June 14, 2026
Let \(C\) be a compact Riemann surface of genus at least two, and let \(G\) be a connected complex reductive group. We study logarithmic \(G\)-Higgs bundles on a pointed curve \((C,D)\) whose residues are nilpotent. We prove that every homogeneous invariant polynomial in the Higgs field loses its leading pole term. Equivalently, the nilpotent-residue logarithmic Hitchin map takes values in a smaller meromorphic Hitchin base. In degree two this gives a logarithmic quadratic one-form over pointed Teichmuller space. We relate this one-form to the variation of the energy for tame nilpotent harmonic bundles, under a positive-decay assumption near the punctures.
Let \(C\) be a compact Riemann surface of genus \(g\geq2\), and let \(G\) be a connected complex reductive group with Lie algebra \(\mathfrak g\). We consider the Betti moduli space \[\mathcal{M}_B(G) = \operatorname{Hom}(\pi_1(C),G)^{\mathrm{red}}/G\] and the Dolbeault moduli space \(\mathcal{M}_{\mathrm{Dol}}(G,C)\) of semistable principal \(G\)-Higgs bundles. On smooth stable loci these spaces are identified by nonabelian Hodge theory [1]–[6]. The Betti space is topological, while the Dolbeault space depends on the complex structure of \(C\). Hitchin studied this variation over Teichmuller space in [7]. For \(G=GL(n,\mathbb{C})\), the one-form which appears in the variation of Hitchin’s energy is \[\phi(Y) = -\frac{1}{2}\int_C\operatorname{tr}(\Phi^2)\mu_Y, \qquad Y\in T_C\mathcal{T}_g,\] where \(\Phi\) is the Higgs field and \(\mu_Y\) is a Beltrami representative of \(Y\). If \(f\) denotes the energy, then \(Y(f)=\operatorname{Re}\phi(Y),\) and by averaging the circle action \((E,\Phi)\longmapsto(E,e^{i\theta}\Phi)\) he obtains a connection satisfying \[d_A\phi=0,\qquad \{\phi,\phi\}=0,\qquad F_A+\frac{1}{8}\{\phi,\bar\phi\}=0.\] For the first part of this paper we use the similar construction using an invariant nondegenerate symmetric bilinear form \(B:\mathfrak g\otimes\mathfrak g\longrightarrow\mathbb{C}.\) The corresponding quadratic one-form is \[\phi_G(Y) = -\frac{1}{2}\int_C B(\Phi,\Phi)\mu_Y.\] For the standard representations of \(GL(n,\mathbb{C}), Sp(2n,\mathbb{C}), \text{ and } SO(n,\mathbb{C}),\) we take \(B(X,Y)=\operatorname{tr}(XY)\). Then the Fourier-coefficient argument gives the same averaged-connection identities. We also compute the formulae for the standard real forms \(SL(2,\mathbb{R})\), \(Sp(2n,\mathbb{R})\), \(SO(p,q)\), and \(SU(p,q)\) (see [8]–[11] for more details about these moduli spaces).
We next consider logarithmic Higgs bundles on a pointed curve. Let \(D=p_1+\cdots+p_r\) be a reduced divisor on \(C\). A logarithmic \(G\)-Higgs bundle on \((C,D)\) is a pair \((P,\Phi)\), where \(P\) is a holomorphic principal \(G\)-bundle and \[\Phi\in H^0(C,\operatorname{ad}(P)\otimes K_C(D))\] (see [12]–[14] for parabolic bundles and parabolic Higgs bundles). Near \(p_i\), after choosing a coordinate \(z\) centred at \(p_i\), the Higgs field has the form \[\Phi= \left( \frac{N_i}{z}+A_i(z) \right)dz, \qquad N_i=\operatorname{Res}_{p_i}(\Phi),\] where \(A_i(z)\) is holomorphic. If \(p\in\mathbb{C}[\mathfrak g]^G\) is homogeneous of degree \(d\), then \(p(\Phi)\) can have a pole of order \(d\) at a marked point. The nilpotency of the residues removes this highest order term.
Theorem 1 (Theorem 7). Let \(G\) be a connected complex reductive group with Lie algebra \(\mathfrak g\), and let \(p\in\mathbb{C}[\mathfrak g]^G\) be homogeneous of degree \(d\geq1\). If \((P,\Phi)\) is a logarithmic \(G\)-Higgs bundle on \((C,D)\) with nilpotent residues, then \[p(\Phi)\in H^0\bigl(C,K_C^d((d-1)D)\bigr).\]
Consequently the nilpotent-residue logarithmic Hitchin map factors through the smaller base \[\bigoplus_j H^0\bigl(C,K_C^{d_j}((d_j-1)D)\bigr),\] where \(d_j\) are the degrees of the basic invariant polynomials. The proof of the Theorem is local. Since \(p\) is homogeneous, \[p(\Phi)=z^{-d}p(N_i+zA_i(z))\,dz^d.\] By the Jacobson-Morozov theorem, homogeneous invariant polynomials of positive degree vanish on nilpotent elements. Hence \(p(N_i)=0\), and the pole order drops by at least one. For \(p(X)=\frac{1}{2}B(X,X),\) the theorem gives \(B(\Phi,\Phi)\in H^0(C,K_C^2(D)),\) and this space is naturally identified with the cotangent space to pointed Teichmuller space i.e. \[T^*_{(C,D)}\mathcal{T}_{g,r} \cong H^0(C,K_C^2(D)).\] Thus \(-\frac{1}{2}B(\Phi,\Phi)\) defines a logarithmic quadratic one-form \(\phi_{G,\log}\) over \(\mathcal{T}_{g,r}\). In Čech-Dolbeault form, if \(Y\in H^1(C,T_C(-D))\) is represented by \((\mu,\xi_1,\ldots,\xi_r)\), then \[\phi_{G,\log}(Y) = -\frac{1}{2} \left[ \int_{C^\circ}B(\Phi,\Phi)\mu + \sum_{i=1}^{r} \operatorname{Res}_{p_i} \bigl(B(\Phi,\Phi)\xi_i\bigr) \right].\] For the trace invariants of \(GL(n,\mathbb{C})\), the pole order \(k-1\) can occur, i.e. there are local logarithmic Higgs fields with nilpotent residue for which \(\operatorname{tr}(\Phi^k)\) has pole order exactly \(k-1\). For \(SO(2m,\mathbb{C})\), the Pfaffian satisfies \[\operatorname{Pf}(\Phi)\in H^0(C,K_C^m((m-1)D)).\] If, moreover, \(\operatorname{rk}N_i\leq 2m-4\) at every marked point \(p_i\), then \[\operatorname{Pf}(\Phi)\in H^0(C,K_C^m((m-2)D)).\] The meromorphic Hitchin fibration over pointed curves is studied in [15].
The analytic part proves the corresponding logarithmic energy formula. We use tame nilpotent harmonic bundles on \(C^\circ=C\setminus D\) and the asymptotic estimates of Simpson and Mochizuki [16], [17]. For deformations with fixed Betti representation and fixed local monodromy conjugacy classes, we assume that the infinitesimal complex gauge term has positive decay in the cusp scale. Fixing the local monodromy conjugacy classes is also the condition which, on the Betti side, specifies the symplectic leaves of the Poisson structure on open character varieties [18]. Under the decay assumption we prove \[Y(f_{\mathrm{par}}) = \operatorname{Re}\phi_{G,\log}(Y).\] The decay condition is used only to control the boundary terms in the integration by parts. The algebraic pole estimate and the construction of \(\phi_{G,\log}\) do not depend on it. For real groups over punctured curves, see [19], [20], and for asymptotic geometry in the parabolic \(SL(2,\mathbb{C})\) case, see [21].
The paper is organized as follows. Section 2 contains the compact quadratic one-form and the averaged connection. In Section 3, we consider the classical groups, real forms, and the Pfaffian Hamiltonian. In Section 4, we consider the parabolic bundles, and Serre-duality notation. In Section 5, we prove the pole estimate for invariant polynomials, the trace-invariant example, and the Pfaffian estimate. In Section 6, we construct the logarithmic quadratic one-form \(\phi_{G,\log}\). In Section 7, we prove the logarithmic energy-variation formula under the cusp decay assumption.
The natural analytic problem left open is to prove the required positive-decay condition from a logarithmic slice theorem for the linearized tame harmonic metric equation with fixed local monodromy.
Let \(C\) be a compact Riemann surface of genus \(g\geq 2\) with the canonical bundle \(K_C\), and let \(G\) be a connected complex reductive group with Lie algebra \(\mathfrak g\). We fix an invariant nondegenerate symmetric bilinear form \[B:\mathfrak g\otimes\mathfrak g\longrightarrow\mathbb{C},\] which is used to define the Atiyah-Bott-Goldman symplectic form, the energy functional, and the quadratic Hamiltonian. For the standard classical groups we consider \(B(X,Y)=\operatorname{tr}(XY)\) in the defining representation.
For a holomorphic principal \(G\)-bundle \(P\longrightarrow C,\) the adjoint action of \(G\) on \(\mathfrak g\) gives the associated adjoint bundle \[\operatorname{ad}(P) = P\times^G\mathfrak g .\]
Definition 1. A principal \(G\)-Higgs bundle on \(C\) is a pair \((P,\Phi),\) where \(P\) is a holomorphic principal \(G\)-bundle and \(\Phi\in H^0(C,\operatorname{ad}(P)\otimes K_C)\) is a holomorphic section, called the Higgs field.
For \(G=GL(n,\mathbb{C})\), this is the usual Higgs bundle \((E,\Phi),\) with \(\Phi\in H^0(C,\operatorname{End}(E)\otimes K_C).\) For \(G=SL(n,\mathbb{C})\), we also have \(\det E\simeq\mathcal{O}_C \text{ and } \operatorname{tr}(\Phi)=0.\) The topological type of a principal \(G\)-bundle over \(C\) is indexed by an element \(d\in \pi_1(G).\) We fix such a topological type throughout. When \(G\) is semisimple, \(\pi_1(G)\) is finite.
Let \(Q\subset G\) be a maximal parabolic subgroup, and let \(P_Q\subset P\) be a holomorphic reduction of structure group to \(Q\). The reduction is said to be preserved by the Higgs field if \[\Phi\in H^0(C,\operatorname{ad}(P_Q)\otimes K_C)\] under the natural inclusion \(\operatorname{ad}(P_Q)\subset \operatorname{ad}(P).\)
Definition 2. The principal Higgs bundle \((P,\Phi)\) is called stable (resp. semistable), if for every such \(\Phi\)-invariant reduction to a maximal parabolic subgroup we have \[\deg(\operatorname{ad}(P_Q))<0 \quad (\text{resp. } \leq 0).\]
Let \(\mathcal{M}^d(G)\) be the moduli space of semistable holomorphic principal \(G\)-bundles over \(C\) of topological type \(d\). It is a normal projective variety, and on the stable locus, \[\dim_{\mathbb{C}}\mathcal{M}^d(G) = (g-1)\dim_{\mathbb{C}}G+\dim_{\mathbb{C}}Z(G),\] where \(Z(G)\) denotes the centre of \(G\). We denote by \(\mathcal{M}_{\mathrm{Dol}}^d(G,C)\) the smooth stable locus of the moduli space of polystable principal \(G\)-Higgs bundles of the same topological type.
For \((P,\Phi) \in \mathcal{M}_{\mathrm{Dol}}^d(G,C)\), the tangent space is the first hypercohomology of the deformation complex \[\operatorname{ad}(P) \xrightarrow{[\Phi,\cdot]} \operatorname{ad}(P)\otimes K_C.\] Thus \[T_{(P,\Phi)}\mathcal{M}_{\mathrm{Dol}}^d(G,C) \cong \mathbb{H}^1\left( C,\operatorname{ad}(P) \xrightarrow{[\Phi,\cdot]} \operatorname{ad}(P)\otimes K_C \right).\] On this smooth stable locus, \[\dim_{\mathbb{C}}\mathcal{M}_{\mathrm{Dol}}^d(G,C) = 2(g-1)\dim_{\mathbb{C}}G+2\dim_{\mathbb{C}}Z(G).\] In particular, if \(G\) is semisimple, then \[\dim_{\mathbb{C}}\mathcal{M}_{\mathrm{Dol}}^d(G,C) = 2(g-1)\dim_{\mathbb{C}}G.\]
For the construction of the moduli spaces of principal bundles we refer to Ramanathan [22], [23]. For Higgs bundles and the nonabelian Hodge correspondence, see Simpson [4]–[6]. The corresponding Betti moduli space is the character variety \[\mathcal{M}_B(G) = \operatorname{Hom}(\pi_1(C),G)^{\mathrm{red}}/G.\] By nonabelian Hodge theory, the smooth stable Dolbeault locus is identified with the smooth locus of the character variety. This is obtained by solving Hitchin’s equations and passing from a harmonic metric to a flat connection [1]–[6]. The Betti moduli space carries the Atiyah–Bott–Goldman symplectic form defined using \(B\) [24], [25]. Under the nonabelian Hodge theory this is one of the Kähler forms of the hyperkähler metric on the smooth Higgs moduli space. For related constructions of Kähler forms on Higgs-bundle moduli spaces, see [26].
Choose a maximal compact subgroup \(K\subset G\). A \(K\)-reduction of \(P\) defines a Chern connection \(A\), and it also defines an adjoint operation on \(\operatorname{ad}(P)\). We denote this adjoint by \(\tau\). For \(G=GL(n,\mathbb{C})\), if \(\Phi=A(z)\,dz,\) then \[\tau(\Phi)=A(z)^*\,d\bar z,\] where \(A(z)^*\) is the Hermitian adjoint. In general, \(\tau(e^{i\theta}\Phi)=e^{-i\theta}\tau(\Phi).\) Let \(A\) be the Chern connection associated to the \(K\)-reduction, and let \[F_A\in\Omega^2(C,\operatorname{ad}(P))\] be its curvature. Its \((0,1)\)-part is denoted by \(\bar\partial_A\). The Hitchin equations are \[F_A+[\Phi,\tau(\Phi)]=0, \qquad \bar\partial_A\Phi=0.\] Here the bracket combines the Lie bracket on \(\operatorname{ad}(P)\) with the wedge product of forms. If we have \(\varphi=\Phi+\tau(\Phi),\) then the associated flat connection is \(\nabla=d_A+\varphi.\) Then the Hitchin equations imply \[d_A\varphi=0, \qquad d_A{*}\varphi=0.\] We will use Hitchin’s sign convention for the energy, i.e. \[f_G = -i\int_C B(\Phi,\tau(\Phi)) = -\frac{1}{2}\int_C B(\varphi,*\varphi). \label{eq:compact-energy}\tag{1}\] For \(G=GL(n,\mathbb{C})\) with \(B(X,Y)=\operatorname{tr}(XY)\), this is the normalization used in Hitchin’s universal construction [7]. The invariant form \(B\) defines the Atiyah-Bott-Goldman symplectic form on the Betti moduli space. Under nonabelian Hodge theory this is one of the Kähler forms of the hyperkähler metric on the smooth Higgs moduli space, and we denote it by \(\omega_1\). With respect to \(\omega_1\), the function \(f_G\) is the moment map for the circle action \[(P,\Phi)\longmapsto (P,e^{i\theta}\Phi).\] Indeed, the adjoint transforms by \[\tau(e^{i\theta}\Phi)=e^{-i\theta}\tau(\Phi),\] so the circle action preserves Hitchin’s equations. Let \(\mathcal{T}=\mathcal{T}_g\) be the Teichmuller space of compact Riemann surfaces of genus \(g\). At \(C\in\mathcal{T}\), its tangent space is \[T_C\mathcal{T} \cong H^1(C,T_C) \cong H^1(C,K_C^{-1}).\] Let \(Y\in T_C\mathcal{T}\) be represented by a Beltrami differential \[\mu_Y\in\Omega^{0,1}(C,T_C).\] The Higgs field gives a quadratic differential \(B(\Phi,\Phi)\in H^0(C,K_C^2).\) Locally, if \(\Phi=s(z)\,dz,\) then \[B(\Phi,\Phi)=B(s(z),s(z))\,dz^2.\] Contracting with \(\mu_Y\) gives a \((1,1)\)-form, and we define \[\phi_G(Y) = -\frac{1}{2}\int_C B(\Phi,\Phi)\,\mu_Y. \label{eq:compact-oneform}\tag{2}\] This is a complex-valued one-form on Teichmuller space with values in fibrewise holomorphic functions on the Higgs moduli space. For \(G=GL(n,\mathbb{C})\), it is Hitchin’s quadratic one-form in [7].
Proposition 2. Let the complex structure of \(C\) vary in the direction \(Y\in T_C\mathcal{T}\), while the corresponding point of the Betti moduli space is fixed. Equivalently, the flat connection \(\nabla=d_A+\varphi\) is fixed. Then \[Y(f_G) = \operatorname{Re}\phi_G(Y) = -\frac{1}{2}\operatorname{Re} \int_C B(\Phi,\Phi)\,\mu_Y.\]
Proof. The proof is similar to Hitchin [7]. Since the Betti point is fixed, the first-order change of \((A,\varphi)\) is a complex gauge variation. Thus one may write \[\dot{A}=d_A\psi_1+[\varphi,\psi_2], \qquad \dot{\varphi}=d_A\psi_2+[\varphi,\psi_1],\] where \(\psi_1\) is compact-valued and \(\psi_2\) is self-adjoint. Differentiating 1 gives \[\dot{f}_G = -\frac{1}{2}\int_C B(\dot{\varphi},*\varphi) - \frac{1}{2}\int_C B(\varphi,\dot{*}\,\varphi) - \frac{1}{2}\int_C B(\varphi,*\dot{\varphi}).\] The terms containing \(d_A\psi_2\) vanish by integration by parts, using \[d_A\varphi=0, \qquad d_A{*}\varphi=0,\] and the terms containing \([\varphi,\psi_1]\) vanish by the invariance of \(B\). Hence \[\dot{f}_G = -\frac{1}{2}\int_C B(\varphi,\dot{*}\,\varphi).\] The variation of the Hodge star in the direction represented by \(\mu_Y\) gives \[-\frac{1}{2}\operatorname{Re} \int_C B(\Phi,\Phi)\,\mu_Y.\] Therefore \[Y(f_G)=\operatorname{Re}\phi_G(Y). \qedhere\] ◻
The character variety \(\mathcal{M}_B(G)\) does not depend on the complex structure of \(C\). Hence over Teichmuller space we have the product fibration \[\mathcal{M}_B(G)\times\mathcal{T} \longrightarrow \mathcal{T}.\] Let \(\nabla_B\) be its product flat connection. The parallel transport for \(\nabla_B\) keeps the Betti representation fixed and changes only the complex structure of the curve. For each \(C\in\mathcal{T}\), nonabelian Hodge theory identifies \[\mathcal{M}_B(G) \cong \mathcal{M}_{\mathrm{Dol}}(G,C).\] The circle action \[(P,\Phi)\longmapsto(P,e^{i\theta}\Phi)\] therefore transports to an action on the fixed Betti fibre. Applying this circle action to the product connection gives a circle of flat symplectic connections on \[\mathcal{M}_B(G)\times\mathcal{T} \longrightarrow \mathcal{T}.\] Their average will be denoted by \(\nabla_A\). If \(\phi_G=\beta_G+i\gamma_G,\) where \(\beta_G\) and \(\gamma_G\) are real Hamiltonian-valued one-forms on \(\mathcal{T}\), then \[\nabla_A=\nabla_B-\frac{1}{2}\gamma_G .\] This is the averaging formula of Hitchin [7]. The corresponding circle of flat connections can be written as \[\nabla_\theta = \nabla_A - \frac{i}{4} \left( e^{2i\theta}\phi_G - e^{-2i\theta}\bar\phi_G \right), \qquad \theta\in\mathbb{R}/2\pi\mathbb{Z}. \label{eq:theta-connection}\tag{3}\] Here \(\phi_G\) and \(\bar\phi_G\) are Hamiltonian-valued one-forms, and hence vertical vector-field-valued one-forms after using \(\omega_1\). For Hamiltonian functions \(h_1,h_2\) on a fibre, with \[\iota_{X_{h_i}}\omega_1=dh_i,\] we write \[\{h_1,h_2\} = \omega_1(X_{h_1},X_{h_2}).\] For Hamiltonian-valued differential forms on the base, this bracket is combined with the exterior product of the base forms.
Theorem 3. On the smooth stable locus of \[\mathcal{M}_B(G)\times\mathcal{T}\longrightarrow\mathcal{T}\] where the nonabelian Hodge correspondence identifies the Betti and Dolbeault moduli spaces, the averaged connection satisfies \[d_A\phi_G=0, \qquad \{\phi_G,\phi_G\}=0, \qquad F_A+\frac{1}{8}\{\phi_G,\bar\phi_G\}=0.\]
Proof. For \(G=GL(n,\mathbb{C})\) and \(B(X,Y)=\operatorname{tr}(XY)\), these are the identities obtained by Hitchin from the circle family of flat connections [7]. The same can be applied here in our case. Indeed, the circle action gives \[B(e^{i\theta}\Phi,e^{i\theta}\Phi) = e^{2i\theta}B(\Phi,\Phi),\] and the form \(B\) is the same one used in the symplectic form, in the energy, and in the Hamiltonian function \(\phi_G\). Therefore the Hamiltonian one-form has circle weight two. Let \[a_\theta = - \frac{i}{4} \left( e^{2i\theta}\phi_G - e^{-2i\theta}\bar\phi_G \right), \qquad \nabla_\theta=\nabla_A+a_\theta .\] The connections \(\nabla_\theta\) are flat for all \(\theta\), since they are obtained from the product flat connection by the circle action. Hence \[0 = F_{\nabla_\theta} = F_A+d_Aa_\theta+\frac{1}{2}\{a_\theta,a_\theta\}.\] This identity is a finite Fourier series in \(e^{2i\theta}\). Comparing its Fourier coefficients gives the three equations. The coefficient of \(e^{2i\theta}\) gives \[d_A\phi_G=0,\] the coefficient of \(e^{4i\theta}\) gives \[\{\phi_G,\phi_G\}=0,\] and the constant coefficient gives \[F_A+\frac{1}{8}\{\phi_G,\bar\phi_G\}=0.\] ◻
We now consider the classical groups and some real forms. For the standard representations of \(GL(n,\mathbb{C}), \;Sp(2n,\mathbb{C}),\text{ and } SO(n,\mathbb{C}),\) we use \[B(X,Y)=\operatorname{tr}(XY).\] Thus the quadratic Hamiltonian is given by \(\operatorname{tr}(\Phi^2)\). For a general invariant bilinear form \(B\), the same formulae hold with \(\operatorname{tr}(\Phi^2)\) replaced by \(B(\Phi,\Phi)\). We shall also use the Hitchin morphism. Let \[p_1,\ldots,p_\ell\] be homogeneous generators of the invariant ring \(\mathbb{C}[\mathfrak g]^G\) with degrees \[d_1,\ldots,d_\ell.\] The Hitchin morphism is \[h_G:\mathcal{M}_{\mathrm{Dol}}(G,C) \longrightarrow \mathcal{A}_G := \bigoplus_{j=1}^{\ell}H^0(C,K_C^{d_j}),\] defined by \[h_G(P,\Phi) = \bigl(p_1(\Phi),\ldots,p_\ell(\Phi)\bigr).\] The vector space \(\mathcal{A}_G\) is called the Hitchin base (see [1], [27] for more details).
An \(Sp(2n,\mathbb{C})\)-Higgs bundle is a triple \((E,\omega,\Phi),\) where \(E\) is a holomorphic vector bundle of rank \(2n\), \[\omega:E\otimes E\longrightarrow \mathcal{O}_C\] is a nondegenerate skew-symmetric form, and \(\Phi\in H^0(C,\operatorname{End}(E)\otimes K_C)\) satisfies \[\omega(\Phi u,v)+\omega(u,\Phi v)=0\] for local sections \(u,v\) of \(E\). Equivalently, \(\Phi\in H^0(C,\mathfrak{sp}(E,\omega)\otimes K_C).\) The quadratic one-form is \[\phi_{Sp}(Y) = -\frac{1}{2}\int_C\operatorname{tr}(\Phi^2)\mu_Y,\] and the Hitchin base is \[\mathcal{A}_{Sp(2n)} = \bigoplus_{j=1}^{n}H^0(C,K_C^{2j}).\]
An \(SO(n,\mathbb{C})\)-Higgs bundle is a triple \((E,q,\Phi),\) where \(E\) is a holomorphic vector bundle of rank \(n\), \[q:E\otimes E\longrightarrow\mathcal{O}_C\] is a nondegenerate symmetric form with fixed orientation, and \(\Phi\in H^0(C,\operatorname{End}(E)\otimes K_C)\) satisfies \[q(\Phi u,v)+q(u,\Phi v)=0\] for local sections \(u,v\) of \(E\). Equivalently, \(\Phi\in H^0(C,\mathfrak{so}(E,q)\otimes K_C).\) The quadratic one-form is \[\phi_{SO}(Y) = -\frac{1}{2}\int_C\operatorname{tr}(\Phi^2)\mu_Y.\] For \(SO(2m+1,\mathbb{C})\), \[\mathcal{A}_{SO(2m+1)} = \bigoplus_{j=1}^{m}H^0(C,K_C^{2j}),\] while for \(SO(2m,\mathbb{C})\), \[\mathcal{A}_{SO(2m)} = \bigoplus_{j=1}^{m-1}H^0(C,K_C^{2j}) \oplus H^0(C,K_C^m),\] the last summand being the Pfaffian invariant.
In both symplectic and orthogonal cases the circle action preserves the defining symplectic or orthogonal condition on \(\Phi\). Hence the averaged connection and the identities of Theorem 3 restrict to every smooth stable \(Sp(2n,\mathbb{C})\) or \(SO(n,\mathbb{C})\)-component. For the classical Hitchin bases and invariant degrees, see [1].
Let \(G_{\mathbb{R}}\) be a real reductive group, let \(H\subset G_{\mathbb{R}}\) be a maximal compact subgroup, and let \[\mathfrak g_{\mathbb{R}} = \mathfrak h\oplus\mathfrak m\] be the Cartan decomposition.
Definition 3. A \(G_{\mathbb{R}}\)-Higgs bundle is a pair \((P_H,\Phi),\) where \(P_H\) is a holomorphic principal \(H^{\mathbb{C}}\)-bundle and \(\Phi\in H^0(C,P_H(\mathfrak m^{\mathbb{C}})\otimes K_C),\) where \[P_H(\mathfrak m^{\mathbb{C}}) = P_H\times^{H^{\mathbb{C}}}\mathfrak m^{\mathbb{C}}\] is the associated vector bundle.
The circle action preserves the real-form Higgs locus, since \[e^{i\theta}\Phi \in H^0(C,P_H(\mathfrak m^{\mathbb{C}})\otimes K_C)\] whenever \[\Phi\in H^0(C,P_H(\mathfrak m^{\mathbb{C}})\otimes K_C).\] Therefore the circle-translated product connections are tangent to every smooth real-form component, and so is their average. Hence the connection \[\nabla_A=\nabla_B-\frac{1}{2}\gamma\] and the identities of Theorem 3 restrict to these components. We now write the quadratic one-form in the standard block descriptions of the real-form Higgs fields.
For the uniformizing component of the \(SL(2,\mathbb{R})\)-Higgs moduli space, choose a square root \(K_C^{1/2}\). The corresponding Higgs bundle is \[E=K_C^{1/2}\oplus K_C^{-1/2}, \qquad \Phi= \begin{pmatrix} 0&q\\ 1&0 \end{pmatrix}, \qquad q\in H^0(C,K_C^2).\] Here the lower-left entry is the natural section of \[\operatorname{Hom}(K_C^{1/2},K_C^{-1/2})\otimes K_C \cong \mathcal{O}_C.\] A direct calculation gives \[\Phi^2= \begin{pmatrix} q&0\\ 0&q \end{pmatrix}, \qquad \operatorname{tr}(\Phi^2)=2q.\] Therefore \[\phi_{SL(2,\mathbb{R})}(Y) = -\int_C q\,\mu_Y.\] Under \(T_C^*\mathcal{T}\cong H^0(C,K_C^2),\) this is the tautological one-form on \(T^*\mathcal{T}\), up to the sign used in our definition of \(\phi_G\).
An \(Sp(2n,\mathbb{R})\)-Higgs bundle is given by \((V,\beta,\gamma),\) where \(V\) is a holomorphic vector bundle of rank \(n\), \[\beta\in H^0(C,\operatorname{Sym}^2V\otimes K_C), \qquad \gamma\in H^0(C,\operatorname{Sym}^2V^*\otimes K_C).\] The Higgs field has the block form \[\Phi= \begin{pmatrix} 0&\beta\\ \gamma&0 \end{pmatrix}.\] Thus \[\Phi^2= \begin{pmatrix} \beta\gamma&0\\ 0&\gamma\beta \end{pmatrix}.\] Therefore, we get \[\operatorname{tr}(\Phi^2) = 2\operatorname{tr}(\beta\gamma).\] Hence \[\phi_{Sp(2n,\mathbb{R})}(Y) = -\int_C\operatorname{tr}(\beta\gamma)\mu_Y.\] For \(n=1\), this agrees with the \(SL(2,\mathbb{R})\) formula. For \(n\geq2\), the Hitchin base contains higher even differentials, but the variation over Teichmuller space uses only the quadratic invariant. For the moduli spaces of \(Sp(2n,\mathbb{R})\)-Higgs bundles, see García-Prada–Gothen–Mundet [9].
An \(SO(p,q)\)-Higgs bundle is a tuple \((V,W,q_V,q_W,\eta),\) where \(V\) and \(W\) are holomorphic orthogonal bundles of ranks \(p\) and \(q\), respectively, \[\det(V)\otimes\det(W)\cong\mathcal{O}_C,\] and \[\eta\in H^0(C,\operatorname{Hom}(W,V)\otimes K_C).\] Using the orthogonal forms, \(\eta\) has a transpose \[\eta^t:V\longrightarrow W\otimes K_C.\] Therefore, the Higgs field is given by \[\Phi= \begin{pmatrix} 0&\eta\\ -\eta^t&0 \end{pmatrix}.\] Then \[\Phi^2= \begin{pmatrix} -\eta\eta^t&0\\ 0&-\eta^t\eta \end{pmatrix},\] and therefore \[\operatorname{tr}(\Phi^2) = -2\operatorname{tr}(\eta\eta^t).\] Thus, for the trace form, \[\phi_{SO(p,q)}(Y) = \int_C\operatorname{tr}(\eta\eta^t)\mu_Y.\] Equivalently, without choosing signs in the block notation, one may write the same one-form invariantly as \[\phi_{SO(p,q)}(Y) = -\frac{1}{2}\int_C B(\Phi,\Phi)\mu_Y.\] For \(SO(p,q)\)-Higgs bundles and higher Teichmuller components, see [10].
An \(SU(p,q)\)-Higgs bundle consists of holomorphic vector bundles \(V\) and \(W\), of ranks \(p\) and \(q\), with \[\det(V)\otimes\det(W)\cong\mathcal{O}_C,\] and Higgs field \[\Phi= \begin{pmatrix} 0&\beta\\ \gamma&0 \end{pmatrix},\] where \[\beta\in H^0(C,\operatorname{Hom}(W,V)\otimes K_C), \qquad \gamma\in H^0(C,\operatorname{Hom}(V,W)\otimes K_C).\] Then \[\Phi^2= \begin{pmatrix} \beta\gamma&0\\ 0&\gamma\beta \end{pmatrix}, \qquad \operatorname{tr}(\Phi^2) = 2\operatorname{tr}(\beta\gamma).\] Hence \[\phi_{SU(p,q)}(Y) = -\int_C\operatorname{tr}(\beta\gamma)\mu_Y.\]
Let \(G=SO(2m,\mathbb{C}),\;m\geq 2.\) The Hitchin base contains the Pfaffian summand \(H^0(C,K_C^m),\) corresponding to the degree \(m\) invariant \[\operatorname{Pf}(\Phi)\in H^0(C,K_C^m).\] For \(m>2\), this invariant is independent of the quadratic invariant. For \(m=2\), the exceptional isomorphism \[\mathfrak{so}(4,\mathbb{C})\cong \mathfrak{sl}_2(\mathbb{C})\oplus\mathfrak{sl}_2(\mathbb{C})\] places the Pfaffian in degree two.
We use Hitchin’s hyperkähler notation. The complex structure \(I\) is the Dolbeault complex structure, \(J\) is the de Rham complex structure, and \(K=IJ\). The corresponding Kähler forms are denoted by \[\omega_1,\qquad \omega_2,\qquad \omega_3,\] and \[\omega^c=\omega_2+i\omega_3\] is the holomorphic symplectic form for the Dolbeault complex structure.
Let \(h\) be a fibrewise holomorphic function on the Higgs moduli space, and let \(Z_h\) be its Hamiltonian vector field with respect to \(\omega^c \;\text{and}\; \iota_{Z_h}\omega^c=dh.\) The hyperkähler identity \[Jdh=2\,\iota_{Z_h}\omega_1\] gives a closed two-form \(dJdh\) (see [7]). It is of type \((1,1)\) for the complex structures in the hyperkähler family.
Let \(\nu\in H^1(C,K_C^{1-m}).\) Since \(\operatorname{Pf}(\Phi)\in H^0(C,K_C^m),\) Serre duality gives a function on the smooth \(SO(2m,\mathbb{C})\)-Higgs moduli space \[h_{\operatorname{Pf},\nu} = \left\langle \operatorname{Pf}(\Phi),\nu \right\rangle .\] The brackets denote the Serre-duality pairing \[H^0(C,K_C^m)\times H^1(C,K_C^{1-m}) \longrightarrow \mathbb{C} .\]
Proposition 4. The function \(h_{\operatorname{Pf},\nu}\) is fibrewise holomorphic. The associated two-form \(dJdh_{\operatorname{Pf},\nu}\) is closed and of type \((1,1)\) on the smooth \(SO(2m,\mathbb{C})\)-locus.
Proof. The Pfaffian is a polynomial invariant of \(\mathfrak{so}(2m,\mathbb{C})\). Hence \(\operatorname{Pf}(\Phi)\), and therefore \(h_{\operatorname{Pf},\nu}\), depends holomorphically on the Higgs field along each Dolbeault fibre. Let \(Z_{h_{\operatorname{Pf},\nu}}\) be the Hamiltonian vector field with respect to \[\omega^c=\omega_2+i\omega_3, \qquad \iota_{Z_{h_{\operatorname{Pf},\nu}}}\omega^c = dh_{\operatorname{Pf},\nu}.\] The hyperkähler identity \[Jdh_{\operatorname{Pf},\nu} = 2\,\iota_{Z_{h_{\operatorname{Pf},\nu}}}\omega_1\] implies that \(dJdh_{\operatorname{Pf},\nu}\) is closed and of type \((1,1)\). If \(h_{\operatorname{Pf},\nu}\) is complex-valued, the statement is understood by applying the same argument to its real and imaginary parts. ◻
For \(m>2\), the Pfaffian Hamiltonian is different from ordinary Teichmuller variation. Its parameter lies in \(H^1(C,K_C^{1-m}),\) whereas \(T_C\mathcal{T}\cong H^1(C,K_C^{-1}).\) Thus the Pfaffian gives a higher Hitchin Hamiltonian. When \(m=2\), the exceptional isomorphism \[\mathfrak{so}(4,\mathbb{C}) \cong \mathfrak{sl}_2(\mathbb{C})\oplus\mathfrak{sl}_2(\mathbb{C})\] puts the Pfaffian in degree two, and the two parameter spaces coincide.
Let \(D=p_1+\cdots+p_r\) be a reduced divisor on \(C\), and put \(C^\circ=C\setminus D.\) A logarithmic Higgs field on \((C,D)\) is allowed simple poles along \(D\). In a coordinate \(z\) centred at \(p_i\), it has the form \[\Phi= \left( \frac{N_i}{z}+A_i(z) \right)dz, \qquad N_i=\operatorname{Res}_{p_i}(\Phi).\] The nilpotency of \(N_i\) controls the leading pole terms of invariant polynomials in \(\Phi\).
Definition 4. A parabolic bundle over \((C,D)\) is a holomorphic vector bundle \(E\) on \(C\), together with, for every \(p_i\in D\), a decreasing filtration \[E_{p_i}=F_{i,1}\supset F_{i,2}\supset\cdots \supset F_{i,\ell_i}\supset F_{i,\ell_i+1}=0\] and real numbers \[0\leq \alpha_{i,1}<\alpha_{i,2}<\cdots<\alpha_{i,\ell_i}<1.\] The numbers \(\alpha_{i,j}\) are called the parabolic weights.
The multiplicity of the weight \(\alpha_{i,j}\) is \[m_{i,j} = \dim(F_{i,j}/F_{i,j+1}).\] The parabolic degree of \(E\) is \[\operatorname{pardeg}(E) = \deg(E) + \sum_{i=1}^{r}\sum_{j=1}^{\ell_i} \alpha_{i,j}m_{i,j},\] and the parabolic slope is \[\operatorname{par}\mu(E) = \frac{\operatorname{pardeg}(E)}{\operatorname{rk}(E)}.\]
If \(F\subset E\) is a holomorphic subbundle, then \(F\) has an induced parabolic structure. At \(p_i\), the filtration is obtained by intersecting with the filtration of \(E_{p_i}\) \[F_{p_i}\cap F_{i,1}\supset F_{p_i}\cap F_{i,2}\supset\cdots\supset F_{p_i}\cap F_{i,\ell_i}\supset0,\] after deleting repetitions. The weights are the corresponding weights of \(E\). The parabolic degree and parabolic slope of \(F\) are computed with this induced parabolic structure.
Definition 5. A logarithmic Higgs field on \(E\) is a section \(\Phi\in H^0(C,\operatorname{End}(E)\otimes K_C(D)).\) A parabolic Higgs bundle is a parabolic vector bundle \(E\) together with a logarithmic Higgs field \(\Phi\) such that, for every \(p_i\in D\), \[\operatorname{Res}_{p_i}(\Phi)(F_{i,j}) \subset F_{i,j} \qquad \text{for all }j.\] It is called strongly parabolic if \[\operatorname{Res}_{p_i}(\Phi)(F_{i,j}) \subset F_{i,j+1} \qquad \text{for all }i,j.\]
Thus, for a strongly parabolic Higgs field, the residue at \(p_i\) is nilpotent with respect to the flag at \(p_i\). In particular, if the flag is full, then the residue is represented by a strictly upper triangular matrix in a basis adapted to the flag.
Definition 6. A parabolic subbundle \(F\subset E\) is called \(\Phi\)-invariant if \(\Phi(F)\subset F\otimes K_C(D).\) A parabolic Higgs bundle \((E,\Phi)\) is called stable (resp. semistable), if for every proper nonzero \(\Phi\)-invariant parabolic subbundle \(F\subset E\), we have \[\operatorname{par}\mu(F)<\operatorname{par}\mu(E), \quad (\text{resp.} \leq)\]
For fixed rank, degree, parabolic weights and multiplicities, the moduli space of semistable parabolic Higgs bundles is a quasi-projective variety. We shall only use its smooth stable locus (see [12]–[14], [28] for more details).
Remark 5. For \(SL(2,\mathbb{C})\), we have \(\det E\simeq\mathcal{O}_C, \;\text{and} \; \Phi\in H^0(C,\operatorname{End}_0(E)\otimes K_C(D)).\) If \((E,\Phi)\) is strongly parabolic and \(N_i=\operatorname{Res}_{p_i}(\Phi),\) then \(N_i\in\mathfrak{sl}_2(\mathbb{C}).\) Since \(N_i\) strictly lowers the parabolic flag, it is nilpotent, in rank two this gives \(N_i^2=0,\) or equivalently \(\operatorname{tr}(N_i^2)=0.\) This observation will be used later for \(\operatorname{tr}(\Phi^2)\).
Definition 7. A logarithmic \(G\)-Higgs bundle on \((C,D)\) is a pair \((P,\Phi),\) where \(P\) is a holomorphic principal \(G\)-bundle on \(C\), and \(\Phi\in H^0(C,\operatorname{ad}(P)\otimes K_C(D)).\)
Let \(p_i\in D\). Choose a coordinate \(z\) centred at \(p_i\), and choose a local trivialization of \(\operatorname{ad}(P)\). Then the Higgs field has the local form \[\Phi= \left( \frac{N_i}{z}+A_i(z) \right)dz,\] where \(A_i(z)\) is holomorphic. The element \(N_i\in\mathfrak g\) is the residue of \(\Phi\) at \(p_i\), and we write \(N_i=\operatorname{Res}_{p_i}(\Phi).\)
Definition 8. We say that the logarithmic Higgs field \(\Phi\) has nilpotent residues if \(\operatorname{Res}_{p_i}(\Phi)\) is nilpotent for every \(p_i\in D\).
For a reductive Lie algebra \(\mathfrak g = \mathfrak z(\mathfrak g)\oplus[\mathfrak g,\mathfrak g],\) where \(\mathfrak z(\mathfrak g)\) is the centre and \([\mathfrak g,\mathfrak g]\) is semisimple, an element \(N\in\mathfrak g\) is called nilpotent if its component in \(\mathfrak z(\mathfrak g)\) is zero and its component in \([\mathfrak g,\mathfrak g]\) is nilpotent.
We fix a smooth stable family \[\mathcal{M}^{\mathrm{nil}}_{\mathrm{Dol}}(G) \longrightarrow \mathcal{T}_{g,r},\] whose fibre over \((C,D)\) is a smooth stable locus in the logarithmic Dolbeault moduli space of \(G\)-Higgs bundles with fixed topological type and nilpotent residues.
Let \(\mathcal{T}_{g,r}\) denote the Teichmuller space of pointed Riemann surfaces of type \((g,r)\). At the point represented by \((C,D)\), one has \[T_{(C,D)}\mathcal{T}_{g,r} \cong H^1(C,T_C(-D)).\] Since \[(T_C(-D))^\vee=K_C(D),\] Serre duality gives \[T^*_{(C,D)}\mathcal{T}_{g,r} \cong H^0(C,K_C^2(D)).\] Thus the cotangent space is the space of meromorphic quadratic differentials with at most simple poles at the marked points.
We shall use two representatives of a tangent vector \(Y\in H^1(C,T_C(-D)).\) The first is a Dolbeault representative \[\nu\in A^{0,1}(C,T_C(-D)),\] and the second is a Čech-Dolbeault representative \[(\mu,\xi_1,\ldots,\xi_r),\] where \(\mu\) is a Beltrami differential on \(C^\circ\), and \(\xi_i\) is a local holomorphic vector field near \(p_i\). The vector \(\xi_i(p_i)\in T_{p_i}C\) is the tangent direction of the marked point \(p_i\).
The following lemma will be used in the proof of the logarithmic energy-variation formula.
Lemma 1. Every class \(Y\in H^1(C,T_C(-D))\) admits a Dolbeault representative \(\nu\in A^{0,1}(C,T_C(-D))\) which is identically zero on a neighbourhood of \(D\).
Proof. Let \(\nu_0\in A^{0,1}(C,T_C(-D))\) be a \(\bar\partial\)-closed representative of \(Y\). Choose pairwise disjoint coordinate discs \(U_i\) around the points \(p_i\). Since \(U_i\) is contractible, the local \(\bar\partial\)-Poincare lemma for the line bundle \(T_C(-D)\) gives \[\eta_i\in A^{0,0}(U_i,T_C(-D))\] such that \[\bar\partial\eta_i=\nu_0|_{U_i}.\] Choose a smooth function \(\chi_i\) on \(C\), supported in \(U_i\), such that \[\chi_i=1 \quad\text{on a smaller disc }U_i'\subset U_i.\] Define \[\nu = \nu_0-\sum_{i=1}^{r}\bar\partial(\chi_i\eta_i).\] Then \(\nu\) differs from \(\nu_0\) by a \(\bar\partial\)-exact term, and hence represents the same class in \(H^1(C,T_C(-D))\). On \(U_i'\), we have \(\chi_i=1\), and therefore \[\nu = \nu_0-\bar\partial\eta_i = 0.\] Thus \(\nu\) vanishes on a neighbourhood of \(D\). ◻
For a meromorphic one-form \(\alpha\) near \(p\), we write \(\operatorname{Res}_{p}(\alpha)\) for the coefficient of \(dz/z\) in a local coordinate \(z\) centred at \(p\). Equivalently, \[\operatorname{Res}_{p}(\alpha) = \frac{1}{2\pi i} \int_{|z|=\varepsilon}\alpha .\] Let \(q\in H^0(C,K_C^2(D)),\) and let \(Y\in H^1(C,T_C(-D))\) be represented by \[(\mu,\xi_1,\ldots,\xi_r),\] where \(\mu\) is a Beltrami differential on \(C^\circ\), and each \(\xi_i\) is a local holomorphic vector field near \(p_i\). Then the Serre pairing is \[\langle q,Y\rangle = \int_{C^\circ}q\,\mu + \sum_{i=1}^{r} \operatorname{Res}_{p_i}(q\,\xi_i). \label{eq:serre-local}\tag{4}\] This is the Čech-Dolbeault form of the pairing \[H^0(C,K_C^2(D))\times H^1(C,T_C(-D)) \longrightarrow \mathbb{C}.\] The residue term depends only on the tangent vector \(\xi_i(p_i)\in T_{p_i}C.\) Indeed, if \(\eta_i\) is a local holomorphic vector field with \(\eta_i(p_i)=0,\) then \(q\,\eta_i\) is holomorphic at \(p_i\), since \(q\) has at most a simple pole. Therefore \[\operatorname{Res}_{p_i}(q\,\eta_i)=0.\]
We use the notion of nilpotent residue fixed in Section 4. The following lemma is about the quadratic invariant associated to an invariant symmetric bilinear form.
We shall use the following consequence of the Jacobson-Morozov theorem: if \(N\) is a nilpotent element of a complex reductive Lie algebra in the sense fixed above, then there exists \(H\in[\mathfrak g,\mathfrak g]\) such that \([H,N]=2N.\) (see [29]).
Proposition 6. Let \(B:\mathfrak g\otimes\mathfrak g\longrightarrow\mathbb{C}\) be an invariant symmetric bilinear form. If \(N\in\mathfrak g\) is nilpotent, then \[B(N,N)=0.\]
Proof. By the definition fixed in Section 4, the central component of \(N\) is zero. Hence \(N\in[\mathfrak g,\mathfrak g].\) Choose \(H\in[\mathfrak g,\mathfrak g]\) such that \([H,N]=2N.\) Using the invariance of \(B\), we get \[2B(N,N) = B([H,N],N) = B(H,[N,N]) = 0.\] Therefore \[B(N,N)=0. \qedhere\] ◻
We also need the corresponding vanishing for homogeneous invariant polynomials.
Theorem 7. Let \(p\in\mathbb{C}[\mathfrak g]^G\) be a homogeneous invariant polynomial of degree \(d\geq 1\). Let \((P,\Phi)\) be a logarithmic \(G\)-Higgs bundle on \((C,D)\). Assume that every residue \(N_i=\operatorname{Res}_{p_i}(\Phi)\) is nilpotent. Then \[p(\Phi)\in H^0\bigl(C,K_C^d((d-1)D)\bigr).\]
Proof. Fix \(p_i\in D\). Choose a coordinate \(z\) centred at \(p_i\), and choose a local trivialization of \(\operatorname{ad}(P)\). Then \[\Phi= \left( \frac{N_i}{z}+A_i(z) \right)dz,\] where \(A_i(z)\) is holomorphic and \(N_i=\operatorname{Res}_{p_i}(\Phi).\) Since \(p\) is homogeneous of degree \(d\), \[p(\Phi) = z^{-d}p(N_i+zA_i(z))\,dz^d.\] We first show that \(p(N_i)=0.\) By the nilpotency of \(N_i\) and the Jacobson-Morozov consequence, there exists \(H_i\in[\mathfrak g,\mathfrak g]\) such that \([H_i,N_i]=2N_i.\) Since \(p\) is invariant under the adjoint action, \[p(N_i) = p(e^{s\operatorname{ad}H_i}N_i) = p(e^{2s}N_i) = e^{2ds}p(N_i)\] for every \(s\in\mathbb{C}\). Since \(d>0\), this forces \(p(N_i)=0,\) which is the constant term of \(p(N_i+zA_i(z)).\) Hence \[p(N_i+zA_i(z))=O(z),\] and therefore \[p(\Phi)=O(z^{-(d-1)})\,dz^d.\] Thus \(p(\Phi)\) has pole order at most \(d-1\) at \(p_i\). Since this holds for every \(p_i\in D\), we obtain \[p(\Phi)\in H^0\bigl(C,K_C^d((d-1)D)\bigr).\] ◻
We shall also use the following local refinement of the same argument.
Lemma 2. Let \(p\in\mathbb{C}[\mathfrak g]^G\) be homogeneous of degree \(d\). Fix \(p_i\in D\). Suppose that \(1\leq r_i\leq d\) and \[(d^j p)_{N_i}=0 \qquad \text{for }0\leq j\leq r_i-1.\] Then \(p(\Phi)\) has pole order at most \(d-r_i\) at \(p_i\).
Proof. With the notation of the above proof, \[p(\Phi) = z^{-d}p(N_i+zA_i(z))\,dz^d.\] Then the vanishing of the first \(r_i\) Taylor terms gives \[p(N_i+zA_i(z))=O(z^{r_i}).\] Hence \[p(\Phi)=O(z^{-(d-r_i)})\,dz^d.\] ◻
Corollary 1. Let \(p_1,\ldots,p_\ell\) be homogeneous generators of \(\mathbb{C}[\mathfrak g]^G\), with degrees \(d_1,\ldots,d_\ell.\) On the nilpotent-residue locus, the logarithmic Hitchin map takes values in \[\bigoplus_{j=1}^{\ell} H^0\bigl(C,K_C^{d_j}((d_j-1)D)\bigr).\] Equivalently, the usual logarithmic Hitchin map factors as \[h_{G,\log}^{\mathrm{nil}}: \mathcal{M}_{\mathrm{Dol}}^{\mathrm{nil}}(G) \longrightarrow \bigoplus_{j=1}^{\ell} H^0\bigl(C,K_C^{d_j}((d_j-1)D)\bigr).\]
Proof. This follows by applying Theorem 7 to each generator \(p_j\). ◻
Corollary 2. Let \(B:\mathfrak g\otimes\mathfrak g\longrightarrow\mathbb{C}\) be an invariant symmetric bilinear form. Let \((P,\Phi)\) be a logarithmic \(G\)-Higgs bundle on \((C,D)\) with nilpotent residues. Then \[B(\Phi,\Phi)\in H^0(C,K_C^2(D)).\]
Proof. The polynomial \[p(X)=\frac{1}{2} B(X,X)\] is invariant and homogeneous of degree two. The result follows from Theorem 7. ◻
For \(G=GL(n,\mathbb{C})\), Theorem 7 applied to \(p(X)=\operatorname{tr}(X^k)\) gives \[\operatorname{tr}(\Phi^k) \in H^0\bigl(C,K_C^k((k-1)D)\bigr)\] whenever the residues of \(\Phi\) are nilpotent. The following local calculation shows that the pole order \(k-1\) cannot be improved in general.
Proposition 8. Let \(G=GL(n,\mathbb{C})\), and let \(2\leq k\leq n\). There are local logarithmic Higgs fields with nilpotent residue for which \(\operatorname{tr}(\Phi^k)\) has a pole of order exactly \(k-1\).
Proof. It is enough to give a local logarithmic Higgs field with nilpotent residue for which the coefficient of the pole of order \(k-1\) is nonzero.
Let \[N=E_{12}+E_{23}+\cdots+E_{k-1,k} \in\mathfrak{gl}_k\subset\mathfrak{gl}_n\] be the nilpotent Jordan block of size \(k\). Choose a holomorphic matrix \(A(z)\) with \(A(0)=E_{k1}.\) Consider locally \[\Phi= \left( \frac{N}{z}+A(z) \right)dz .\] The coefficient of \(z^{-(k-1)}dz^k\) in \(\operatorname{tr}(\Phi^k)\) is obtained by taking exactly one factor \(A(0)\) and \(k-1\) factors \(N\). Hence it is \[\sum_{j=0}^{k-1} \operatorname{tr} \bigl( N^jA(0)N^{k-1-j} \bigr).\] By cyclicity of the trace, this equals \[k\,\operatorname{tr}\bigl(N^{k-1}A(0)\bigr).\] Since \[N^{k-1}=E_{1k}, \qquad A(0)=E_{k1},\] we get \[\operatorname{tr}\bigl(N^{k-1}A(0)\bigr) = \operatorname{tr}(E_{11}) = 1.\] Thus the coefficient is \(k\neq0\). Therefore a pole of order \(k-1\) can occur. ◻
Remark 9. For \(k=2\), the nilpotent residue removes the double pole of \(\operatorname{tr}(\Phi^2)\), but the simple pole may remain. This is exactly the local behaviour used later for strongly parabolic logarithmic \(SL(2,\mathbb{C})\)-Higgs fields.
Let \(G=SO(2m,\mathbb{C}).\) The Pfaffian is the degree \(m\) invariant polynomial \[\operatorname{Pf}:\mathfrak{so}(2m,\mathbb{C})\longrightarrow\mathbb{C}.\] For a logarithmic \(SO(2m,\mathbb{C})\)-Higgs field, the invariant \(\operatorname{Pf}(\Phi)\) is therefore a meromorphic \(m\)-differential. Theorem 7 gives the pole estimate below. Under an additional rank condition on the residues, the first Taylor term of the Pfaffian also vanishes, and the pole order drops by one more.
Corollary 3. Let \((P,\Phi)\) be a logarithmic \(SO(2m,\mathbb{C})\)-Higgs bundle with nilpotent residues. Then \[\operatorname{Pf}(\Phi) \in H^0\bigl(C,K_C^m((m-1)D)\bigr).\] If, for every \(p_i\in D\), \[\operatorname{rk}\operatorname{Res}_{p_i}(\Phi)\leq 2m-4,\] then \[\operatorname{Pf}(\Phi) \in H^0\bigl(C,K_C^m((m-2)D)\bigr).\]
Proof. The first assertion follows from Theorem 7, since \(\operatorname{Pf}\) is homogeneous of degree \(m\). For the second assertion, put \(N_i=\operatorname{Res}_{p_i}(\Phi).\) It is enough, by Lemma 2, to prove \[d(\operatorname{Pf})_{N_i}=0.\] Choose a local orthogonal trivialization near \(p_i\), so that \(N_i\) is represented by a skew-symmetric \(2m\times 2m\) matrix. For a skew-symmetric \(2m\times 2m\) matrix \(X\), the first partial derivatives of \(\operatorname{Pf}(X)\) are, up to sign, Pfaffians of the \((2m-2)\times(2m-2)\) skew-symmetric matrices obtained by deleting two corresponding rows and columns. If \[\operatorname{rk}(N_i)\leq 2m-4,\] then each such \((2m-2)\times(2m-2)\) matrix is singular. Its determinant is zero, and hence its Pfaffian is zero. Therefore \[d(\operatorname{Pf})_{N_i}=0.\] Then by Lemma 2, with \(p=\operatorname{Pf}\), \(d=m\), and \(r_i=2\), we get the required bound. ◻
Remark 10. The quadratic invariant and the Pfaffian enter the logarithmic theory in different degrees. The quadratic invariant gives \[B(\Phi,\Phi)\in H^0(C,K_C^2(D)),\] which is the cotangent space to pointed Teichmuller space. The Pfaffian gives \[\operatorname{Pf}(\Phi)\in H^0(C,K_C^m((m-1)D)),\] and, under the rank condition above, \[\operatorname{Pf}(\Phi)\in H^0(C,K_C^m((m-2)D)).\] Thus, for \(m>2\), the Pfaffian belongs to the higher Hitchin Hamiltonians rather than to the ordinary Teichmuller cotangent direction. In the exceptional case \(m=2\), the Pfaffian also has degree two.
Let \[\pi:\mathcal{M}^{\mathrm{nil}}_{\mathrm{Dol}}(G) \longrightarrow \mathcal{T}_{g,r}\] be the smooth stable family fixed in Section 4. Thus the fibre over \((C,D)\) consists of logarithmic \(G\)-Higgs bundles \((P,\Phi), \;\text{where} \; \Phi\in H^0(C,\operatorname{ad}(P)\otimes K_C(D)),\) with nilpotent residues. By Corollary 2, \[B(\Phi,\Phi)\in H^0(C,K_C^2(D)).\] On the other hand, \[T^*_{(C,D)}\mathcal{T}_{g,r} \cong H^0(C,K_C^2(D)).\] Hence the quadratic differential \(-\frac{1}{2}B(\Phi,\Phi)\) is a cotangent vector to pointed Teichmuller space at \((C,D)\). We denote the corresponding one-form on the family by \(\phi_{G,\log}.\) Thus, for \[Y\in T_{(C,D)}\mathcal{T}_{g,r} \cong H^1(C,T_C(-D)),\] we set \[\phi_{G,\log}(Y) = -\frac{1}{2} \left\langle B(\Phi,\Phi),Y\right\rangle .\]
Theorem 11. The one-form \(\phi_{G,\log}\) is a holomorphic section of \(\pi^*\Omega^1_{\mathcal{T}_{g,r}}\) over \(\mathcal{M}^{\mathrm{nil}}_{\mathrm{Dol}}(G).\) If \(Y\) is represented by a Dolbeault form \[\nu\in A^{0,1}(C,T_C(-D))\] which vanishes near \(D\), then \[\phi_{G,\log}(Y) = -\frac{1}{2}\int_C B(\Phi,\Phi)\nu .\]
Proof. The first assertion follows from the polynomial dependence of \(B(\Phi,\Phi)\) on the Higgs field, together with the Serre-duality identification \[H^0(C,K_C^2(D)) \cong T^*_{(C,D)}\mathcal{T}_{g,r}.\] Thus \(\phi_{G,\log}\) varies holomorphically along the smooth relative moduli space. If \(\nu\) vanishes near \(D\), then the product \(B(\Phi,\Phi)\nu\) is a smooth \((1,1)\)-form on \(C\). In this case the Serre pairing is represented by the integral \[\left\langle B(\Phi,\Phi),Y\right\rangle = \int_C B(\Phi,\Phi)\nu,\] which gives the required formula. ◻
Let \(Y\in H^1(C,T_C(-D))\) be represented by \((\mu,\xi_1,\ldots,\xi_r),\) where \(\mu\) is a Beltrami differential on \(C^\circ\), and \(\xi_i\) is a local holomorphic vector field near \(p_i\).
Proposition 12. Let \(q=B(\Phi,\Phi),\) then \[\phi_{G,\log}(Y) = -\frac{1}{2} \left[ \int_{C^\circ} B(\Phi,\Phi)\mu + \sum_{i=1}^{r} \operatorname{Res}_{p_i} \bigl(B(\Phi,\Phi)\xi_i\bigr) \right].\]
Proof. By Corollary 2, we have \[B(\Phi,\Phi)\in H^0(C,K_C^2(D)).\] From the definition of \(\phi_{G,\log}\), we get \[\phi_{G,\log}(Y) = -\frac{1}{2} \left\langle B(\Phi,\Phi),Y\right\rangle .\] Therefore, by applying the local form of the Serre pairing from 4 , we get \[\left\langle B(\Phi,\Phi),Y\right\rangle = \int_{C^\circ} B(\Phi,\Phi)\mu + \sum_{i=1}^{r} \operatorname{Res}_{p_i} \bigl(B(\Phi,\Phi)\xi_i\bigr).\] This proves the formula. ◻
We now give the following results which will be used later.
Proposition 13. Let \(q\in H^0(C,K_C^2(D)).\) Then \(\operatorname{Res}_{p_i}(q\,\xi_i)\) depends only on the tangent vector \(\xi_i(p_i)\in T_{p_i}C.\)
Proof. Let \(\xi_i'\) be another local holomorphic vector field near \(p_i\) such that \(\xi_i'(p_i)=\xi_i(p_i).\) Then \(\eta_i=\xi_i-\xi_i'\) vanishes at \(p_i\). In a coordinate \(z\) centred at \(p_i\), write \[\eta_i=z\,g(z)\frac{\partial}{\partial z}\] with \(g\) holomorphic. Since \(q\) has at most a simple pole, locally \[q= \left( \frac{a_i}{z}+h_i(z) \right)dz^2\] with \(h_i\) holomorphic. Hence \[q\,\eta_i = \left( a_i g(z)+z h_i(z)g(z) \right)dz,\] which is holomorphic at \(p_i\). Therefore \[\operatorname{Res}_{p_i}(q\,\eta_i)=0.\] The residue is unchanged when \(\xi_i\) is replaced by \(\xi_i'\). ◻
The integral term in Proposition 12 is also well-defined for bounded Beltrami representatives.
Proposition 14. Let \(q\in H^0(C,K_C^2(D)).\) If \(\mu\) is bounded near \(p_i\), then \[\int q\,\mu\] is locally absolutely convergent near \(p_i\).
Proof. Choose a coordinate \(z\) centred at \(p_i\). Locally, \[q= \left( \frac{a}{z}+h(z) \right)dz^2, \qquad \mu= b(z)d\bar z\otimes\frac{\partial}{\partial z},\] where \(h\) is holomorphic and \(b\) is bounded. Then \[|q\,\mu| \leq C|z|^{-1}\,dx\,dy\] near \(z=0\). In polar coordinates this is bounded by \(C r^{-1}r\,dr\,d\theta,\) which is integrable near \(r=0\). This proves the local convergence. ◻
The circle action \((P,\Phi)\longmapsto (P,e^{i\theta}\Phi)\) preserves the nilpotent-residue locus. Indeed, \[\operatorname{Res}_{p_i}(e^{i\theta}\Phi) = e^{i\theta}\operatorname{Res}_{p_i}(\Phi),\] and scalar multiples of nilpotent elements are nilpotent.
Proposition 15. Under this circle action, \[\phi_{G,\log} \longmapsto e^{2i\theta}\phi_{G,\log}.\]
Proof. The invariant bilinear form \(B\) gives \[B(e^{i\theta}\Phi,e^{i\theta}\Phi) = e^{2i\theta}B(\Phi,\Phi).\] Since \(\phi_{G,\log}\) is obtained from \[-\frac{1}{2}B(\Phi,\Phi)\] by the Serre-duality pairing with \(H^1(C,T_C(-D))\), the one-form has weight two under the circle action. ◻
Let \(p_1,\ldots,p_\ell\) be homogeneous generators of \(\mathbb{C}[\mathfrak g]^G,\) with degrees \(d_1,\ldots,d_\ell.\) For logarithmic Higgs fields with arbitrary residues, the invariant \(p_j(\Phi)\) has, a priori, pole order at most \(d_j\) along \(D\). Thus the usual logarithmic Hitchin map takes values in \[\bigoplus_{j=1}^{\ell} H^0(C,K_C^{d_j}(d_jD)).\] By Corollary 1, on the nilpotent-residue locus this map factors through the smaller space \[\bigoplus_{j=1}^{\ell} H^0(C,K_C^{d_j}((d_j-1)D)).\] The degree-two invariant is the part relevant to pointed Teichmuller space. Indeed, for \(p_B(X)=\frac{1}{2}B(X,X),\) we have \[B(\Phi,\Phi)\in H^0(C,K_C^2(D)).\] Therefore, together with \[H^0(C,K_C^2(D)) \cong T^*_{(C,D)}\mathcal{T}_{g,r},\] this gives the logarithmic quadratic one-form \(\phi_{G,\log}.\)
We prove the energy-variation formula for tame nilpotent harmonic bundles on the punctured curve \(C^\circ=C\setminus D.\) The algebraic construction of \(\phi_{G,\log}\) was independent of harmonic metric estimates. The energy variation uses the tame asymptotic behaviour near the punctures, together with the decay assumption stated below.
A tame harmonic bundle on \(C^\circ\) is a flat \(G\)-bundle together with a harmonic metric whose associated Higgs field has logarithmic growth at the punctures. In this paper we use only the nilpotent-residue case. Thus we consider a logarithmic Higgs bundle \((P,\Phi), \;\text{with}\; \Phi\in H^0(C,\operatorname{ad}(P)\otimes K_C(D)),\) together with a harmonic metric \(h\) on \(C^\circ\), such that every residue of \(\Phi\) is nilpotent.
Let \(d_A\) be the Chern connection determined by the harmonic metric \(h\), and let \(\tau\) denote the adjoint operation induced by \(h\). If we take \(\varphi=\Phi+\tau(\Phi),\) then the associated flat connection is \(\nabla=d_A+\varphi.\) Near a puncture, in a coordinate \(z\), the Higgs field has the form \[\Phi= \left( \frac{N}{z}+A(z) \right)dz,\] where \(N\) is nilpotent and \(A(z)\) is holomorphic. With respect to the complete cusp metric on \(C^\circ\), \[|\varphi|_{h,\mathrm{cusp}}=O(1)\] near each puncture (see [16], [17]).
We consider variations with fixed local monodromy conjugacy classes. The infinitesimal variation of the harmonic metric is then expressed by a complex gauge term. The required decay of this term is stated in Assumption 1.
Near a point of \(D\), write \[z=re^{i\theta}, \qquad x=(-\log |z|^2)^{-1}.\] Then \(x\to0\) as \(z\to0\). The Poincare metric near the puncture is quasi-isometric to \[\frac{dx^2}{x^2}+x^2d\theta^2.\] The vector fields of bounded length for this metric are generated locally by \(x\partial_x, \;\text{and} \; x^{-1}\partial_\theta.\) We denote by \(C_{\mathrm{cusp}}^{k,\alpha}\) the Hölder space defined using these vector fields and the cusp distance. For \(\delta>0\), the weighted space \[x^\delta C_{\mathrm{cusp}}^{k,\alpha}\] consists of sections \(u\) such that \[x^{-\delta}u\in C_{\mathrm{cusp}}^{k,\alpha}.\]
Assumption 1. For the variation under consideration, the local monodromy conjugacy classes are fixed. The infinitesimal complex gauge term \(\psi\) satisfies, near each puncture, \(\psi\in x^\delta C_{\mathrm{cusp}}^{2,\alpha}\) for some \(\delta>0, \; 0<\alpha<1.\)
This assumption is used only in the integration-by-parts argument below, where it makes the boundary terms tend to zero.
Define \[f_{\mathrm{par}} = -\frac{1}{2}\int_{C^\circ}B(\varphi,*\varphi),\] provided the integral is absolutely convergent.
Proposition 16. For a tame nilpotent harmonic bundle as above, the integral defining \(f_{\mathrm{par}}\) is absolutely convergent.
Proof. It is enough to prove convergence near each point of \(D\). By the tame nilpotent estimate, \[|\varphi|_{h,\mathrm{cusp}}=O(1)\] near a puncture. Since \(B\) is fixed, there is a constant \(C>0\) such that \[|B(\varphi,*\varphi)| \leq C|\varphi|_{h,\mathrm{cusp}}^2\,d\mathrm{vol}_{\mathrm{cusp}}.\] Thus the local contribution to the energy is bounded by a constant multiple of the cusp area form.
In a punctured coordinate disc, the cusp area form is comparable to \[\frac{dr\,d\theta}{r(\log r)^2}.\] Hence the local integral is bounded by \[C\int_0^\epsilon \frac{dr}{r(\log r)^2},\] which is finite. Since \(D\) is finite, the integral over \(C^\circ\) is absolutely convergent. ◻
Choose pairwise disjoint coordinate discs around the points of \(D\), and put \[C_\varepsilon = C\setminus\bigcup_{i=1}^{r}\{|z_i|<\varepsilon\}.\] The first-variation formula is first applied on the compact surface \(C_\varepsilon\). The integration-by-parts terms coming from the complex gauge variation are supported on \(\partial C_\varepsilon\). The following estimate is the only place where Assumption 1 is used.
Lemma 3. Under Assumption 1, the boundary terms arising from the complex gauge variation over \(C_\varepsilon\) tend to zero as \(\varepsilon\to0\).
Proof. It is enough to estimate the contribution near one puncture. Let \(x=(-\log |z|^2)^{-1}.\) On the circle \(|z|=\varepsilon\), we have \(x=x_\varepsilon\), where \(x_\varepsilon\to0\) as \(\varepsilon\to0\). The cusp length of this circle is \(O(x_\varepsilon).\) The tame nilpotent estimate gives \[|\varphi|_{h,\mathrm{cusp}}=O(1).\] By Assumption 1, \[|\psi|_h=O(x_\varepsilon^\delta)\] on \(|z|=\varepsilon\), for some \(\delta>0\). The boundary terms produced by the gauge part of the variation are finite sums of integrals of the form \[\int_{|z|=\varepsilon} B(\psi,\iota_n\varphi)\,ds_{\mathrm{cusp}}, \qquad \int_{|z|=\varepsilon} B(\psi,\iota_n *\varphi)\,ds_{\mathrm{cusp}},\] where \(n\) is the cusp unit normal. On \(|z|=\varepsilon\), the contraction with the cusp unit normal satisfies \[|\iota_n\varphi|_{h}\leq |\varphi|_{h,\mathrm{cusp}}, \qquad |\iota_n *\varphi|_{h}\leq |\varphi|_{h,\mathrm{cusp}}.\] Hence, for a boundary term of the first type, we have \[\begin{align} \left| \int_{|z|=\varepsilon} B(\psi,\iota_n\varphi)\,ds_{\mathrm{cusp}} \right| &\leq C \sup_{|z|=\varepsilon}|\psi|_h\, \sup_{|z|=\varepsilon}|\varphi|_{h,\mathrm{cusp}}\, \operatorname{length}_{\mathrm{cusp}}(|z|=\varepsilon) \\ &\leq C\,x_\varepsilon^\delta\cdot 1\cdot x_\varepsilon = O(x_\varepsilon^{1+\delta}). \end{align}\] The same estimate applies to the term with \(\iota_n *\varphi\). Since \(\delta>0\), these boundary integrals tend to zero as \(\varepsilon\to0\). Summing over the finitely many punctures proves the lemma. ◻
Let \[Y\in T_{(C,D)}\mathcal{T}_{g,r} \cong H^1(C,T_C(-D)).\] By Lemma 1, choose a representative \[\nu\in A^{0,1}(C,T_C(-D))\] which vanishes on a neighbourhood of \(D\).
Theorem 17. Let \((P,\Phi,h)\) be a tame nilpotent harmonic bundle on \(C^\circ\). Consider a first-order deformation in the direction \[Y\in T_{(C,D)}\mathcal{T}_{g,r}\] with fixed Betti representation and fixed local monodromy conjugacy classes. Assume that Assumption 1 holds for the corresponding infinitesimal complex gauge term. Then \[Y(f_{\mathrm{par}}) = -\frac{1}{2}\operatorname{Re} \int_C B(\Phi,\Phi)\nu.\] Equivalently, \[Y(f_{\mathrm{par}}) = \operatorname{Re}\phi_{G,\log}(Y).\]
Proof. Choose disjoint coordinate discs around the points of \(D\), and put \[C_\varepsilon = C\setminus\bigcup_i\{|z_i|<\varepsilon\}.\] For sufficiently small \(\varepsilon\), the form \(\nu\) vanishes in a neighbourhood of \(\partial C_\varepsilon\). Define the truncated energy by \[f_{\mathrm{par},\varepsilon} = -\frac{1}{2} \int_{C_\varepsilon}B(\varphi,*\varphi).\] On \(C_\varepsilon\), all fields are smooth and the surface is compact with boundary. Since the Betti representation is fixed, the infinitesimal variation of the flat connection is a complex gauge variation, as in Proposition 2. Then by the first-variation formula on \(C_\varepsilon\), we get \[Y(f_{\mathrm{par},\varepsilon}) = -\frac{1}{2}\operatorname{Re} \int_{C_\varepsilon}B(\Phi,\Phi)\nu + \mathcal{B}_\varepsilon,\] where \(\mathcal{B}_\varepsilon\) is the sum of the boundary terms coming from the gauge variation. There is no boundary contribution from the variation of complex structure, because \(\nu\) vanishes near \(\partial C_\varepsilon\). By Lemma 3, \[\mathcal{B}_\varepsilon\longrightarrow0 \qquad \text{as }\varepsilon\to0.\] Since \(\nu\) vanishes near \(D\), the integral \[\int_{C_\varepsilon}B(\Phi,\Phi)\nu\] is independent of \(\varepsilon\) for all sufficiently small \(\varepsilon\), and is equal to \[\int_C B(\Phi,\Phi)\nu.\] The finite-energy estimate, together with the tame bounds and the decay of the gauge term in Assumption 1, allows the first variation of \(f_{\mathrm{par},\varepsilon}\) to pass to the limit as \(\varepsilon\to0\). Equivalently, the contribution from the cusp region \(C^\circ\setminus C_\varepsilon\) to the first variation tends to zero. Therefore, \[Y(f_{\mathrm{par}}) = -\frac{1}{2}\operatorname{Re} \int_C B(\Phi,\Phi)\nu.\] The equality \[Y(f_{\mathrm{par}}) = \operatorname{Re}\phi_{G,\log}(Y)\] now follows from Theorem 11. ◻
Combining this with the local expression of \(\phi_{G,\log}\) gives the residue form of the variation formula.
Corollary 4. Let \(Y\in H^1(C,T_C(-D))\) be represented by \((\mu,\xi_1,\ldots,\xi_r),\) where \(\mu\) is a Beltrami differential on \(C^\circ\), and \(\xi_i\) is a local holomorphic vector field near \(p_i\). Under the hypotheses of Theorem 17, \[Y(f_{\mathrm{par}}) = -\frac{1}{2}\operatorname{Re} \left[ \int_{C^\circ}B(\Phi,\Phi)\mu + \sum_{i=1}^{r} \operatorname{Res}_{p_i} \bigl(B(\Phi,\Phi)\xi_i\bigr) \right].\]
Proof. By Theorem 17, \[Y(f_{\mathrm{par}}) = \operatorname{Re}\phi_{G,\log}(Y).\] The formula follows from Proposition 12. ◻
Remark 18. The pole-order theorem and the construction of \(\phi_{G,\log}\) do not use Assumption 1. The assumption is used only in the proof of the energy-variation formula, to make the gauge boundary terms vanish.
The author is supported by the INSPIRE faculty fellowship (Ref No.: IFA22-MA 186) funded by the DST, Govt. of India.