weights is only possible under further numerical constraints, but a weaker lift satisfying the SU(2)-constraints can still be achieved in terms of an extra marked point z0 with parabolic weights α01 = α02 = 1/2 (so that β0 = 0). This operation leaves the moduli problem unchanged, since it forces any flag over E|z0 to be trivial and the corresponding residue Resz0 Φ to vanish.4 The alternative convention −1/2 < −α′ i < α′ i < 1/2 for the SU(2)-constraints is related to the former in terms of the transformations β′ i = 1− βi, i = 1, 2, 3, 4. 4An operation on parabolic bundles discussed in the literature is the parabolic tensor product. We don’t consider it here as it does not preserve the underlying associated bundle End(E). Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 7 3 Geometry of P(Aut(E))-actions and stratifications in QP(E) Let E → CP1 be a holomorphic vector bundle of rank 2 and degree d. A basic holomorphic invari- ant associated to E is its Harder–Narasimhan filtration. In genus 0, the Birkhoff–Grothendieck theorem postulates the existence of an isomorphism of vector bundles E ∼= O(m1)⊕O(m2), m1 ≥ m2, in such a way that d = m1 +m2. Every rank 2 bundle E is either semi-stable (i.e., it splitting coefficients satisfy m1 = m2) or otherwise admits a unique filtration of the form E1 ⊂ E2 = E, deg(E1) > deg(E2)/2. In other words, every isomorphism E ∼= O(m1) ⊕ O(m2) is a refinement of the Harder–Nara- simhan filtration of E. When E is not semi-stable, we have that E1 ∼= O(m1), and a choice of Birkhoff–Grothendieck splitting for E is equivalent to a choice of complementary sub-bundle O(m2) ⊂ E. The integers m1 ≤ m2 are holomorphic invariants for E. E is called evenly-spit if m1−m2 ≤ 1, or equivalently if H1(CP1,End(E)) = 0, where End(E) := E⊗E∨. Consequently, on any holomorphic family F → B × CP1 of rank 2 and degree d holomorphic vector bundles E → CP1, the existence of evenly-split bundles is an open condition. Pointwise multiplication endows the space H0(CP1,End(E)) with an associative algebra structure. The group of global bundle automorphisms of E is the Lie group of invertible elements Aut(E) ⊂ H0 ( CP1,End(E) ) . Its projectivization is defined as P(Aut(E)) := Aut(E)/Z(Aut(E)), where its center Z(Aut(E)) consists of all nonzero multiples of the identity. Aut(E) ∼= GL(2,C) in the special case when E is semi-stable. Otherwise, when m1 > m2, Aut(E) is (m1−m2+3)-dimensional and preserves the Harder–Narasimhan filtration of E. Its unipotent radical is the normal subgroup of unipotent automorphisms of E, and will be denoted by R(Aut(E)). R(Aut(E)) is maximally abelian, has codimension 2 in Aut(E), and acts freely and transitively on the space Lm2(E). In particular, the subgroup R(Aut(E)) ∩ Z(Aut(E)) is trivial. Consider a point z ∈ CP1. For any F ∈ P(E|z), let P(F ) ⊂ GL(E|z) be the parabolic subgroup stabilizing the flag F ⊂ E|z, and R(F ) its unipotent radical. The evaluation maps Aut(E) 7→ Aut(E)|z ⊂ gl(E|z) determine a subgroup of GL(E|z). This subgroup coincides with GL(E|z) when m1 = m2, in which case the evaluation map is an isomorphism. Otherwise, whenm1 > m2, Aut(E)|z coincides with the parabolic subgroup P(E1|z) that is induced by the Harder–Narasimhan filtration of E. In this case we define the affine line Vz(E) := P(E|z)\{E1|z}. Since there is an isomorphism Aut(E)|z ∼= P(E1|z), and P(R(E1|z)) ∼= C acts freely and transi- tively on Vz(E), the cell decomposition P(E|z) = {E1|z} ⊔ Vz(E) is the partition into two orbits determined by the action of P(Aut(E)|z) on P(E|z). 8 C. Meneses 3.1 Line sub-bundles and interpolation of quasi-parabolic structures For any E, we will denote the space of all line sub-bundles L ⊂ E of degree j by Lj(E). Given an isomorphism E ∼= O(m1)⊕O(m2), Lj(E) is modeled by the Zariski open set Lj(O(m1)⊕O(m2)) ⊂ P ( H0 ( CP1,O(m1 − j) ) ⊕H0 ( CP1,O(m2 − j) )) generated by pairs of holomorphic sections (s1, s2) that are not simultaneously zero and have disjoint zero sets. In the particular case when m1 = m2 = m, we have the isomorphisms Lm(E) ∼= CP1, Lm−1(E) ∼= CP3\Segre ( CP1 × CP1 ) . On the other hand, when m1 > m2 there are no line sub-bundles L ⊂ E of degree j for m1 > j > m2, and Lm2(E) ∼= Am1−m2+1 ⊂ CPm1−m2+1. Given I ⊂ {1, 2, 3, 4} and j ≤ m2, we define the O(j)-interpolation locus on QP(E), relative to I, as BI,j(E) := { (F1, F2, F3, F4) ∈ QP(E) : ∃L ∈ Lj(E), Fk = L|zk ∀k ∈ I } , (3.1) which satisfies BI′,j(E) ⊂ BI,j(E) whenever I ⊂ I ′. (3.2) For any vector bundle E, there is a natural action of P(Aut(E)) on its spaces of line sub-bund- les Lj(E). In the particular case when m1 = m2 = m, this reduces to the standard action on the projective line Lm(E) by Möbius transformations, and moreover, Lm−1(E) is a principal homogeneous space for P(Aut(E)). Otherwise, in the case when m1 > m2, Lm2(E) is a principal homogeneous space for R(Aut(E)). Since the action of Aut(E) on the spaces Lj(E) commutes with the operations of evaluation, every BI,j(E) is invariant under the induced P(Aut(E))- action. In the particular case when I = {1, 2, 3, 4}, it follows from the general properties of polynomial interpolation under an arbitrary choice of bundle isomorphism E ∼= O(m1)⊕O(m2) that if dimLj(E) < 4, then the induced line bundle evaluation map evLj(E) : Lj(E) → QP(E), L ⊂ E 7→ ( L|z1 , L|z2 , L|z3 , L|z4 ) is a holomorphic embedding, and evLj(E)(Lj(E)) = B{1,2,3,4},j(E). In addition, when m1 > m2, O(m1)-interpolation relative to a subset I endows QP(E) with the following P(Aut(E))-invariant stratification QP(E) = ⊔ I⊂{1,2,3,4} BI,m1(E), BI,m1(E) := {∏ i∈I {E1|zi} } × { ∏ j∈Ic Vzi(E) } . 3.2 Combinatorial stratification of orbit spaces in QP(E) We will now describe the action of P(Aut(E)) on the spaces QP(E). Due to the structural differences of the groups P(Aut(E)) depending on whether E is semi-simple or not, the treatment is split into the two cases m1 = m2 and m1 > m2. In the first case, we define the configuration space locus to be the following Zariski open subset of QP(E) C(E) := ⋂ |I|=2 Bc I,m(E) ∼= Conf4 ( CP1 ) . Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 9 Proposition 3.1. If m1 −m2 ≤ 2, then P(Aut(E)) acts freely on the Zariski open subset U(E) := QP(E) ∖ ⋃ I⊔I′={1,2,3,4} { BI,m1(E) ∩ BI′,m2(E) } , which is a P(Aut(E))-principal homogeneous space if m1 −m2 = 2, and empty if m1 −m2 > 2. There is a P(Aut(E))-invariant “cross-ratio” map cr : U(E) → CP1 identifying the U(E)-orbit space with a projective line with three (resp. four) double points if m1 = m2 (resp. m1−m2 = 1), and such that cr(U′(E)) = CP1\{z1, 0, 1,∞} for U′(E) :=  C(E) ∖ B{1,2,3,4},m−1(E), m1 = m2 = m, B∅,m1(E) ∖ ⊔ |I′|≥3 BI′,m2(E), m1 −m2 = 1. Proof. Case m1 = m2 = m. All nonempty loci BI,m(E) provide an invariant characterization of the trivialization-dependent identities Fi = Fj , ∀ i, j ∈ I, since for any |I| ≥ 2 BI,m(E) = ⋂ I′⊂I, |I′|=2 BI′,m(E). Therefore, C(E) is the locus in U(E), where P(Aut(E)) acts properly, in such a way that C(E)/P(Aut(E)) ∼= CP1\{0, 1,∞}. Moreover, the six connected components in U(E)\C(E) = ⊔ |I|=2 { BI,m(E) ∩U(E) } are principal homogeneous spaces for P(Aut(E)), and it follows that there is a unique P(Aut(E))- invariant map cr : U(E) → CP1 normalized in the following manner cr ( B{1,2},m(E) ∩U(E) ) = cr ( B{3,4},m(E) ∩U(E) ) = 0, cr ( B{1,3},m(E) ∩U(E) ) = cr ( B{2,4},m(E) ∩U(E) ) = 1, cr ( B{1,4},m(E) ∩U(E) ) = cr ( B{2,3},m(E) ∩U(E) ) = ∞. In turn, since B{1,2,3,4},m−1(E) is also a principal homogeneous space for P(Aut(E)), we have that cr(U′(E)) = CP1\{z1, 0, 1,∞}, since under the previous normalization cr ( B{1,2,3,4},m−1(E) ) = (z1, z2; z3, z4) = z1. Case m1 > m2. In general, we have that for any partition I ⊔ I ′ = {1, 2, 3, 4} and |I| > 2−m1 +m2, BI,m1(E) ∩ BI′,m2(E) = BI,m1(E). Moreover, when |I| ≤ 2 − m1 + m2, the P(Aut(E))-action on BI,m1(E)\BIc,m2(E) is free and proper, as it can be factored through the following intermediate affine quotients{ BI,m1(E)\BIc,m2(E) } /R(Aut(E)) ∼= C3−m1+m2−|I|\{0}, 10 C. Meneses on which P(Aut(E))/R(Aut(E)) ∼= C∗ acts in the standard way. Since we can re-express U(E) = ⊔ |I|≤2−m1+m2 { BI,m1(E)\BIc,m2(E) } , it follows that U(E) is empty if m1 −m2 > 2, a principal homogeneous space for P(Aut(E)) if m1 −m2 = 2, and moreover, that there exists a P(Aut(E))-invariant map U(E) → CP1 when m1 −m2 = 1, which determines an isomorphism U′(E)/P(Aut(E)) ∼= CP1\{z1, 0, 1,∞}, and the complement U(E)\U′(E) consists of eight connected components, namely B∅,m1(E) ∩ B{j,k,l},m2 (E) and B{i},m1 (E)\B{j,k,l},m2 (E), each of which is a principal homogeneous space for P(Aut(E)). These eight orbits determine four different pairs of compactification points for U′(E)/P(Aut(E)). ■ Proposition 3.2. (i) The elements of the sets X(E) := ⋃ |I|=2−m1+m2 { BI,m1(E) ∩ BIc,m2(E) }∖ B{1,2,3,4},m1 (E), Y(E) := ⋃ |I|=3−m1+m2, |I|=1−m1+m2 { BI,m1(E) ∩ BIc,m2(E) }∖ B{1,2,3,4},m1 (E) have pointwise stabilizer subgroups P(Aut(E))(F1,F2,F3,F4) ∼= C∗. (ii) If m1 −m2 ≤ 2 then X(E) and Y(E) are degeneration loci for the orbits in U(E)\U′(E), where U′(E) = ∅ if m1 −m2 = 2, i.e., X(E) ∪Y(E) ⊂ U(E)\U′(E). Proof. The first claim is verified in analogy to the proof of Proposition 3.1. Notice that when m1 > m2, {BI,m1(E) ∩ BIc,m2(E)} ∩ B{1,2,3,4},m1 (E) = ∅, X(E) = ∅ if m1 − m2 > 2, and the stabilizer of any point in X(E) or Y(E) is biholomorphic to P(Aut(E))/R(Aut(E)) ∼= C∗. The second claim is trivial when m1 −m2 = 2. When m1 = m2 = m, the result follows since for any partition I ⊔ I ′ = {1, 2, 3, 4}, |I| = 2{ BI,m(E) ∩ BI′,m(E) } ⊂ BI,m(E) ∩U(E) ∩ BI′,m(E) ∩U(E), while when |I| = 3 BI,m(E) ⊂ ⋂ I′⊂I, |I′|=2 BI′,m(E) ∩U(E). The case m1 −m2 = 1, |I| = 1, follows since for each partition {i} ⊔ {j, k, l} = {1, 2, 3, 4} B{i},m1 (E) ∩ B{j,k,l},m2 (E) ⊂ B∅,m1(E) ∩ B{j,k,l},m2 (E) ∩ B{i},m1 (E)\B{j,k,l},m2 (E), and is trivial when |I| = 2 or |I| = 0 from the definition of U′(E). ■ The stratification of QP(E) can be completed by also grouping quasi-parabolic structures with stabilizer subgroups of higher dimension. We don’t describe these additional strata explicitly since they won’t appear in subsequent constructions. Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 11 4 Structural results and stratification of QPH(E) The construction of geometric models for Harder–Narasimhan strata will be based on the fol- lowing series of results on the structure of spaces of quasi-parabolic Higgs bundles. For any fixed holomorphic rank 2 bundle E → CP1, let QPH(E) denote the space of all quasi-parabolic Higgs bundles (E·,Φ) with quasi-parabolic structure supported on D. We can identify QPH(E) with an algebraic subvariety of the product QP(E)×H0 ( CP1,End(E)⊗KCP1(D) ) characterized as the locus of all points (F1, F2, F3, F4,Φ) satisfying the incidence constraints Reszi Φ ∈ n(E|zi), Fi ⊂ ker(Reszi Φ), i = 1, 2, 3, 4. Equivalently, if we denote by Int(E) the intersection of the four quadrics in the space{ Φ ∈ H0 ( CP1,End(E)⊗KCP1(D) ) : tr(Φ)|zi = 0, i = 1, 2, 3, 4 } defined by the equations det(Φ)|zi = 0, i = 1, 2, 3, 4, then QPH(E) is modeled by the blow-up of Int(E) along each of the codimension 2 loci Reszi Φ = 0, i = 1, 2, 3, 4. (4.1) There is a natural action of Aut(E) on QPH(E), whose point stabilizers contain Z(Aut(E)), descending to an action of the quotient group P(Aut(E)). The proof of the following lemma is obvious. Lemma 4.1. There is a bijective correspondence between isomorphism classes of quasi-parabolic Higgs bundles {(E·,Φ)} and P(Aut(E))-orbits in QPH(E). Since in the present case dimH0(CP1,K2 CP1(D)) = 1, we will define the bulk as QPHC∗(E) := det−1 ( H0 ( CP1,K2 CP1(D) ) \{0} ) , and the nilpotent locus as QPH0(E) := det−1(0). For the embedding ι : QP(E) ↪→ QPH0(E) defined as ι := {0} × id, we will denote Q(E) := ι(QP(E)), which is also characterized as the fixed-point locus of the holomorphic involution defined as τ(F1, F2, F3, F4,Φ) := (F1, F2, F3, F4,−Φ). (4.2) More generally, QPH(E) is equipped with a C∗-action, defined for any c ∈ C∗ as c · (F1, F2, F3, F4,Φ) = (F1, F2, F3, F4, cΦ), c ∈ C∗. (4.3) Since both τ and this C∗-action are P(Aut(E))-equivariant, they descend to an involution and a C∗-action on the space of P(Aut(E))-orbits of QPH(E). While they are only trivial on Q(E), we will verify that their descents possess additional fixed loci. The next results characterize the structure of QPH(E) in terms of the stratification QPH(E) = QPH0(E) ⊔QPHC∗(E). 12 C. Meneses Lemma 4.2. A nonzero parabolic Higgs field Φ preserves a line sub-bundle L(Φ) ⊂ E if and only if it is nilpotent, in which case L(Φ) is unique and determined by ker(Φ). Proof. Given Φ ̸= 0, the equation Φ · L = sL with s ∈ H0(CP1,KCP1(D)) requires that s2 = −detΦ ∈ H0(CP1,K2 CP1(D)), from which s ≡ 0, i.e., Φ is nilpotent. Conversely, if Φ is nilpotent, we can reconstruct L(Φ) in a unique way. Since the zero set of Φ as a holomorphic section of End(E) ⊗ KCP1(D) is finite, we can associate an effective divisor (Φ) to Φ. The complement of such zero set is the Zariski open set U of points z ∈ CP1, where ker(Φ)(z) ⊂ E|z (or ker(Reszi Φ) if z = zi, i = 1, 2, 3, 4) is 1-dimensional. Hence ker(Φ)|U is a line sub-bundle of E|U . Let σ be a holomorphic section of the line bundle [(Φ)] such that (σ) = (Φ). Then by construction σ−1Φ is a holomorphic section of EndE ⊗KCP1(D − (Φ)) such that ker(σ−1Φ)(z) is 1-dimensional ∀ z ∈ CP1. Therefore L(Φ) := ker(σ−1Φ) is a line sub-bundle of E such that L|U = ker(Φ)|U and invariant under Φ over all CP1. ■ Two important conclusions follow from Lemma 4.2: when it exists, an element of QPHC∗(E) can never be unstable, and moreover, the elements in QPH0(E)\Q(E) can be classified according to their unique invariant line sub-bundles, inducing a finer stratification of QPH0(E) that we will describe in detail. Lemma 4.3. QPHC∗(E) is nonempty if and only if m1 − m2 ≤ 2. Given a line sub-bundle L ⊂ E, there is a nonzero nilpotent parabolic Higgs field Φ such that L(Φ) = L if and only if 2(deg(L) + 1) ≥ deg(E). (4.4) Proof. Any given choice of bundle isomorphism E ∼= O(m1)⊕O(m2) induces an isomorphism End(E)⊗KCP1(D) ∼= ( O(2) O(m1 −m2 + 2) O(m2 −m1 + 2) O(2) ) , from which it follows that a parabolic Higgs field on E gets characterized as a holomorphic section Φ = ( u2 −vm1−m2+2 wm2−m1+2 −u2 ) (4.5) whose pointwise evaluation on D moreover satisfies the set of nilpotency constraints det(Φ)|zi = ( −u22 + vm1−m2+2wm2−m1+2 )∣∣ zi = 0, i = 1, 2, 3, 4. It follows that the map det is surjective if and only if m1 −m2 ≤ 2, from which the first claim follows. To prove the second claim, first assume that m1 > m2 and L = E1. A nonzero nilpotent parabolic Higgs field Φ preserving E1 is characterized in terms of the identity wm2−m1+2 ≡ 0 (which is independent of the choice of isomorphism E ∼= O(m1)⊕O(m2)) since the nilpotency constraints imply that u2 ≡ 0 necessarily. Since Φ is identified with a nonzero holomorphic section of O(m1 −m2 + 2), it follows that there is always some Φ such that L(Φ) = E1. Otherwise, assume that L ̸= E1 if m1 > m2, or equivalently, that deg(L) ≤ m2. Given a nonzero nilpotent parabolic Higgs field Φ such that L(Φ) = L, we can choose σk ̸= 0 ∈ H0(CP1, [(Φ)]) for some 0 ≤ k ≤ 2−m1 +m2, such that (σk) = (Φ) and express Φ = σk ( u2−k −v2−m2+m1−k w2−m1+m2−k −u2−k ) , (4.6) such that w2−m1+m2−k ̸= 0 if m1 > m2, and the following equation holds in H0(CP1,O(4− 2k)) u22−k = v2−m2+m1−kw2−m1+m2−k. (4.7) Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 13 The holomorphic sections u2−k and w2−m1+m2−k could have a (simple) common zero only in the exceptional case when m1 = m2 = m and k = 0, when a parabolic Higgs field takes the form Φ = ( u′1v ′ 1 −u′21 v′21 −u′1v ′ 1 ) . (4.8) Since L(Φ) is generated by (u′1, v ′ 1), it follows that L(Φ) ∼= O(m−1). Otherwise, we can assume that u2−k and w2−m1+m2−k don’t have a common zero. Then L(Φ) is generated by (u2−k, w2−m1+m2−k) ∈ H0 ( CP1,O(2− k)⊕O(2−m1 +m2 − k) ) and deg(L(Φ)) = m1 − 2 + k ≥ m1 − 2. If m1 − m2 = 2, then k = 0 and deg(L) = m2. If m1 −m2 = 0 (resp. 1) and k = 0, 1 (resp. 0), then w2−m1+m2−k would have at least one zero, and it would follow from (4.7) that u2−k and w2−m1+m2−k have a common zero, a contradiction. Therefore k = 2 (resp. 1), or equivalently, deg(L(Φ)) = m2. In all of the four possible cases, we conclude that k = 2(deg(L(Φ)) + 1)− deg(E). Therefore, k ≥ 0 is equivalent to the lower bound (4.4), and the claim follows. Conversely, consider first any line sub-bundle L ⊂ E such that deg(L) = m2, generated by a holomorphic section (um1−m2 , w0) with w0 ̸= 0, and assume that (4.4) is satisfied, i.e., m1 −m2 ≤ 2. It readily follows that there exists a holomorphic section v2m1−2m2 ∈ H0 ( CP1,O(2m1 − 2m2) ) , (v2m1−2m2) = 2(um1−m2), solving (4.7), for k = 2−m1+m2. Any Φ constructed according to (4.6), for an arbitrary choice of σ2−m1+m2 ∈ H0(CP1,O(2 −m1 +m2))\{0}, would be such that L(Φ) = L. The remaining case m1 = m2 = m, deg(L) = m− 1 follows analogously. ■ The next result, on the stratifications of QPH0(E) for any bundle splitting type, is immediate from Lemma 4.3. Notice that the C∗-action on QPH(E) preserves its stratification, and that of QPH0(E) in particular. Corollary 4.4. Let R(E) (resp. Sj(E)) denote the locus of quasi-parabolic Higgs bundles such that Φ ̸= 0, for which L(Φ) = E1 (resp. L(Φ) ∈ Lj(E)). QPH0(E) is stratified as QPH0(E) =  Q(E) ⊔ Sm(E) ⊔ Sm−1(E) if E ∼= O(m)⊕O(m), Q(E) ⊔ R(E) ⊔ Sm(E) if E ∼= O(m+ 1)⊕O(m), Q(E) ⊔ R(E) ⊔ Sm−1(E) if E ∼= O(m+ 1)⊕O(m− 1), Q(E) ⊔ R(E) if m1 −m2 ≥ 3. Let Pj(E) → Lj(E) be the tautological principal C∗-bundle associated to the canonical projective embeddings of Lj(E). For any I ⊂ {1, 2, 3, 4}, BlI(CP1) will denote the rational nodal curve resulting from blowing-up CP1 ⊂ CP2 at zi ∀i ∈ I. In particular, Bl{1,2,3,4}(CP1) is the D4-configuration. Consider the P(Aut(E))-equivariant maps LE,j : Sj(E) → Lj(E), (E·,Φ) 7→ L(Φ), and in the case k = 2(j + 1)− deg(E) > 0, also the P(Aut(E))-invariant maps DivE,j : Sj(E) → Sk(CP1), (E·,Φ) 7→ (Φ). 14 C. Meneses Proposition 4.5. The restriction of the C∗-action to each Sj(E) turns it into a principal C∗- bundle, whose base is modeled by the 3-dimensional subvariety of the product Lj(E)× S2(j+1)−deg(E) ( CP1 ) ×QP(E), defined by the incidence constraints Fi = L(Φ)|zi whenever zi ̸∈ (Φ), and projection given by the map LE,j ×DivE,j ×par. In particular, there is an isomorphism Sj(E) ∼= { Pj(E), 2(j + 1)− deg(E) = 0, Bl{1,2,3,4}(CP1)× Pj(E), 2(j + 1)− deg(E) = 1. In general, par(Sj(E)) = ⋃ |I|=2(j+1)−deg(E) BI,j(E). Proof. The map LE,j ×DivE,j ×par is invariant under the C∗-action on QPH(E) by definition. The first claim follows from the definition of Sj(E) as a stratum of QPH(E), and the expression of the parabolic Higgs field Φ of any element (E·,Φ) ∈ Sj(E) in one of the two canonical forms (4.6) and (4.8) under a choice of isomorphism E ∼= O(m1) ⊕ O(m2) as in proof of Lemma 4.3, with k = 2(j + 1)− deg(E), and in such a way that (Φ) = (σk). ■ Proposition 4.6. (i) If E is evenly-split, then par ( QPHC∗(E) ) = U′(E) ⊔X(E), and par ( QPH0(E)\Q(E) ) = X(E) ⊔Y(E). (ii) If m1 −m2 = 2, then par ( QPHC∗(E) ) = par(Sm−1(E)) = X(E). (iii) If m1 −m2 ≥ 2, then par|R(E) is surjective. Proof. (i) Assume that E is evenly-split. We will treat both cases independently using the parametrization (4.5) of parabolic Higgs fields on a quasi-parabolic bundle E· depending on a choice of isomorphism E ∼= O(m1) ⊕ O(m2). When m1 = m2 = m, for a partition {1, 2, 3, 4} = {i} ⊔ {j, k, l}, consider the triple of Lagrange interpolating sections {sj , sk, sl} spanning H0(CP1,O(2)) relative to the triple {zj , zk, zl}. Since the “residue evaluation” map n(Fj)⊕ n(Fk)⊕ n(Fl) → sl(E|zi), (ϕj , ϕk, ϕl) 7→ (ϕjsj + ϕksk + ϕlsl)|zi is an isomorphism, there is a line in n(Fj) ⊕ n(Fk) ⊕ n(Fl) mapping to n(Fi). This implies that any quasi-parabolic Higgs field Φ is expressed as ϕjsj + ϕksk + ϕlsl, and consequently par(QPH(E)\Q(E)) = QP(E). If par(E·,Φ) ∈ BI,m(E) for some |I| ≥ 3 and Φ ̸= 0, then after representing Φ = ϕjsj + ϕksk + ϕlsl as before for some {j, k, l} ⊂ I, we see that dim(Span{ϕj , ϕk, ϕl}) = 1, and conse- quently (E·,Φ) ∈ Sm(E). An analogous argument for par(E·,Φ) ∈ B{1,2,3,4},m−1(E) implies that (E·,Φ) ∈ Sm−1(E). Finally, if par(E·,Φ) ∈ BI,m(E) for some (E·,Φ) ∈ QPHC∗(E) and |I| = 2, then the same argument implies that par(E·,Φ) ∈ {BI,m(E)∩BIc,m(E)}\B{1,2,3,4},m(E) ⊂ X(E). Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 15 This last possibility occurs when the orbit of (E·,Φ) contains an element with parabolic Higgs field Φ = ( 0 −v2 w2 0 ) , (4.9) with (v2) = zi+zj and (w2) = zk+zl for a partition {i, j}⊔{k, l} = {1, 2, 3, 4}. The P(Aut(E))- stabilizers of these points are trivial, and their projection exhaust X(E). The surjectivity of par|QPH(E)\Q(E) and the definition of U′(E) (Proposition 3.1) imply that QPHC∗(E) necessa- rily projects onto U′(E) ⊔ X(E). The characterization of par(QPH0(E)\Q(E)) follows from Corollary 4.4 and Proposition 4.5. When E ∼= O(m+1)⊕O(m), we recall that the divisor (w1) associated to the expression (4.5) is independent of the choice of Birkhoff–Grothendieck splitting and an invariant of any point QPH(E)\Q(E). The projection of the Zariski open subset of QPHC∗(E) for which (w1) ∈ CP1\{z1, 0, 1,∞} equals U′(E) (defined in Proposition 3.1), since for any (F1, F2, F3, F4) ∈ U′(E) and any u2 such that zi ̸∈ (u2) there exists a unique v3 such that ker(Reszi Φ) = Fi, i = 1, 2, 3, 4, and on the other hand, an element QPH(E)\Q(E) projecting to BI,m(E) for any |I| ≥ 3 necessa- rily belongs to Sm(E). In turn, the projection of the complementary loci of points in QPHC∗(E) for which (w1) = zi, i = 1, 2, 3, 4, exhaust the exceptional locus X(E) in Proposition 3.2, via the P(Aut(E))-orbits of elements (E·,Φ) ∈ QPHC∗(E) of the form Φ = ( 0 −v3 w1 0 ) , (4.10) such that (v3) = zj+zk+zl for the partition {i}⊔{j, k, l} = {1, 2, 3, 4}. Moreover, it follows that the P(Aut(E))-stabilizers of these points are trivial as well. Finally, it also follows from (4.5) and Proposition 4.5 that any quasi-parabolic structure in the complement of U′(E) can be always represented as the projection of an element in either R(E) or S0(E), from which the full claim follows. (ii) Whenm1−m2 = 2, a point in QPHC∗(E)⊔S0(E), satisfies w0 ̸= 0, and is fully determined by Φ. Under a choice of isomorphism E ∼= O(m + 1) ⊕ O(m − 1), the P(Aut(E))-orbit of Φ always contains an element of the form Φ = ( 0 −v4 w0 0 ) , from which the claim follows. (iii) The result follows from (4.5) and the constraints wm2−m1+2 = 0 and u2 = 0 under a choice of isomorphism E ∼= O(m1)⊕O(m2). ■ 5 Combinatorial description of parabolic weight polytopes We will present a thorough account of the combinatorial structure on [0, 1]4 that encodes wall- crossing phenomena for the toy model, and refines an explicit embedding of the convex polytope known as the 4-demicube or demitesseract.5 In order to emphasize the features resulting from the inclusion of parabolic Higgs fields into the moduli problem, we will construct this structure from first principles. Consider the functions βI : [0, 1]4 → R, I ⊂ {1, 2, 3, 4}, 5The latter was described by Bauer [1] under the SU(2)-constraints (requiring d to be even in the case of 4 marked points), and later by Biswas [5] for parabolic degree 0, in relation to the moduli problem for parabolic bundles of rank 2. 16 C. Meneses which by definition satisfy the relations βI + βIc = 0 and βI′ ≤ βI whenever I ′ ⊂ I, as well as the bounds −|Ic| ≤ βI ≤ |I|. (5.1) Without any loss of generality (see Remark 2.1), we will reformulate the necessary and sufficient conditions on parabolic weights for the existence of stable parabolic bundles in terms of β-weights and the inequalities (2.1). Proposition 5.1 ([1, 5]). There exists a rank 2 semi-stable parabolic bundle E∗ of degree d with respect to βββ ∈ [0, 1)4 if and only if for any I ⊂ {1, 2, 3, 4} such that d ≡ |I| − 1 (mod 2), βI(βββ) ≤ |I| − 1. (5.2) E∗ is necessarily stable if all inequalities are strict. Proposition 5.1 states necessary and sufficient conditions on βββ ∈ [0, 1)4 to grant the semi- stability of generic quasi-parabolic structures on evenly-split bundles, since then |I| = d− 2 deg(L) + 1 = dimP ( H0 ( CP1, E ⊗ L−1 )) is the minimum number of flags in any E· that can be interpolated by some line sub-bundle L ⊂ E, for which we have that d ≡ |I| − 1 (mod 2). The inequalities (5.2) only depend on the parity of d, and each possibility will be treated independently. In the case when d is even, these inequalities are equivalent to the following interval bounds −2 ≤ βI ≤ 0, |I| = 1 ⇔ 0 ≤ βI′ ≤ 2, |I ′| = 3, (5.3) while when d is odd, we obtain the interval bounds −3 ≤ β∅ ≤ −1 ⇔ 1 ≤ β{1,2,3,4} ≤ 3 (5.4) −1 ≤ βI ≤ 1, |I| = 2. (5.5) For both degree parities, the total number of independent interval bounds is equal to four. Given I ⊂ {1, 2, 3, 4}, let χI : {1, 2, 3, 4} → {0, 1} be its characteristic function. Since β−1 I (−|Ic|) = 4∑ i=1 χIc(i) ei, the map I 7→ vI := β−1 I (−|Ic|) determines a bijective correspondence between subsets I ⊂ {1, 2, 3, 4} and vertices in ∂[0, 1]4. A vertex vI ∈ ∂[0, 1]4 will be called even (resp. odd) if |I| is even (resp. odd). The parity of a vertex is also equal to the parity of its number of nonzero entries. More generally, we have the following convexity result, whose proof is straightforward. Lemma 5.2. Any I ⊂ {1, 2, 3, 4} induces a partition of the set of vertices in ∂[0, 1]4 by the level sets of βI . For every j = 0, . . . , 4, β−1 I (j − |Ic|) contains exactly ( 4 j ) vertices, namely, those whose Hamming distance to vI is j. Each level set is a convex hull of its set of vertices. The partitions induced by I and Ic coincide. The subset of the power set P ({1, 2, 3, 4}) containing sets of even (resp. odd) cardinality will be denoted by P0({1, 2, 3, 4}) (resp. P1({1, 2, 3, 4})). Definition 5.3. An even (resp. odd) partition set is any subset of P0({1, 2, 3, 4}) (resp. P1({1, 2, 3, 4})) containing exactly one element of the pair {I, I ′} for every partition {1, 2, 3, 4} = I ⊔ I ′ by even (resp. odd) subsets. Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 17 Proposition 5.4. Let B0 (resp. B1) be the convex 4-polytope in [0, 1]4 determined by the interval bounds (5.3) (resp. (5.4)–(5.5)). B0 is the convex hull of all even vertices in ∂[0, 1]4. Similarly, B1 is the convex hull of all odd vertices in ∂[0, 1]4. Consequently, both are isomorphic to the 4-demicube. ∂B• is a triangulation with 16 tetrahedral cells, parametrized by partition sets I ⊂ P•({1, 2, 3, 4}) of given parity, as the convex hulls of the sets {vI ∈ B• : I ∈ I}. Proof. The first two statements are straightforward from the explicit form of the interval bounds (5.3)–(5.5). Moreover, 3-cells in each boundary ∂B• correspond to convex hulls of sets of vertices in B• whose Hamming distance is exactly 2. The latter are in bijective correspondence with partitions sets I ⊂ P•({1, 2, 3, 4}) of the given parity, whose cardinalities are equal to 4, as I ↔ {vI ∈ B• : I ∈ I}. ■ Definition 5.5. A semi-stability wall is any hyperplanar region in [0, 1]4 of the form HI,k = β−1 I (k), k ∈ {|I| − 1, |I| − 2, |I| − 3}. A wall HI,k is even (resp. odd) if k is even (resp. odd). The walls HI,|I|−2 will be called interior ; otherwise, they will be called boundary. The definition of semi-stability walls is based on the condition (2.1) and the interval bounds (5.1). Depending on their parity, semi-stability walls are subsets of either B0 or B1. Since HI,|I|−3 = HIc,|Ic|−1, (5.6) even (resp. odd) boundary semi-stability walls are in bijection with subsets I of odd (resp. even) cardinality under the correspondence I 7→ HI,|I|−3. In addition, there are four interior semi- stability walls for each parity, corresponding to the four partitions I⊔I ′ = {1, 2, 3, 4} into subsets of the same cardinality parity, in the form HI,|I|−2 = HI′,|I′|−2. (5.7) For both parities, v1/2 := (1/2, 1/2, 1/2, 1/2) is the intersection of all four interior semi-stability walls. Tables 1 and 2 list all semi-stability walls for both parities. Table 1. List of even semi-stability walls. Boundary H{i},−2 = H{j,k,l},2, {i} ⊔ {j, k, l} = {1, 2, 3, 4} H{j,k,l},0 = H{i},0, {i} ⊔ {j, k, l} = {1, 2, 3, 4} Interior H∅,−2 = H{1,2,3,4},2 H{i,j},0 = H{k,l},0, {i, j} ⊔ {k, l} = {1, 2, 3, 4} Table 2. List of odd semi-stability walls. Boundary H∅,−3 = H{1,2,3,4},3 H{i,j},−1 = H{k,l},1, {i, j} ⊔ {k, l} = {1, 2, 3, 4} H{1,2,3,4},1 = H∅,−1 Interior H{i},−1 = H{j,k,l},1, {i} ⊔ {j, k, l} = {1, 2, 3, 4} 18 C. Meneses Definition 5.6. The even (resp. odd) parabolic weight polytope is the refinement W0 (resp. W1) of [0, 1]4 resulting after the inclusion of all semi-stability walls into B0 (resp. B1). An even (resp. odd) open chamber is the interior of a 4-cell in W0 (resp. W1), or equivalently, any connected component in (0, 1)4\ ∪ HI,k, k ∈ 2Z (resp. k ∈ Z\2Z). An open chamber inside B• is called interior, and exterior otherwise. A tetrahedral cell in ∂B• is of type A if it is also contained in ∂[0, 1]4, and of type B otherwise. Proposition 5.7. There are exactly 8 tetrahedral cells of type A in ∂B•. Tetrahedral cells of type B correspond to the 8 boundary semi-stability walls of a given parity. Proof. A tetrahedral cell belongs to ∂B• ∩ ∂[0, 1]4 if and only if it is the convex hull of a set of 4 vertices of a given parity with a fixed coordinate component value βi = 0, 1, i = 1, 2, 3, 4, yielding 8 possibilities for each parity. It follows from Lemma 5.2 and Proposition 5.4 that the 8 remaining tetrahedral cells in ∂B• are exhausted by the boundary semi-stability walls HI,|I|−3 for all I ⊂ {1, 2, 3, 4} of a given parity. ■ Corollary 5.8 (classification of exterior chambers). At most one inequality in either (5.3) or (5.4)–(5.5) can fail to hold on (0, 1)4, and each boundary semi-stability wall HI,|I|−3 de- termines a unique exterior chamber whose closure contains it. Every exterior chamber to B0 (resp. B1) is the interior of the convex hull of HI,|I|−3 and the vertex vI for a unique I ⊂ {1, 2, 3, 4} of odd (resp. even) cardinality. Proof. The first claim follows from the convexity of the regions βI < |I|−3. Similarly, it follows from the identity (5.6) that there is a bijective correspondence between exterior chambers and regions β−1 I ((−|Ic|, |I| − 3)) for each I ⊂ {1, 2, 3, 4}, in such a way that their chamber parity is given by the parity of |I| − 3. ■ On the other hand, interior chambers on each parabolic weight polytope are effectively clas- sified in terms of the convex geometry of interior semi-stability walls. This is formalized in the next two results. Proposition 5.9. Every 2-cell in the boundary of a given tetrahedral cell in ∂B• is the inter- section of the latter and an interior semi-stability wall of the given parity. Consequently, the restriction W•|B• is equal to the pyramid of B• over the apex v1/2. Proof. By Lemma 5.2, the interior semi-stability wall corresponding to a partition I ⊔ I ′ = {1, 2, 3, 4} as in (5.7) is the convex hull of 6 vertices, obtained by removing {vI , vI′} (of pairwise Hamming distance 4) from the set of 8 vertices of given parity. In this set of 6 vertices, there exist exactly 8 triples of vertices whose pairwise Hamming distance is equal to 2, so that each triple spans a 2-cell in ∂B•. It follows from Proposition 5.4 that every tetrahedral cell in ∂B• is uniquely expressed as the convex hull of one of these triples and either vI or vI′ , in such a way that its intersection with each of the 4 interior semi-stability walls in B• retrieves the 2-cells in its boundary. The second claim is immediate from the convexity of B•, since v1/2 is the common intersection of all interior semi-stability walls of given parity. ■ Corollary 5.10 (classification of interior chambers). Every interior chamber is the intersection of a choice of side regions {βI < |I| − 2: I ∈ I} to all interior semi-stability walls for a unique partition set I ⊂ P•({1, 2, 3, 4}). Equivalently, it is the interior of the convex hull of the apex v1/2 and the tetrahedral cell in ∂B• spanned by {vI′ : I ′ ∈ I}. Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 19 Proof. It follows from Proposition 5.9 that interior chambers in each polytope B• can be parametrized by a choice of side of all interior semi-stability walls. Any of these choices is defined and parametrized by a unique partition set under the correspondence I 7→ ⋂ I∈I RI , RI := β−1 I ( (−|Ic|, |I| − 2) ) , given that for any partition {1, 2, 3, 4} = I ⊔ I ′, the open regions RI and RI′ satisfy vI ∈ RI , vI′ ∈ RI′ , RI ∪RI′ = [0, 1]4, RI ∩RI′ = HI,|I|−2 = HI′,|I′|−2, from which the first claim follows. Similary, it follows from Proposition 5.4 that the closure of any interior chamber contains a unique tetrahedral cell in ∂B•, which is equal to the intersection of ∂B• and the former. Any additional vertex in W• would necessarily arise from the intersection of interior semi-stability walls of a given parity. Since their common intersection equals v1/2, the second claim follows. ■ We will stick to the following notational convention. Interior chambers will be denoted as CI , where I is the partition set parametrizing each of them under the previous correspondence. On the other hand, the exterior chamber to the boundary semi-stability wall HI,|I|−3 will be denoted as CI . The intersection of ∂B• and the closure of an open chamber of the same parity is a tetrahedral cell and characterizes an interior chamber uniquely. Since tetrahedral cells of type B are exterior semi-stability walls, they characterize uniquely both an interior and an exterior chamber as the common boundary of their closures. The type of an interior chamber in W• (or equivalently, its associated partition set) is defined as the type of its tetrahedral cell. Reflection along an exterior semi-stability wall HI,|I|−3 exchanges the vertex vI and the apex v1/2, and determines a bijection between its exterior chamber and its interior chamber of type B. We will say that two interior chambers CI and CI′ are neighboring if a reflection along a semi- stability wall HI,|I|−2 = HI′,|I′|−2 transforms one into the other. The proof of the next corollary is immediate from Corollaries 5.8 and 5.10. Corollary 5.11 (combinatorial wall-crossing). Every interior semi-stability wall of given parity HI,|I|−2 = HI′,|I′|−2 is the union of 4 tetrahedral cells in W•, each of which is the common boundary of one of the four possible pairs of neighboring interior chambers {CI , CI′} of opposite type whose partitions sets satisfy I\{I} = I ′\{I ′}. Every even (resp. odd) exterior semi-stability wall HI,|I|−3 is the common boundary between the exterior chamber CI and the even (resp. odd) interior chamber CI(I) of type B, where I ′ ∈ I(I) ⇔ vI′ ∈ HI,|I|−3. Tables 3 and 4 provide explicit lists of partition sets for each parity, together with the tetra- hedral cells in ∂B· for the corresponding open chambers they parametrize, according to Propo- sition 5.4. Notice that the collections of partition sets of a given type are closed under the operation of taking complements in P•({1, 2, 3, 4}). The group of symmetries of [0, 1]4 is called hexadecachoric group, denoted as B4. It is a group of order 384 generated by permutation of coordinates and reflections along the hyperplanes βi = 1/2, i = 1, 2, 3, 4. Its restriction to reflections along pairs of hyperplanes leads to a distin- guished index 2 subgroup, the Coxeter group D4. 20 C. Meneses Table 3. Even partition sets, i = 1, 2, 3, 4, {j, k, l} = {1, 2, 3, 4}\{i}. Partition set I Tetrahedral cell in ∂B0 Type A {∅, {k, l}, {j, l}, {j, k}} B0 ∩ {βi = 1} {{1, 2, 3, 4}, {i, j}, {i, k}, {i, l}} B0 ∩ {βi = 0} Type B {∅, {i, j}, {i, k}, {i, l}} H{i},−2 {{1, 2, 3, 4}, {k, l}, {j, l}, {j, k}} H{j,k,l},0 Table 4. Odd partition sets, i = 1, 2, 3, 4, {j, k, l} = {1, 2, 3, 4}\{i}. Partition set I Tetrahedral cell in ∂B1 Type A {{i}, {i, k, l}, {i, j, l}, {i, j, k}} B1 ∩ {βi = 0} {{j, k, l}, {j}, {k}, {l}} B1 ∩ {βi = 1} Type B {{1}, {2}, {3}, {4}} H∅,−3 {{i}, {j}, {i, j, l}, {i, j, k}} H{i,j},−1 {{2, 3, 4}, {1, 3, 4}, {1, 2, 4}, {1, 2, 3}} H{1,2,3,4},1 Corollary 5.12. D4 is the group of symmetries of both W0 and W1. Both polytopes are iso- morphic under the action of the group B4/D4 ∼= Z2. Moreover, D4 acts transitively on the sets of exterior chambers, and on the sets of interior chambers of type A and B respectively. The stabilizer of any open chamber is the permutation group of the vertices of its characteristic tetrahedral cell. Definition 5.13. The wall-crossing graph of W• is the 1-skeleton of its dual polytope. It assigns a vertex to every open chamber in W•, and an edge to any pair of open chambers whose closures intersect at a tetrahedral cell. Figure 2. The wall-crossing graph for both W0 and W1. Square vertices correspond to the 8 exterior open chambers. White round vertices correspond to the 8 interior open chambers of type A, while black round vertices to the 8 interior open chambers of type B. Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 21 The wall-crossing graph of W• inherits a coloring from the three possible types of open chambers. The qualitative type of wall-crossing in W• is then measured by the possible vertex types that are connected by a given edge in the wall-crossing graph. Up to the action of the symmetry groupD4, there are only two qualitative types of wall-crossing: (i) between an exterior chamber and its neighboring chamber of type B, whose closures intersect in a boundary wall, and (ii) between neighboring chambers of type A and B, whose closures intersect in an interior wall (Figure 2). 6 Conditional stability and basic building blocks The construction of geometric models for all possible Harder–Narasimhan strata as spaces of stable orbits under P(Aut(E))-actions, as well as the characterization of their wall-crossing under variation of parabolic weights, will be achieved by linking the classification of conditionally stable strata (Definition 6.3) to the combinatorics of weight polytopes. This problem depends on a choice of degree parity for a rank 2 vector bundle. We will consider the two possibilities simultaneously. In order to classify the geometry and topology of all orbit spaces of conditionally stable strata, which in general contain non-Hausdorff loci even after restricting the type of invariant sub-bundles, their “taming” is achieved by dissecting them into elementary Hausdorff irreducible components. The details are summarized in Definition 6.6 and Propositions 6.5, 6.7 and 6.9. We begin with the following corollary. Corollary 6.1. If m1−m2 ≤ 2, then for any βββ := (β1, β2, β3, β4) ∈ (0, 1)4, the set QPHs βββ(E) of stable elements in QPH(E) is nonempty, Zariski open, and contains QPHC∗(E). If m1−m2 ≥ 4, then QPHs βββ(E) = ∅ with respect to any βββ ∈ [0, 1)4. Proof. The first claim follows from Lemmas 4.2 and 4.3, since QPHC∗(E) ̸= ∅ if and only if m1 −m2 ≤ 2. If m1 −m2 ≥ 4, Lemma 4.3 implies that QPH(E) = QPH0(E) = Q(E) ⊔ R(E), and every element in QPH(E) would be destabilized by E1. ■ Remark 6.2. The absence of weight constraints for the existence of moduli spaces of semi-stable parabolic Higgs bundles in genus 0, following from Corollary 6.1, is already implicit, in terms of the general non-abelian Hodge correspondence, in Simpson’s solution of the Deligne–Simpson problem [31]. This is a sharp contrast with moduli spaces of semi-stable parabolic bundles [1, 5], for which Φ ≡ 0. Corollary 6.1 states that for even degree d = 2m, the only Birkhoff–Grothendieck splitting types that can support stable parabolic Higgs bundles are O(m)⊕O(m) and O(m+ 1)⊕O(m− 1), while for odd degree d = 2m+ 1, these are O(m+ 1)⊕O(m) and O(m+ 2)⊕O(m− 1). It follows from general principles that stability of quasi-parabolic Higgs bundles is an invariant along a given open chamber, and moreover, that the subset of points in QPH(E) that are stable with respect to every choice of open chamber in W• is Zariski open. Nevertheless, we will verify that the intersection of the former Zariski open subset with QPH0(E) is always empty with respect to any choice of degree parity, i.e., it either coincides with QPHC∗(E) when m1−m2 ≤ 2, or is empty when m1 − m2 ≥ 3. For any choice of open chamber C ⊂ W•, we will denote the stable locus associated to any βββ ∈ C by QPHs C(E). 22 C. Meneses Definition 6.3. A point in QPH(E) will be called conditionally stable if it is stable with respect to some choice of open chamber C ⊂ W• of corresponding degree parity. The different substrata of conditionally stable quasi-parabolic Higgs bundles in QPH0(E) will be accordingly denoted by Qcs(E), Rcs(E), and Scsj (E). The subset of conditionally stable parabolic Higgs bundles in QPH(E) is equal to the com- plement of the Zariski closed subset of points that are either unstable or strictly semi-stable for any choice of weights βββ ∈ [0, 1)4. By Corollary 6.1, the latter is a subset of QPH0(E). The next lemma is the key to characterize conditionally stable loci systematically. The characterization is performed in Proposition 6.5. Lemma 6.4. (E·,Φ) ∈ QPH0(E)\Q(E) is not stable with respect to any βββ ∈ [0, 1)4 if and only if − ∣∣IcL(Φ),E· ∣∣ ≥ deg(E)− 2 deg(L(Φ)). (6.1) A point (E·, 0) ∈ Q(E) is not stable with respect to any βββ ∈ [0, 1)4 if and only if there is a pair of line sub-bundles L,L′ ⊂ E, L ∼= O(m1) and L′ ∼= O(m2), satisfying {1, 2, 3, 4} = IL,E· ∪ IL′,E· . (6.2) Proof. Since −|Ic| ≤ βI , the inequality (6.1) for some (E·,Φ) ∈ QPH0(E)\Q(E) implies that the inequality (2.1) would never be satisfied in the strict sense for any given choice of βββ ∈ [0, 1)4. The converse statement follows after taking the infimum of (2.1) over the values of βIL(Φ),E· . Similarly, the condition (6.2) implies that for any βββ ∈ [0, 1)4, (E·, 0) ∈ Q(E) is not stable with respect to at least one sub-bundle of the pair {L,L′}. On the other hand, if (6.1) holds for a point (E·, 0) ∈ Q(E) and some L ∼= O(m1), then it follows from the properties of line bundle interpolation that there always exists L′ ∼= O(m2) such that the pair L,L′ satisfies (6.2). Therefore, to prove the remaining implication, it suffices to assume that (E·, 0) ∈ Q(E) is not stable for any βββ ∈ [0, 1)4 while (6.1) doesn’t hold for any L ∼= O(m1). We will treat the possibilities m1 = m2 and m1 > m2 separately. If we assume that m1 > m2, instability of (E·, 0) for all βββ ∈ [0, 1)4, and |IE1,E· | < 4−m1 +m2, then the instability of (E·, 0) for βIE1,E· < m2 −m1 implies the existence of a destabilizing L′ ∼= O(m2), and we obtain IL′,E· = IcE1,E· . In particular, (6.2) holds. Similarly, if we assume that m1 = m2 = m and L ∼= O(m) with |IL,E· | < 4, then the instability of (E·, 0) for all βIL,E· < 0 implies the existence of L′ ∼= O(m) such that IL′,E· = IcL,E· . The case when (F1, F2, F3, F4) ∈ B{1,2,3,4},m(E) holds trivially, since this is the only instance for which IL,E· ∩ IL′,E· ̸= ∅, and then (6.2) holds if we let L′ = L. ■ Proposition 6.5. Let6 RI(E) := R(E) ∩ par−1(BI,m1(E)), SI,j(E) := Sj(E) ∩ par−1(BI,j(E)). (i) The different types of conditionally stable substrata are classified as follows: Qcs(E) = { ι(U(E)) if m1 −m2 ≤ 2, ∅ if m1 −m2 = 3, 6The notation for SI,j(E) is redundant when 2(deg(L(Φ))+1)−deg(E) = 4, that is, when E ∼= O(m)⊕O(m) or E ∼= O(m+ 1)⊕O(m− 1) and j = m− 1, since then S{1,2,3,4},m−1(E) = Sm−1(E). Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 23 Rcs(E) = ⊔ |I|≤3−m1+m2 RI(E), Scsj (E) = {⊔ |I|=2,3 SI,m(E) if m1 = m2 = m, j = m, Sj(E) otherwise. (ii) An element in Qcs(E) is stable with respect to any choice of interior open chamber if and only if it lies in ι(U′(E)). (iii) P(Aut(E)) acts freely on Rcs(E). If m1 = m2 = m then P(Aut(E)) acts freely on Scsm(E) and P(Aut(E)) × C∗ acts freely and transitively on Scsm−1(E). If m1 − m2 = 1, 2 then P(Aut(E)) acts freely and transitively in the factor Pm2(E) of Sm2(E). Proof. The first and second claims follow from Lemma 6.4. The third claim follows from Proposition 4.5, the definition of the subspaces BI,j(E) ⊂ QP(E) in (3.1) together with prop- erty (3.2), the definition of the strata BI,m1(E) when m1 > m2, and the specializations of the canonical form (4.6) (including (4.8)) in each case. Notice that ⊔ |I|≤2RI(E) = ⊔ |I|=1,2RI(E) when m1 −m2 = 1, since R∅(E) = ∅. ■ Definition 6.6. The (possibly empty) basic building blocks of a subset I ⊂ {1, 2, 3, 4}, denoted by KI (resp. LI whenever j = m2), are the P(Aut(E))-orbit spaces of RI(E) (resp. SI,j(E)). In turn, the individual orbits NI are unambiguously defined as NI := ι(BI,j(E) ∩U(E))/P(Aut(E)). When E is evenly-split we will denote the “bulk” of orbits of quasi-parabolic bundles in QP(E) that are stable with respect to any interior open chamber as NB• := ι(U′(E))/P(Aut(E)) ∼= CP1\{z1, 0, 1,∞}. Propositions 4.5 and 6.5 imply that the orbit space of Scsm−1(E) consists of a single basic building block, which is a P(Aut(E))-principal bundle with base a point when E ∼= O(m+ 1)⊕ O(m− 1) and C∗ when E ∼= O(m)⊕O(m). We will denote these as L{1,2,3,4},0 and L{1,2,3,4},1 respectively. Even though they are associated to different bundle splitting types, they glue into a punctured line (Proposition 7.4). For notational simplicity, we will denote the resulting punctured line by L{1,2,3,4}. Proposition 6.7. (i) There is a bijective correspondence between basic building blocks NI (resp. KI , resp. LI) and subsets I ⊂ {1, 2, 3, 4}. (ii) When nonempty, the action of P(Aut(E)) on SI,m2(E) and RI(E) is free and proper. Apart from L{1,2,3,4},0 = pt and L{1,2,3,4},1 ∼= C∗, the nonempty basic building blocks are classified as follows: KI ∼= { C, |I|+m1 −m2 = 2, CP1, |I|+m1 −m2 = 3, LI ∼= { C, |I| −m1 +m2 = 2, CP1, |I| −m1 +m2 = 3. Moreover, for any |I| = 3−m1 +m2, there is an isomorphism KI ⊔ { ⊔ I′⊂I, |I\I′|=1 KI′ } ∼= BlI ( CP1 ) 24 C. Meneses and for any |I| = 3 +m1 −m2, there is an isomorphism LI ⊔ { ⊔ I′⊂I, |I\I′|=1 LI′ } ∼= BlI ( CP1 ) . Proof. The first claim is already implicit in Proposition 3.1 for the basic building blocks of the form NI , and the remaining possibilities are implicit in the second claim. In terms of this correspondence, the resulting orbits are individual points. The second claim is a consequence of the P(Aut(E))-equivariance of par combined with Propositions 4.5 and 6.5. When I is such that |I| = 3−m1 +m2 (resp. |I| = 3 +m1 −m2), the divisor (Φ) associated to any element in RI(E) (resp. SI,m(E)) takes the form (Φ) = z + ∑ j∈Ic zj , in such a way that if z = zi for any i ∈ I, then Fi = L(Φ)|zi . In this case, the correspondence (E·,Φ) 7→ z determines the isomorphism KI ∼= CP1 (resp. LI ∼= CP1). Moreover, when I ′ ⊂ I is such that |I ′| = 2 − m1 + m2 (resp. |I ′| = 2 + m1 − m2), letting I\I ′ = {i} we have that z = zi and Fi ̸= L(Φ)|zi . Then (E·,Φ) 7→ Fi gives a biholomorphism KI′ ∼= C (resp. LI′ ∼= C), and moreover, the biholomorphism KI ⊔ KI′ ∼= Bl{i} ( CP1 ) ( resp. LI ⊔ LI′ ∼= Bl{i} ( CP1 )) , from which the claim follows. ■ There are three non-Hausdorff orbit spaces of conditionally stable strata in QPH0(E)\Q(E), resulting from fixing a type of kernel lines, as a consequence of the pathological orbit space topology of U(E)\U′(E) and Y(E). These are parametrized by the incidence properties of certain projective plane configurations. When m1 = m2, the topology of the union of basic building blocks LI is determined by the incidence properties of the complete quadrilateral (6243) (Figure 3), with all basic building blocks of the form L{j,k,l} (resp. all pairs of the form {L{i,j},L{k,l}} for any partition {i, j} ⊔ {k, l} = {1, 2, 3, 4}) identified under the quotient topology in the full orbit space. When m1 − m2 = 1, the same is true for all basic building blocks KI and the incidence properties of the complete quadrangle (4362) (Figure 4), with all basic building blocks of the form K{k,l} being identified under the quotient topology in the full orbit space. When m1 − m2 = 2, the basic building blocks KI are superimposed as an identification of four blow-ups of a line at a point (Figure 5). A full list of conditionally stable loci grouped by QPH0(E)-stratum type, and their induced orbit space type, is compiled in Table 5. For each degree parity, there is a bijective correspondence between nontrivial basic building blocks and subsets I ⊂ {1, 2, 3, 4}. We conclude with a combinatorial rule to organize their families according to their biholomorphism type. When the parities of |I| and d differ, basic building blocks are biholomorphic to CP1 and in bijective correspondence with the sets of exterior semi-stability walls. Concretely, the collec- tions are{ {KI : |I| = 1} and {LI : |I| = 3} if d is even, K∅, {KI : |I| = 2} and L{1,2,3,4} if d is odd. (6.3) Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 25 Table 5. Master kit of orbit spaces for all possible conditionally stable loci, following from Proposi- tions 6.5 and 6.7. m1 −m2 Conditionally stable locus Orbit space 0 Qcs(E) = ι(U(E)) CP1 w. 3 double points Scsm(E) = ⊔ |I|=2,3 SI,m(E) (6243)-family (Figure 3) Scsm−1(E) = Sm−1(E) C∗ 1 Qcs(E) = ι(U(E)) CP1 w. 4 double points Rcs(E) = ⊔ |I|=1,2 RI(E) (4362)-family (Figure 4) Scsm(E) = Sm(E) D4-config. (Figure 1) 2 Qcs(E) = ι ( B∅(E)\B{1,2,3,4},0(E) ) point Rcs(E) = ⊔ |I|=0,1 RI(E) (1− 4)-family (Figure 5) Scsm−1(E) = Sm−1(E) point 3 Qcs(E) = ∅ – Rcs(E) = R∅(E) CP1 a b Figure 3. (a) The complete quadrilateral or (6243)-configuration. (b) Maximal Hausdorff unions of basic building blocks in the orbit space of Scs0 (E) when E ∼= O(m) ⊕ O(m), parametrized by the sets {{j, k}, {j, l}, {k, l}, {j, k, l}}, j ̸= k, j ̸= l, k ̸= l. a b Figure 4. (a) The complete quadrangle or (4362)-configuration. (b) Maximal Hausdorff union of basic building blocks in the orbit space of Rcs(E) when E ∼= O(m + 1) ⊕ O(m), parametrized by the sets {{k}, {l}, {k, l}}, k ̸= l. 26 C. Meneses ∅ Figure 5. The (non-Hausdorff) orbit space of Rcs(E) when E ∼= O(m + 1) ⊕ O(m − 1), formed by union of the punctured line K∅ and the quadruple line ∪4 i=1K{i}, with maximal Hausdorff subspaces K{i} ⊔ K∅ ∼= Bl{i}(CP1), i = 1, 2, 3, 4. In turn, a nontrivial basic building block is biholomorphic to C if the parities of |I| and d coincide. It is convenient to group these in pairs corresponding to partitions I ⊔ I ′ = {1, 2, 3, 4} of even and odd type, as those pairs are in bijective correspondence with interior semi-stability walls. This will be the basis to the classification of interior wall-crossing. The resulting pairs are {{ L{i,j},L{k,l} } and { K∅,L{1,2,3,4} } if d is even,{ K{i},L{j,k,l} } if d is odd. (6.4) We will also group N{1,2,3,4} := ι(B{1,2,3,4},m−1(E))/P(Aut(E)) when m1 = m2 = m, and N∅ := ι(U(E))/P(Aut(E)) when m1 − m2 = 2. With the exception of the previous pair of orbits occurring over different bundles, whose topology will be considered subsequently, all other pairs of the form {NI ,NIc} are orbits in the same ambient space having intersecting closures. When E evenly-split, these are the orbits determining double points in the orbit space of Qcs(E), locally compactifying the corresponding puncture sphere model of NB• . ∅ ∅ a b Figure 6. Nilpotent cone assembly kits for even degree interior chambers CI of type A. Left (resp. right) columns correspond to E ∼= O(m) ⊕ O(m) (resp. E ∼= O(m + 1) ⊕ O(m − 1)). (a) I = {∅, {k, l}, {j, l}, {j, k}}. (b) I = {{1, 2, 3, 4}, {i, j}, {i, k}, {i, l}}. Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 27 Definition 6.8. The nilpotent cone assembly kit of an open chamber C ⊂ W• of given parity is the collection of stable components in all conditionally stable orbit spaces associated to all strata in the two admissible nilpotent loci. Proposition 6.9 (open chamber stable orbit loci). (i) The point NI is stable with respect to an interior chamber CI if and only if I ∈ I. (ii) A I-basic building block is stable with respect to an interior chamber CI if and only if I ∈ I, and stable with respect to an exterior chamber CI′ if and only if I ∈ I(I ′) and I ̸= I ′. Proof. Immediate from the definition of stability and Corollaries 5.8, 5.10 and 5.11. ■ Figures 6–11, built from Definition 6.8 and Propositions 6.7 and 6.9, compile the geometry and combinatorics of all nilpotent cone assembly kits in both parities, with central spheres represented as horizontal lines. In particular, when d is odd, there is a single Harder–Narasimhan stratum for all open chambers (corresponding to E ∼= O(m+1)⊕O(m)), except for the exterior chamber C∅. ∅ ∅ a b Figure 7. Nilpotent cone assembly kits for even degree interior chambers CI of type B. Left (resp. right) columns correspond to E ∼= O(m) ⊕ O(m) (resp. E ∼= O(m + 1) ⊕ O(m − 1)). (a) I = {∅, {i, j}, {i, k}, {i, l}}. (b) I = {{1, 2, 3, 4}, {k, l}, {j, l}, {j, k}}. 7 Proof of main results and further properties 7.1 Proof of Theorem 1.1 We will combine all previous results to construct all Harder–Narasimhan strata MC(E) of a given open chamber C as P(Aut(E))-orbit spaces. The missing information to do so is con- tained in Proposition 7.1, where we relate the geometry of bulk orbit spaces and basic building blocks. 28 C. Meneses ∅ a b Figure 8. Nilpotent cone assembly kits for even degree exterior chambers CI . Left (resp. right) columns correspond to E ∼= O(m)⊕O(m) (resp. E ∼= O(m+ 1)⊕O(m− 1)). (a) I = {i}. (b) I = {j, k, l}. a b Figure 9. Nilpotent cone assembly kits for odd degree interior chambers of type A, all in E ∼= O(m+ 1)⊕O(m). (a) I = {{i}, {i, k, l}, {i, j, l}, {i, j, k}}. (b) I = {{j, k, l}, {j}, {k}, {l}}. Proposition 7.1 (orbit space models in QPH(E)). (i) The group P(Aut(E)) acts freely on QPH(E)\ι(U(E)c). (ii) For E evenly-split, P(Aut(E)) acts properly on QPHC∗(E), turning it into a principal P(Aut(E))-bundle over (ΣD\{w1}) × C∗ when E ∼= O(m) ⊕ O(m) and ΣD × C∗ when E ∼= O(m+ 1)⊕O(m). (iii) If m1 −m2 = 2, the map par× det determines the isomorphism QPHC∗(E) ⊔ S0(E) ∼= Pm−1(E)×H0 ( CP1,K2 CP1(D) ) , inducing the biholomorphism{ QPHC∗(E) ⊔ S0(E) } /P(Aut(E)) ∼= H0 ( CP1,K2 CP1(D) ) . Proof. (i) Propositions 3.1 and 3.2 establish the claim over Q(E). The triviality of P(Aut(E))- stabilizers for points in QPH(E)\Q(E) follows from Proposition 4.6(i). Namely, the claim is Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 29 a b c Figure 10. Nilpotent cone assembly kits for odd degree interior chambers of type B, all in E ∼= O(m + 1) ⊕ O(m). (a) I = {{1}, {2}, {3}, {4}}. (b) I = {{i}, {j}, {i, j, l}, {i, j, k}}. (c) I = {{2, 3, 4}, {1, 3, 4}, {1, 2, 4}, {1, 2, 3}}. ∅ a b c Figure 11. Nilpotent cone assembly kits for odd degree exterior chambers CI . Left (resp. right) columns correspond to E ∼= O(m + 1) ⊕ O(m) (resp. E ∼= O(m + 2) ⊕ O(m − 1)). 1) I = ∅. 2) I = {i, j}. 3) I = {1, 2, 3, 4}. trivial over par−1(U′(E)), and on QPHC∗(E) ∩ par−1(X(E)) it follows from the explicit form of the residual C∗-automorphism action on the canonical forms (4.9)–(4.10). Corollary 4.4, Proposition 4.5, and the canonical form (4.5) for elements in R(E) when m1 − m2 = 1 imply the claim over QPH0(E)\Q(E). 30 C. Meneses (ii) Both claims follow in analogy to (i), after restriction to QPHC∗(E). For any q ∈ H0(CP1,K2 CP1(D))\{0}, the orbits in det−1(q) ∩ par−1(X(E)) are fixed by τ , and the limit properties of X(E) with respect to U(E) described in Proposition 3.2 imply that the orbit space of det−1(q) is biholomorphic to the punctured elliptic curve ΣD\{w1} when m1 = m2 = m, and to ΣD when m1 −m2 = 1. (iii) In analogy to (ii), the divisor of any parabolic Higgs field Φ is necessarily trivial for both strata in QPHC∗(E) ⊔ S0(E) when m1 −m2 = 2, and every point is uniquely determined by Φ, which admits a unique decomposition of the form Φ = Φ0 +Φ1, det(Φ0) = 0, Φ1 ∈ H0 ( CP1,End(E)⊗KCP1 ) . Moreover, the map Φ1 7→ det(Φ0 +Φ1) is a linear isomorphism to H0(CP1,K2 CP1(D)) for any fixed Φ0, from which the first claim follows. The second claim follows from Proposition 6.5. ■ Proof of Theorem 1.1. The case m1 − m2 = 3 follows from Propositions 6.5, 6.7 and 6.9, since QPHC∗(E) = ∅ (Lemma 4.3), and is trivial for all open chambers other than C∅, in which case the Harder–Narasimhan stratum is a projective line (Figures 9–11). The case m1 −m2 = 2 can be concluded from Propositions 6.9 and 7.1(iii) (see Figures 6–8; Figure 12 illustrates the topology of the Harder–Narasimhan stratum when its nilpotent cone component is a nodal rational curve). When E is evenly-split, both claims would follow as a consequence of Proposition 7.1(i) and (ii) once the orbit limits of QPHC∗(E) as det(Φ) → 0 are established. Proposition 6.9 ensures that for every possible choice of open chamber, this limit would be a rational curve, possibly noncompact and disconnected, with at most nodal points. The possibilities for both degree parities are illustrated in detail in the left column of Figures 6–11. By construction, the resulting orbit spaces are smooth surfaces. ■ ∅ ∅ ∅ Figure 12. Maximal Harder–Narasimhan strata in the splitting E ∼= O(m+1)⊕O(m− 1) correspond- ing to the exterior chambers C{i}, consisting of the image of a Hitchin section and two nilpotent cone irreducible components. 7.2 Proof of Theorems 1.2 and 1.3 In order to understand the geometry of gluing of strata, we will present a series of results on orbit space limits associated to jumping families of even (resp. odd) degree bundles, i.e., 1-parameter families F = {Et : t ∈ C} for which E0 ∼= O(m+ 1)⊕O(m− 1) ( resp. E0 ∼= O(m+ 2)⊕O(m− 1) ) Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 31 and Et is evenly-split if t ∈ C∗. A holomorphic family of cocycles {gt01 : t ∈ C} with respect to the standard affine cover {U0 = CP1\{∞}, U1 = CP1\{0}} of CP1, with coordinates z ∈ U0, ζ ∈ U1 related on the intersection U01 as ζ = 1/z, can be given as follows gt01(z) =  ( zm+1 0 t zm−1 ) , d even,( zm+2 0 t zm−1 ) , d odd. That the corresponding vector bundles Et in each degree parity are evenly-split when t ∈ C∗, is elucidated in the matrix factorizations of the form gt01 = gt0g01(g t 1) −1, ( zm+1 0 t zm−1 ) = ( 1 −z 0 −t )( zm 0 0 zm )( z−1 −1 −t 0 )−1 , d even, ( zm+2 0 t zm−1 ) = ( 1 −z2 0 −t )( zm+1 0 0 zm )( z−1 −1 −t 0 )−1 , d odd, (7.1) inducing the cocycle equivalences gt01 ∼= g01 with the standard evenly-split cocycle. The impli- cations of the existence of jumping families on the limiting behavior of the relevant orbit spaces of quasi-parabolic structures are gathered in the next lemma. Lemma 7.2. Let {Et : t ∈ C} be a jumping family of vector bundles of arbitrary degree. Then lim t→0 Y ( Et ) ⊂ Y ( E0 ) . If d is even, then lim t→0 B{1,2,3,4},m−1 ( Et ) = X ( E0 ) . Proof. It follows from the definition of Y(E) in Proposition 3.2, when E is not evenly-split, that Y(E) =  4⊔ i=1 {{E1|zi}} × {∏ j ̸=i P(E|zi) } , d even, QP(E), d odd. The first claim follows from the cocycle factorizations (7.1), after acting with the local change of trivializations gt0 and gt0 on the elements of Y(E), E evenly-split. If d is even, the second statement is verified in the same way after normalizing elements in B{1,2,3,4},1(E t), t ∈ C∗, in the form (z1, 0, 1,∞), under the identification C (Et) ∼= Conf4(L0(E t)) and the isomorphism L0(E t) ∼= CP1. ■ We will introduce additional notation for special components of orbit spaces in QPHC∗(E). Recall that for m1 −m2 ≤ 2, the connected components of the sets X(E) are parametrized by I ⊔ I ′ = {1, 2, 3, 4} of the corresponding degree parity. Concretely, these are {i, j} ⊔ {k, l}, m1 = m2 = m, {i} ⊔ {j, k, l}, m1 −m2 = 1, ∅ ⊔ {1, 2, 3, 4}, m1 −m2 = 2. 32 C. Meneses For every partition I⊔I ′ = {1, 2, 3, 4}, we will denote its corresponding orbit space in QPHC∗(E) by H{I,I′}. In all cases, the isomorphism H{I,I′} ∼= C∗ defined by the determinant map Φ 7→ det(Φ) readily follows from Proposition 7.1. Moreover, in the case when m1 −m2 = 2, we have that H{∅,{1,2,3,4}} = QPHC∗(E)/P(Aut(E)). The next corollary is a consequence of Lemma 7.2 (see Figure 12). Corollary 7.3. Every orbit in H{∅,{1,2,3,4}} is the limit of a holomorphic family of orbits in QPHC∗(E)/P(Aut(E)), with E ∼= O(m)⊕O(m). Proposition 7.4 (limits of P(Aut(E))-orbit families). (i) If d is even, there is a jumping family such that lim t→0 N t {1,2,3,4} = N 0 ∅ , while lim t→0 ⊔ L t {j,k,l} = ⊔ K 0 {i}, and moreover, there is a family of orbits {(Et · ,Φ t)} ∈ L{1,2,3,4},1 such that lim t→0 {( Et · , 1 t Φt )} = L 0 {1,2,3,4},0. In particular, L{1,2,3,4},1 ∼= C∗ is compactified by the orbits L{1,2,3,4},0 and N{1,2,3,4}. (ii) If d is odd, then there is a jumping family for which lim t→0 L t {1,2,3,4} = K 0 ∅ . Proof. Let E = O(m)⊕O(m) and consider any nonzero parabolic Higgs field of the form Φ = ( u1v1 −u21 v21 −u1v1 ) , (u1) ̸= (v1) whose kernel line is an arbitrary element of L1(E), and consider the family of quasi-parabolic Higgs bundles {(Et,Φt) : t ∈ C∗} that is induced in terms of the local form Φt|U0 := tgt0(Φ|U0) ( gt0 )−1 . Since lim t→0 ker ( Φt ) = E0 1 , it follows from Lemma 7.2 that the corresponding P(Aut(E0))-orbit equals L 0 {1,2,3,4},0. The remaining claims follow analogously from Lemma 7.2 and Proposition 6.7, given that par ( ⊔ |I|=3+m1−m2 SI,m(E) ) = Y(E) when E is evenly-split, and otherwise par ( ⊔ |I|=3−m1+m2 RI(E) ) = Y(E). ■ It follows from Propositions 7.1 and 7.4 that for each choice of degree parity, the non-Hausdorff phenomena on conditionally stable loci concentrates on the basic building block data (6.3) and (6.4). This is stated in the following corollary. Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 33 Definition 7.5. The central sphere SI for an interior open chamber CI is the associated moduli space of stable parabolic bundles NCI . The central sphere SI for an exterior open chamber CI is its corresponding stable basic building block (6.3). Corollary 7.6. The union of all exterior central spheres SI of a given degree parity determines a line of 8-tuple points. The pair (6.4) of a given partition I ⊔ I ′ = {1, 2, 3, 4} determines a punctured line of double points compactified by {NI ,NI′}. The union of all conditionally stable loci of given degree parity conforms a D4-configuration whose nodes are 10-tuple points, its central punctured sphere is formed by 9-tuple points, and its 4 complementary components are formed by double points. Proof of Theorem 1.2. It follows from Proposition 7.1 together with Corollary 7.3 that when d is even, H{∅,{1,2,3,4}} glues into the bulk QPHC∗(E) for m1 = m2 = m as a branch locus component under τ over the ramification point w1, in a compatible way with respect to the map det. This determines an isomorphism for both degree parities between det−1(H0(CP1,K2 CP1(D)) \{0}) and ΣD×C∗ which is independent of the choice of open chamber. Similarly, it follows from Proposition 7.4 that for all open chambers, the different components in the nilpotent cone as- sembly kit (whose explicit classification is compiled in Figures 6–11) glue into a D4-configuration (Figure 1), on which det extends to 0 as a consequence of Proposition 7.1. The resulting explicit geometry of the map det implies the isomorphism MC ∼= Mtoy for all degrees and all choices of open chambers. ■ Proof of Theorem 1.3. Since wall-crossing is concentrated along the locus det−1(0) in each MC , all that remains to be done after having established Theorems 1.1 and 1.2 is to understand the resulting transformations in det−1(0) when either an interior or exterior semi- stability wall is crossed. The claim for both possibilities follows if we combine Corollaries 5.11 and 5.12 with Corollary 7.6, as the latter ensures that all nilpotent cone components that can be pairwise exchanged according to the combinatorial rules dictated by the former share the same topology in their respective geometric model. ■ Remark 7.7. In [6, Section 6.7], Blaavand considers parabolic Higgs bundles of parabolic degree zero on vector bundles of degree −1, subject to the constraints α11 = α21 = α31 = 0 and α12 + α22 + α32 + α41 + α42 = 1. Those constraints confine the parabolic weights βββ to lie in the open chamber C{1,2,3,4}. Consequently, the moduli spaces that he considers correspond to the geometric model MC{1,2,3,4} . Remark 7.8. The isomorphisms for the different open chamber geometric models are also implicit in Corollary 5.12. In fact, all the constructed geometric models are obviously isomorphic as complex surfaces, independently of the choice of degree parity. One standard strategy to construct isomorphisms between moduli spaces associated to parabolic structures is known in the literature under the name of elementary transformations [19, 20]. We have chosen not to invoque that notion in this work, as our method of construction of geometric models is not compatible with it: the groups and associated orbits are fundamentally different for each choice of degree parity. 7.3 The Hitchin fibration, C∗-action, and Hitchin sections Each geometric model MC can be endowed with a product of transversal projections π1 × π2 : MC → CP1 ×H0 ( CP1,K2 CP1(D) ) , where π1 maps onto the central sphere of MC , while π2 is the standard Hitchin elliptic fibration π2([(E∗,Φ)]) = det(Φ). 34 C. Meneses By Proposition 7.1, the nonzero fibers π−1 2 (q), q ̸= 0 are isomorphic to the elliptic curve ΣD of the pair (CP1, D), with an involution induced from (4.2) that we will keep denoting by τ . It follows from Proposition 7.1 that the resulting branch loci are parametrized by the connected components of the union of the sets X(E) of a given degree parity, and consequently, correspond to the four components H{I,I′} parametrized by partitions I⊔I ′ = {1, 2, 3, 4} of the given parity. The zero fiber π−1 2 (0) is called the nilpotent cone of the geometric model MC . In general the Zariski open sets MC\π−1 2 (0) are independent of the choice of open chamber C. The definition of π1 requires careful attention to details and will be presented in several stages. In essence, the restriction π1 × π2|MC\π−1 2 (0) is defined as the induced branched cover MC\π−1 2 (0) 2:1−−→ CP1 ×H0 ( CP1,K2 CP1(D) ) \{0}, (7.2) under identification of the first factor in the image with the central sphere, and the extension to the nilpotent cone is then defined as its collapse to the former. We will make this precise in terms of the global properties of the C∗-action induced by (4.3) on any geometric model MC , described in Propositions 7.9 and 7.10. Yet another consequence of Propositions 4.6 and 6.5 – in analogy to the proof of Proposi- tion 6.7 – is that for any basic building block in a pair (6.4), there is exactly one P(Aut(E))-orbit contained in par−1(X(E)). In the special case when m1 −m2 = 2 and I = {1, 2, 3, 4}, this orbit coincides with L{1,2,3,4},0. Depending on the basic building block they belong to in a given pair, we will denote them as K + I ( resp. L + I ) . For any partition I ⊔ I ′ = {1, 2, 3, 4}, Corollary 7.6(ii) implies that the pair {K + I ,L + I′ } (or {L + I ,L + I′ } if |I| = |I ′| = 2) conforms a double point. Proposition 7.9. Any central sphere is pointwise-fixed by the C∗-action. For any complemen- tary nilpotent cone component in a pair (6.4), K + I (resp. L + I ) is fixed by the C∗-action, while KI\K + I (resp. LI\L + I ) is a C∗-principal homogeneous space. Moreover, in the latter case, any orbit {(E·,Φ)} ∈ KI\K + I (resp. LI\L + I ) satisfies lim c→∞ c · {(E·,Φ)} = K + I ( resp. L + I ) , while for any partition I ⊔ I ′ = {1, 2, 3, 4} of given parity and any orbit {(E·,Φ)} ∈ H{I,I′}, lim c→0 c · {(E·,Φ)} = { K + I ,L + I′ } ( or { L + I ,L + I′ } if |I| = |I ′| = 2 ) . Proof. The first statement is clear when the central sphere is a moduli space of stable parabolic bundles. Otherwise, a conditionally stable P(Aut(E))-orbit is fixed under the induced C∗-action if and only if for any choice of c ∈ C∗, the equation (F1, F2, F3, F4, cΦ) = (g · F1, g · F2, g · F3, g · F4,Ad(g)(Φ)) (7.3) can be solved for some g ∈ P(Aut(E)). We will independently consider the two families of basic building blocks according to their stratum type in QPH0(E). Under a choice of bundle isomor- phism E ∼= O(m1)⊕O(m2), every orbit in a basic building block LI contains a representative (F1, F2, F3, F4,Φ) whose parabolic Higgs field takes the form Φ = ( 0 0 w2−m1+m2 0 ) , m1 −m2 = 0, 1, 2. Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 35 In all cases, the set of these canonical forms is a principal homogeneous space for a subgroup in P(Aut(E)) isomorphic to C∗. It follows from Proposition 6.7 and the definition of Y(E) for every admissible splitting type, that when an orbit belongs to the central sphere SI of an exterior chamber CI , the equation (7.3) can be solved on its subset of canonical form elements, and the C∗- action leaves the orbit invariant. In the case of a nilpotent cone’s complementary component, the same holds for the special orbit L + I by the definition of X(E), while the action of the subgroup stabilizing canonical forms is trivial on quasi-parabolic structures associated to LI\L + I . Hence it coincides with the standard C∗-action, and determines a principal homogeneous space structure on it. The final claim, on the orbit limits under the C∗-action, follows once again from the limiting properties of X(E) and Y(E) with respect to the corresponding P(Aut(E)) action on the loci U(E). On the other hand, in the case of conditionally stable basic building blocks KI , the parabolic Higgs field of all elements in every given orbit is always in the normal form Φ = ( 0 −v2+m1−m2 0 0 ) , m1 −m2 = 1, 2, 3. Such parabolic Higgs field’s normal forms are R(Aut(E))-invariant by definition, and the residual group of bundle automorphisms acting on the underlying space {E·} of quasi-parabolic bundles associated to the orbit is isomorphic to C∗. The same argument as before also verifies the claim in this case. ■ Proposition 7.10. For every interior open chamber CI of given parity, the elliptic fibration π2 possesses four natural holomorphic sections, extending the connected components of the branch locus of (7.2) to π−1 2 (0), parametrized by the subsets I ∈ I, σI : H0 ( CP1,K2 CP1(D) ) → MCI such that σI ( H0 ( CP1,K2 CP1(D) ) \{0} ) = H{I,I′}, σI(0) = K + I′ ( or L + I ) . For every exterior chamber CI′′, an analogous construction holds in terms of its neighboring interior chamber CI(I′′) of type B. Proof. Recall that (4.9) and (4.10) establish correspondences between nonzero elements q ∈ H0 ( CP1,K2 CP1(D) ) \{0} and orbits of parabolic Higgs bundles with parabolic Higgs field Φ = ( 0 −v2+m1−m1 w2−m1+m2 0 ) in terms of factorizations q = v2+m1−m1w2−m1+m2 , parametrized by partitions I⊔I ′ = {1, 2, 3, 4} induced by fixing the divisors (v2+m1−m1) and (w2−m1+m2). For each degree parity, there are exactly four partitions, and for any such partition, we can assume without loss of generality that I ∈ I. By Corollary 7.6 and Proposition 7.9, the induced isomorphism H0(CP1,K2 CP1(D))\{0} ∼= H{I,I′} can be extended uniquely to a section σI as σI(0) := K + I (or σI(0) := L + I if that were the case for I). Since exterior wall-crossing leaves all basic building blocks in the pairs (6.4) invariant, in the case of an exterior chamber CI′′ , the corresponding holomorphic sections can be defined to be the same as the corresponding sections of MCI(I′′) . ■ 36 C. Meneses We will refer to the previous holomorphic sections as the Hitchin sections of the geometric model MC . When d is even, three of the Hitchin sections are defined in the Zariski open Harder– Narasimhan stratum of MC , while the fourth one belongs to the complementary stratum. This fourth section corresponds to the holomorphic section constructed by Gothen–Oliveira [11]. In turn, when d is odd, all Hitchin sections always belong to the Zariski open Harder–Narasimhan stratum of MC . Corollary 7.11. Under the conventions of Proposition 7.10, for every open chamber C ⊂ W• there is a biholomorphism MC ∖⊔ σI ( H0 ( CP1,K2 CP1(D) )) ∼= T ∗CP1 (7.4) mapping the C∗-action into the standard cotangent bundle C∗-action. The limit c → 0 collapses the Zariski open set MC\ ⊔ σI′(H 0(CP1,K2 CP1(D))) to the central sphere, and every component H{I,I′} to the point K + I (or L + I ). Remark 7.12. For each interior chamber CI , the points {NI : I ∈ I} ⊂ SI correspond to isomorphism classes of stable parabolic bundles supporting nilpotent Higgs fields. Altogether, they are called the wobbly locus in the literature. By Corollary 7.6, the induced loci on all central spheres recovers the pair (CP1, D) for any open chamber. Summarizing, we conclude that the projection π1 for an interior chamber CI can be defined as an extension of the forgetful map π1([(E∗,Φ)]) = [E∗] over T ∗SI ⊂ MCI , under the implicit isomorphism SI ∼= CP1. In virtue of Corollary 7.6, π1 is also defined for each exterior chamber CI as a projection onto the corresponding exterior central sphere SI ∼= CP1. In both cases, the restriction π1|π−1 2 (0) is the map collapsing the nilpotent cone to its central sphere SC along the wobbly locus. In view of the restrictions (7.2) and (7.4) and Corollary 7.11, every geometric model MC is the simultaneous extension of each of the two complex surfaces T ∗CP1 and ΣD × C∗ towards the Hitchin section loci and the nilpotent cone, respectively, in such a way that the intersection of any elliptic fiber of π2 and the Hitchin section loci corresponds to the set of Weierstrass points of the former. In the case when C is an interior chamber and NC ̸= ∅, π1 corresponds to the dominant abelianization morphism constructed in [2] for the only possible spectral curve ΣD, whose very stable locus equals NB• . The C∗-action on MC corresponds to the standard C∗-action on both surfaces. The characterization of the limits of this action follows from Propositions 7.9 and 7.10 (Figure 13). There is always a bijective correspondence between C∗-fixed points not in the central sphere and points in the wobbly locus. 7.4 Weight degeneration to semi-stability walls We will finish this section with a brief discussion of an interpretation of the notion of parabolic S-equivalence for any choice of parabolic weights along the intersection of any given semi- stability wall with the interior of [0, 1]4. Strictly speaking, it is necessary to exclude the points in ∂[0, 1]4, as they lead to a degeneration of the moduli problem by definition, to a strictly smaller number of marked points in CP1. Recently, Godinho–Mandini [10] have studied the geometry of the special degeneration βββ = 0 of the Zariski open set of holomorphically trivial parabolic Higgs bundles (in the case when d = 0), in terms of a correspondence with the so-called null hyperpolygons. Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 37 Figure 13. The topology of the union of the nilpotent cone and the four Hitchin sections, indicating in red the four 0-dimensional and one 1-dimensional components of fixed points of the global C∗-action in the geometric model, to which MC collapses in the limit c → 0. Proposition 7.13. (i) For any βββ ∈ H̊I,|I|−3, all points in the central spheres SI and SI(I) = NCI(I) are strictly semi-stable. (ii) For any βββ ∈ H̊I,|I|−2 = H̊I′,|I′|−2 and a corresponding pair {CI , CI′} of neighboring interior chambers, all points in the associated pair (6.4), as well as the pair {NI ,NI′}, are strictly semi-stable. Additionally, the orbits of ι(BI,j(E) ∩ BI′,j(E)) ⊂ ι(X(E)) are strictly semi- stable. Proof. The result is a straightforward consequence from Lemma 6.4, Proposition 6.9, and the definition of strict semi-stability for parabolic Higgs bundles. ■ Corollary 7.14 (parabolic S-equivalence on geometric models). For any βββ in the interior of a semi-stability wall, the pointwise identification of strictly semi-stable components as in Proposition 7.13 determines a surface biholomorphic to Mtoy. A Residue models We will describe another approach to explicitly construct moduli spaces of stable parabolic Higgs bundles based on residue evaluations. We will refer to the resulting orbit spaces M ev C (E) as residue models for Harder–Narasimhan strata. The desingularization by blow-up of the cone of nilpotent endomorphisms n(V ) of a 2- dimensional vector space V is constructed as the smooth complex hypersurface ñ(V ) := { (ϕ, F ) ∈ n(V )× P(V ) : F ⊂ ker(ϕ) } . The restriction to ñ(V ) of the projection onto the second factor is again a projection which we will denote by pr, and endows ñ(V ) with the structure of a holomorphic line bundle of degree −2, identifying its exceptional divisor E ∼= P(V ) with the zero section of the latter. The fiber n(F ) over a given line F ∈ P(V ) is identified with the Lie algebra of nilpotent endomorphisms of V whose kernel contains F . ñ(V ) is the fiber product of n(V )\{0} (viewed as a principal C∗-bundle) under the standard C∗-representation ρ on C, ñ(V ) = ( n(V )\{0} ) ×ρ C. Our goal is to parametrize the spaces QPH(E) and their stratifications in terms of the geometry of a natural evaluation map associated to D. Consider the maps evzi : QPH(E) → ñ(E|zi), (E·,Φ) 7→ (Reszi Φ, Fi), i = 1, 2, 3, 4. 38 C. Meneses We define the total evaluation to be the map Ev = ∏4 i=1 evzi . Letting Pr be the projection map associated to the product of projections ∏4 i=1 przi , we obtain the following commutative diagram QPH(E) Ev // par ## 4∏ i=1 ñ(E|zi) Pr {{ QP(E) (A.1) which is equivariant with respect to the induced P(Aut(E))-actions. In other words, the total evaluation map is derived from the residue map on parabolic Higgs fields ResE : Int(E) → 4∏ i=1 n(E|zi), Φ 7→ (Resz1 Φ,Resz2 Φ,Resz3 Φ,Resz4 Φ), by taking simultaneous blow-ups on its domain and codomain along the loci defined by the identities (4.1). Remark A.1. The surface ñ(V ) can be also described in terms of explicit coordinate charts.7 These coordinates determine another way to parametrize QPH(E) from local trivializations on E, from which a projective model for Harder–Narasimhan strata can be constructed repeating the ideas of this section (cf. [18]). We do not pursue that possibility here, since a coordinate- invariant approach is self-sufficient for our general purposes. For every z ∈ CP1, there is a natural action of the group Aut(E)|z on the space n(E|z) by conjugation. Since the action of central elements is trivial, there is an induced action of P(Aut(E|z)) on ñ(E|z) and P(E|z), making the projection prz equivariant. When m1 = m2 the induced action of P(Aut(E)|z) on P(E|z) is transitive with stabilizer P(Aut(E|z))F = P(P(F )) for any F ∈ P(E|z). Moreover, since the action of P(Aut(E)|z) on n(E|z)\{0} is also transitive, and such that for any ϕ ∈ n(E|z)\{0}, its stabilizer is equal to the subgroup P(Aut(E|z))ϕ = P(R(F )), this action partitions the surface ñ(E|z) into a total of two orbits, corresponding to the excep- tional divisor Ez and its complement, which we will denote by Oz, that is ñ(E|z) = Oz ⊔ Ez, Oz ∼= n(E|z)\{0}. 7A choice of isomorphism V ∼= C2 determines an identification n(V ) ∼= Z2\C2 under the explicit parametrization (Z0, Z1) 7→ ( Z0Z1 −Z2 0 Z2 1 −Z0Z1 ) , which is invariant under the Z2-action given by (Z0, Z1) 7→ (−Z0,−Z1), and associates the kernel line [Z0 : Z1] ∈ P(C2) whenever (Z0, Z1) ̸= (0, 0). Thus, the introduction of the standard affine charts U0, U1 on P(C2) induces two affine charts W0,W1 ∼= C2 for the blown-up surface Z̃2\C2, defined on their domain intersections in the complement of the exceptional divisor by the maps (Z0/Z1, Z 2 1 ) 7→Z2 1 ( Z0/Z1 −(Z0/Z1) 2 1 −Z0/Z1 ) = Z2 0 ( Z1/Z0 −1 (Z1/Z0) 2 −Z1/Z0 ) 7→ ( Z1/Z0, Z 2 0 ) . Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles 39 When m1 > m2, the prz-inverse image of each of the two orbits Vz(E) ⊔ {E1|z} = P(E|z) is in turn partitioned into the two P(Aut(E)|z)-orbits which result from the intersection with Ez and its complement Oz. This way, the action of P(Aut(E)|z) on ñ(E|z) partitions it into a total of four orbits (Figure 14), that we will denote by Oz,1, Oz,2, Oz,3 and Oz,4, where Oz,1 := pr−1 z (Vz(E))\Ez = ( Ez ∪ n(E1|z) )c is Zariski open and a principal homogeneous space for P(Aut(E)|z). The remaining orbits Oz,2 := n(E1|z)\Oz,4, Oz,3 := Ez\Oz,4, Oz,4 := Ez ∩ n(E1|z) are respectively isomorphic to C∗, C, and a single point, since the stabilizer of any point (ϕ,E1|z) ∈ Oz,2 is equal to P(R(E1|z)). Figure 14. Orbit stratification of ñ(E|z) when m1 > m2 ∀z ∈ CP1. Given that Ev(Q(E)) = ∏4 i=1 Ezi , the inclusion ι : QP(E) → QPH(E) satisfies Pr◦Ev◦ι = id. Since the sum of an arbitrary End(E)-valued holomorphic differential and any parabolic Higgs field is again a parabolic Higgs field, there is an additional affine action of the (possibly trivial) vector space H0(CP1,End(E) ⊗ KCP1) on QPH(E). The orbit of any (E·,Φ) ∈ QPH(E) is the level set Ev−1(Ev(E·,Φ)), and Ev is precisely the invariant map for this action, endowing QPH(E) with the structure of an (possibly trivial) affine bundle over its image. Proposition A.2. (i) If E is evenly-split, then Ev is a P(Aut(E))-equivariant holomorphic embedding, whose image is the smooth 5-dimensional variety of residue constraints. (ii) When m1−m2 = 2, QPHC∗(E)⊔Sm−1(E) and Q(E)⊔R(E) are H0(CP1,End(E)⊗KCP1)- invariant, Ev(QPHC∗(E) ⊔ Sm−1(E)) ⊂ ∏4 i=1 Ozi,1 and Ev(Q(E) ⊔ R(E)) = ∏4 i=1 Oc zi,1 . (iii) When m1 −m2 = 3, Ev(R∅(E)) = ∏4 i=1 Ozi,2. Proof. (i) Ev is injective if and only if E is evenly-split. The definition of Ev implies that, in general, dEv has constant rank at any point (E·,Φ) ∈ QPH(E). Consequently, Ev is a holomor- phic embedding if E is evenly-split. (ii) Since nonzero elements in H0(CP1,End(E) ⊗KCP1) preserve E1, and these exist if and only ifm1−m2 ≥ 2, while QPH(E) = Q(E)⊔R(E) whenm1−m2 ≥ 3, H0(CP1,End(E)⊗KCP1)- invariance is trivial unless m1−m2 = 2, and then it follows from (4.5), since QPHC∗(E)⊔S0(E) is characterized by w0 ̸= 0, which is unchanged under addition of End(E)-valued holomorphic differentials, and is independent of the isomorphism E ∼= O(m + 1) ⊕ O(m − 1). The second claim is verified from (4.5) in analogy to the proof of Proposition 4.6(ii) and (iii). (iii) When m1 −m2 = 3 by definition we have that Ev(R∅(E)) = ∏4 i=1 Ozi,2. ■ 40 C. Meneses Corollary A.3. Assume that QPHs C(E) ̸= ∅. (i) If E evenly-split, there is an isomorphism MC(E) ∼= M ev C (E), where M ev C (E) is the P(Aut(E))-orbit space of Ev(QPHs C(E)) ⊂ 4∏ i=1 ñ(E|zi). (ii) When m1 −m2 = 2, M ev C (E) := Ev(QPHs C(E))/R(Aut(E)) ∼= MC(E). (iii) When m1 −m2 = 3, R(Aut(E)) acts freely and transitively on Ev(QPHs C(E)) and MC(E) ∼= P ( H0 ( CP1,End(E)⊗KCP1 )) . Proof. (i) The claim follows from the equivariance of the commutative diagram (A.1), Propo- sition A.2(i), and Theorem 1.1. (ii)–(iii) When m1 − m2 ≥ 2, the affine H0(CP1,End(E) ⊗ KCP1)-action on QPH(E) is nontrivial and commutes with the action of the subgroup R(Aut(E)). R(Aut(E)) acts trivially on H0(CP1,End(E) ⊗ KCP1), and the induced action of P(Aut(E))/R(Aut(E)) on it equals the square of its standard C∗-action. When m1 − m2 = 2, it follows from the classification of nilpotent cone assembly kits and Proposition A.2(ii) that the action of P(Aut(E)) on the fibers of Ev|QPHs C(E) is transitive, and the result follows. When m1 − m2 = 3 there is a single case when QPHs C(E) ̸= ∅, namely C = C∅, for which QPHs C(E) = R∅(E). Every element in R∅(E) can be uniquely expressed as the sum of an element in ι(B∅,m+2(E)) and a nonzero element in H0(CP1,End(E) ⊗ KCP1). The result follows from Proposition A.2(iii), since R(Aut(E)) acts freely and transitively on ∏4 i=1 Ozi,2. ■ Acknowledgements I would like to thank Hartmut Weiß, whose encouragement and support where crucial in promp- ting the appearance of the present manuscript, Steven Rayan for providing many insightful remarks, and the anonymous referee for the careful revision of the manuscript and the construc- tive criticism provided. This work was supported by the DFG SPP 2026 priority programme “Geometry at infinity”. 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institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1815-0659
language English
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publishDate 2022
publisher Інститут математики НАН України
record_format dspace
spelling Meneses, Claudio
2026-01-09T12:50:58Z
2022
Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study. Claudio Meneses. SIGMA 18 (2022), 062, 41 pages
1815-0659
2020 Mathematics Subject Classification: 14H60; 14D22; 32G13; 22E25
arXiv:2012.13389
https://nasplib.isofts.kiev.ua/handle/123456789/211725
https://doi.org/10.3842/SIGMA.2022.062
We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces of semi-stable parabolic Higgs bundles in genus 0, which is encoded into the combinatorics of weight polytopes. We also present a comprehensive analysis of the geometric models' behavior under variation of parabolic weights and wall-crossing, which is concentrated on their nilpotent cones.
I would like to thank Hartmut Weiß, whose encouragement and support were crucial in prompting the appearance of the present manuscript, Steven Rayan for providing many insightful remarks, and the anonymous referee for the careful revision of the manuscript and the constructive criticism provided. This work was supported by the DFG SPP 2026 priority programme “Geometry at infinity”.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study
Article
published earlier
spellingShingle Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study
Meneses, Claudio
title Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study
title_full Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study
title_fullStr Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study
title_full_unstemmed Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study
title_short Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study
title_sort geometric models and variation of weights on moduli of parabolic higgs bundles over the riemann sphere: a case study
url https://nasplib.isofts.kiev.ua/handle/123456789/211725
work_keys_str_mv AT menesesclaudio geometricmodelsandvariationofweightsonmoduliofparabolichiggsbundlesovertheriemannsphereacasestudy