Proof. The fibers of \(q\) are finite, so proper base change gives \(R^j q_*\mathbb Q=0\) for \(j>0\). Each fiber is one \(G\)-orbit, hence \((q_*\mathbb Q)^G=\mathbb Q\). Averaging makes invariants exact, and Leray gives (1), also after extending coefficients. Pullback respects mixed Hodge structures. If \(V\) is smooth and projective, strictness of the weight filtration and the displayed injection into \(H^i(V,\mathbb Q)\) give purity. ◻

Proof. For a projective bundle \(p:\mathbb P(E)\to S\) of relative dimension \(e-1\), put \(\xi=c_1(\mathcal O_{\mathbb P(E)}(1))\). The projective-bundle formula is \[H^q(\mathbb P(E),\mathbb Q) =\bigoplus_{i=0}^{e-1} \xi^i p^*H^{q-2i}(S,\mathbb Q).\] The powers of \(\xi\) restrict to a basis on each fiber, which proves the formula by Leray–Hirsch. Every displayed map is algebraic.

For a blowup \(b:\widetilde X\to X\) of a center of codimension \(c\), let \(j:E_Z\hookrightarrow\widetilde X\) be the exceptional divisor and \(p:E_Z=\mathbb P(N_{Z/X})\to Z\). With \(\xi=c_1(\mathcal O_{E_Z}(1))\), the blowup formula is \[H^q(\widetilde X,\mathbb Q)=b^*H^q(X,\mathbb Q) \oplus\bigoplus_{i=1}^{c-1} j_*\bigl(\xi^{i-1}p^*H^{q-2i}(Z,\mathbb Q)\bigr).\] For completeness, the map of relative pairs \((\widetilde X,E_Z)\to(X,Z)\) induces an isomorphism in relative cohomology, since it is an isomorphism off the closed subsets and the quotient spaces obtained by collapsing those subsets agree. On \(E_Z\), the projective-bundle formula and \(j^*j_*\eta=-\xi\eta\) show that subtracting the displayed exceptional terms leaves a class whose restriction comes from \(Z\). The long exact sequences of the pairs then show that the remaining class comes from \(b^*H^q(X)\). The sum is direct: \(b_*b^*=\operatorname{id}\), \(b_*\) kills these exceptional terms, and their restrictions have independent positive powers of \(\xi\) on \(E_Z\). All maps in the formula are algebraic. Companion results 3(i) prove both supply assertions. The formulas contain only finitely many summands, proving the last assertion. ◻

Proof. The Fermat hypersurfaces are smooth. The dimension-zero assertion is immediate. The plane curve \(F_n^1\) is connected: distinct positive-degree components in \(\mathbb P^2\) would meet by Bézout’s theorem, making their union singular. For this curve, the coordinate power map is the quotient by \(G=(\mu_n)^3/\mu_n\) onto the line \(z_0+z_1+z_2=0\) in \(\mathbb P^2\), branched only at its three coordinate points. The invariant part of first cohomology vanishes by (1). For a nontrivial character \(\chi\) of \(G\), its summand of the finite pushforward of the constant sheaf is a rank-one local system off the branch points where \(\chi\) is nontrivial on inertia, and has zero stalk at those points. At a point where \(\chi\) kills inertia it extends across the point, as one sees by passing to the quotient by \(\ker\chi\). If there are \(r\le3\) active points, its Euler characteristic is \(2-r\). The connected curve has no nontrivial character in degrees zero or two, so \[\dim_k H^1(F_n^1,k)_\chi=r-2\le1.\] Here a character occurring in first cohomology necessarily has \(r\ge2\). The Hodge group of the Jacobian commutes with the algebraic deck action. Over \(k\) its centralizer on first cohomology is a product of multiplicative groups, since every occurring character has multiplicity one. The connected Hodge group is therefore a torus. Thus \(\operatorname{Jac}(F_n^1)\) is CM. The Abel map supplies the curve’s first cohomology from this Jacobian, and point correspondences supply degrees zero and two.

We recall the explicit domination associated with Shioda and Katsura [21], including its resolution so that the maps used for supply are algebraic. For \(a,b\ge1\), choose \(\varepsilon\in\mathbb C\) with \(\varepsilon^n=-1\). There is a rational map \[ \begin{split} F_n^a\times F_n^b&\dashrightarrow F_n^{a+b},\\ (x,y)&\longmapsto \bigl((x_i y_{b+1})_{0\le i\le a}, (\varepsilon x_{a+1}y_j)_{0\le j\le b}\bigr). \end{split} \tag{2}\] Summing the \(n\)th powers of the two blocks gives the target equation. To see dominance, take a target point whose first block has nonzero power sum. Rescale its projective coordinates so that this sum is \(-1\); the second block then has power sum \(1\). Set \(x_{a+1}=y_{b+1}=1\), use the first block for \((x_0,\ldots,x_a)\) and \(\varepsilon^{-1}\) times the second for \((y_0,\ldots,y_b)\). These are preimages on the two Fermat hypersurfaces. Source and target have equal dimension, so the map is generically finite.

The base locus is the smooth codimension-two subvariety \[Z=\{x_{a+1}=y_{b+1}=0\} \simeq F_n^{a-1}\times F_n^{b-1}.\] Near each of its points some \(x_i\) and some \(y_j\) in the displayed blocks are units in projective coordinates. The base ideal of (2) is consequently exactly the ideal generated by \(x_{a+1},y_{b+1}\). Blowing up \(Z\) removes this ideal and extends the map to a morphism \(f:\widetilde X\to F_n^{a+b}\) from a smooth projective variety.

Induct on the target dimension, using \(a=1\), \(b=d-1\) in dimension \(d\ge2\). The product \(F_n^1\times F_n^{d-1}\) and the center \(F_n^0\times F_n^{d-2}\) are supplied by finitely many correspondence maps from powers of \(\operatorname{Jac}(F_n^1)\), by the induction hypothesis and external products. The center may be disconnected; the blowup formula and supply apply to its finitely many smooth components. Lemma 7 gives the same supply for \(\widetilde X\). If \(e>0\) is the generic degree of \(f\), the projection formula gives \(f_*f^*=e\,\operatorname{id}\) on rational cohomology. Thus \(f_*\) is surjective, and composing the supply maps with it supplies the full target cohomology. Every source remains a power of the same Jacobian, and each stage uses finitely many correspondence maps in each degree. ◻

Proof. The group-algebra character idempotents and all their scalar conjugates preserve the Hodge decomposition, so \(V_\chi\) is admissible and its determinant has the meaning fixed in Section 2. We use only the closure assertions of Companion results 3(i), not the Hodge-class conclusion.

Let \(g\) be the genus of \(B\) and let \(D=t_1+\cdots+t_r\) be its reduced branch divisor. The images of the branch meridians have product one in \(\Delta\), by the relation in the punctured surface group and commutativity of \(\Delta\). Thus \(r=0\) or \(r\ge2\). The \(\chi\)-part of \(q_*k_C\) is a rank-one local system on \(B\setminus D\), with zero stalk at every point of \(D\), because \(\chi\) is faithful on each nontrivial inertia group. Its Euler characteristic is \(2-2g-r\). Since \(C\) is connected, the nontrivial character does not occur in \(H^0(C,k)\) or \(H^2(C,k)\). Hence \[ n:=\dim_k V_\chi=2g+r-2. \tag{3}\] There is no nontrivial connected unramified cover of a genus-zero base, whose fundamental group is trivial. The numerical cases \(n=0\) are therefore \(g=0,r=2\) and \(g=1,r=0\); in either case the determinant is the unit supplied by a point. We assume \(n>0\) from now on.

Proof. Suppose first that \(\chi\ne1\) and put \(K=\ker\chi\). The quotient \(C_\chi=C/K\) is a smooth connected projective curve; locally the quotient of a smooth curve by its finite stabilizer has parameter \(t^e\). The residual group \(P/K\) is cyclic and \(\chi\) is faithful on it. Its action on \(C_\chi\) is effective. Indeed, choose a point of \(C\) with trivial \(P\)-stabilizer, outside the finitely many fixed points of nonidentity automorphisms. If a coset acted trivially on \(C/K\), it would send this point into its \(K\)-orbit, forcing the coset to be the identity.

Equivariantly for the residual action, (1) identifies \[H^1(C_\chi,k)_\chi\xrightarrow{\;\sim\;}H^1(C,k)_\chi\] by quotient pullback. A point of \(B\) is branched in \(C_\chi\to B\) exactly when its original inertia has nontrivial image in \(P/K\). Lemma 9 applies to this connected cyclic quotient and gives its determinant supply; (3) gives the dimension formula. The tensor power of the rational graph pullback sends that determinant line to \(D(H^1(C,k)_\chi)\), by Companion results 3(i). If the character space is zero both lines are the unit. This includes a quotient that becomes unramified although the original cover is branched.

For \(\chi=1\), quotient pullback identifies the character space with \(H^1(B,k)\). Choose a point \(b\in B\). The rational algebraic class \[[\Delta_B]-[b\times B]-[B\times b]\] is the \(H^1(B)\otimes H^1(B)\) component of the diagonal and, up to the cup-pairing convention, is the inverse symplectic tensor. If \(g=g(B)\), antisymmetrizing its \(g\)th external power gives a nonzero multiple of the top exterior power of this nondegenerate alternating tensor. It therefore generates \(D(H^1(B,k))\) by a rational algebraic class. For \(g=0\) this is the unit. Pullback on the curve slots proves the assertion.

For the last assertion, let \(\mathcal O(P)\) be the finite set of orders of nontrivial characters of \(P\), and set \[M_P=\prod_{m\in\mathcal O(P)}\operatorname{Jac}(F_m^1),\] with the empty product interpreted as a point. The faithful lemma uses powers of the indicated factor for a character of order \(m\), and the trivial case uses a point. Each such factor power is supplied from the corresponding power of \(M_P\) by its algebraic factor inclusion and projection. The product \(M_P\) is CM. This proves the uniform choice of sources. ◻

Proof. A rational tensor is real. In weight \(2a\), it has type \((a,a)\) exactly when it belongs to \(F^a\): reality then also places it in \(\overline{F^a}\), whose intersection with \(F^a\) is the \((a,a)\) part. Membership in the holomorphic subbundle \(F^a\) is the vanishing of the induced section of its quotient bundle, so \(Z_t\) is closed analytic. There are countably many rational tensors. A proper analytic subset of the connected complex manifold is nowhere dense, and Baire’s theorem proves the nonemptiness assertion. Odd tensor degrees have no balanced vectors, since the circle element \(-1\) acts on them by \(-1\).

At a point in the displayed complement, the balanced rational tensors are exactly those balanced at every marked parameter. The latter spaces are preserved by \(\Gamma\), because it acts equivariantly on the whole variation. By Companion results 3(iii), their pointwise stabilizer in the symplectic group is \(J_s\). Thus every element of \(\Gamma\) normalizes \(J_s\).

An individual globally balanced tensor has finite \(\Gamma\)-orbit. We use the finite-action argument of [1]. If its denominator is \(q\) in the integral tensor lattice, every translate still lies in \(q^{-1}H_{\mathbb Z}^{\otimes 2a}\), and its tensor-polarization norm is unchanged by transport. At a fixed fiber all its translates lie in the real balanced subspace. The tensor polarization, with its fixed weight sign, is definite on that subspace. A fixed norm sphere there is compact and contains only finitely many points of this lattice. The orbit is therefore finite. The connected algebraic group \(L^\circ\) fixes every such tensor, so Companion results 3(iii) give \(L^\circ\subset J_s\). ◻

Proof. It is enough to prove the Lie algebra assertion, since the groups in the conclusion are connected. Extending scalars from \(k\) to \(\mathbb C\) preserves irreducibility, semisimplicity, and the rank condition, so we work over \(\mathbb C\). In the symplectic alternative the invariant alternating form descends to \(k\): invariant forms are the kernel of linear equations over \(k\), and the determinant is nonzero on a suitable alternating form with coefficients in \(k\). Equality of the connected group images then descends under scalar extension. The center acts by scalars. The Jordan decomposition in an algebraic Lie algebra shows that a nilpotent element has zero central component and nilpotent components in its simple ideals. Write the irreducible representation of the semisimple image as a tensor product \(\bigotimes_i V_i\) of nontrivial irreducibles for its simple ideals.

Let \(N\) be the given rank-one nilpotent. In particular \(N^2=0\). If two of its simple-ideal components \(N_i,N_j\) were nonzero, choose length-two terminal Jordan strings in \(V_i,V_j\) and kernel vectors in the other factors. On the tensor product of the top vectors of these strings, \(N^2\) is \(2N_i\otimes N_j\ne0\). This is impossible. If just \(N_i\) is nonzero, its rank on \(V\) is \(\operatorname{rank}(N_i)\prod_{j\ne i}\dim V_j\). Rank one therefore forces the semisimple image to be simple.

Put \(N\) in the nilradical of a Borel subalgebra of this simple image. A generic one-parameter subgroup of its maximal torus, together with a scalar rescaling, degenerates \(N\) to a nonzero root vector \(x_\alpha\). The locus of matrices of rank at most one is closed, so \(x_\alpha\) has rank one. Every root vector for a root of the same length has rank one, by Weyl conjugacy. Restriction to the corresponding root \(\mathfrak{sl}_2\) consequently consists of one standard doublet and trivial representations: the rank of the raising operator on an irreducible \(\mathfrak{sl}_2\)-module of dimension \(a\) is \(a-1\). Thus each reflection in a root of that length exchanges exactly two weights, both of multiplicity one, and fixes every other weight.

Here is the root-system check that finishes the argument. We use the usual classification and weight lattices of reduced irreducible root systems, as in [9]. The roots of a specified length form a full-rank simply laced root system, not necessarily a closed root subsystem of the ambient root system: the whole system in simply laced type; \(D_\ell\) or \(A_1^\ell\) in types \(B_\ell\) and \(C_\ell\); \(D_4\) in type \(F_4\); and \(A_2\) in type \(G_2\). Use the small-rank conventions \(D_3=A_3\) and \(D_2=A_1\sqcup A_1\).

The simple reflections of this root system act as distinct transpositions on the weights. They are distinct because the weights span the dual Cartan space. Orthogonal simple roots give disjoint transpositions, while adjacent simple roots give transpositions sharing one endpoint. A branching vertex of the Dynkin diagram would require three disjoint transpositions each meeting the same two-element set, which is impossible. A connected full-rank path \(A_\ell\) therefore acts on precisely \(\ell+1\) moved weights as successive transpositions on a path. All other weights, being fixed by this full-rank root system, are zero. The moved weights form its standard regular simplex: their successive differences are the simple roots with a consistent choice of sign, as follows by pairing with adjacent simple coroots. In full type \(A_\ell\) the highest weight is thus the first or last fundamental weight. Irreducibility gives the standard representation or its dual. If this path arose instead from \(G_2\), \(B_3\), or \(C_3\), the full Weyl group contains negation, whereas this simplex, for \(\ell\ge2\), is not stable under negation. Those cases are excluded.

It remains to consider the orthogonal-axis root system \(A_1^\ell\). Its transpositions are disjoint. A moved weight for one axis is fixed by every other axis reflection, so lies on that axis; the doublet condition makes its two weights \(\pm\alpha/2\). For the long roots \(2e_i\) in \(C_\ell\), this gives the weights \(\pm e_i\), and the highest weight is that of the symplectic standard representation. For the short roots \(e_i\) in \(B_\ell\), the vectors \(\pm e_i/2\) are not weights when \(\ell\ge2\). In the rank-two \(D_2\) root system the half-roots \((\pm e_1\pm e_2)/2\) are not weights of \(C_2\); in \(B_2\) they give its spin representation, which is the standard representation under \(B_2=C_2\). Rank one gives the ordinary \(\mathfrak{sl}_2\) standard. These are exactly the alternatives in the statement. ◻

Proof. The eigenspace on the unbranched cover is the cohomology of the corresponding rank-one local system on the punctured base, by averaging over the finite deck group. Filling an inactive puncture gives ordinary cohomology on the filled base. At an active puncture the local complex is \(k\xrightarrow{t_i-1}k\), which is acyclic, so no local term changes the character cohomology. The cellular cochain model, or the standard crossed-homomorphism model of degree-one group cohomology, gives Equation (15). Euler characteristic gives Equation (14); in the unramified case both degree zero and degree two vanish for a nontrivial rank-one system.

For the twist, conjugate the two meridians by their product \(\delta=g_ig_j\) and fix the exterior generators. The identity \[z(fuf^{-1})=(1-t(u))z(f)+t(f)z(u)\] gives Equation (16) by direct subtraction. In particular \(\ell_\alpha(u_\alpha)=t_it_j-1\). All such disk twists preserve \(\phi\): they act trivially on the abelianization of the punctured surface group. For other mapping classes preserving \(t\), a positive power preserves \(\phi\), which suffices for all Lie-algebra statements.

Set \(s_i=z(g_i)/(1-t_i)\). These are the fixed points of the affine transformations represented by the meridians. The fixed-hyperplane condition is \(s_i=s_j\) after transport along \(\alpha\). More explicitly, if \(\alpha\) is represented by \(p_i^{-1}xp_j\) in terms of based access paths and a based loop \(x\), it is \[ s_i=z(x)+t(x)s_j. \tag{17}\] This description is unchanged by endpoint windings. For \(r\ge3\) the image vector in Equation (16) is nonzero modulo coboundaries, since another active meridian has zero coordinate in it. When \(r=2\), if that vector is a coboundary, all exterior multipliers are one; the surface relation then also makes \(\ell_\alpha\) zero.

The polarization pairs opposite character spaces perfectly and gives a nondegenerate Hermitian form on \(W_{\chi,\mathbb C}\), preserved by the monodromy. If \(T=1+u\ell\) is a nontrivial isometry, every fixed vector is orthogonal to \((T-1)W=ku\). Both spaces have codimension one, proving the fixed-hyperplane assertion. A common zero intersection of fixed hyperplanes makes the directions span by nondegeneracy. A nonzero invariant subspace contains a vector moved by one transformation, hence contains its image direction. Nonorthogonality then propagates this inclusion along the graph and gives the whole space.

Finally, if a simple closed curve has trivial \(t\)-monodromy, lift its annulus to the cyclic cover and twist simultaneously about its lifts. The ordinary intersection formula restricts on a character space to rank at most one: the projected lift classes are scalar multiples of each other. If evaluation of the cocycle on one closed lift is nonzero, the intersection-dual projected class is nonzero as well, by the perfect opposite-character pairing. The resulting transformation is a nontrivial rank-one unipotent. Its logarithm, or a nonzero integral multiple obtained by taking a power, belongs to the Lie algebra of connected monodromy. ◻

Proof. The unipotent radical of the identity component is normal in the full closure. Its common fixed subspace is nonzero and is invariant under the full group, so it is all of \(V\); the radical is trivial in the given faithful image. The identity component is therefore reductive. Its isotypic summands are permuted transitively by the full group. A rank-one nilpotent can act nontrivially on only one isotypic summand, and there with multiplicity one; that summand has dimension at least two. Transitivity gives the same two properties for every summand. A scalar multiple of a rank-one change cannot permute two summands of dimension at least two: its difference from that scalar identity would have rank at least two. Thus all the generators preserve every summand. Irreducibility forces a single summand, with multiplicity one, as required. ◻

Proof. Take the based handle bouquet on the handle side of a small vertex disk, and fan the access paths \(p_i\) to the punctures out on its other side. Choose the first two access paths to avoid the interior of the standard \(2,3\) arc when that arc is present. Concatenate \(p_1^{-1}xp_2\). Outside the vertex disk its pieces are embedded and disjoint. In the vertex disk there are two germs from the puncture side and two from the handle side. Choose the orientation of \(x\) so that the incoming and outgoing germs can be joined in noncrossing pairs. Replacing these joins by small smooth arcs removes the vertex and gives the required embedded arc. Its regular neighborhood with the two endpoint disks is a disk; with the \(2,3\) arc added it is a regular neighborhood of an embedded three-vertex tree, also a disk. This explains why the consecutive-meridian formulas apply to the winding and standard arcs together. Auxiliary marks can be avoided by small perturbations.

Use the standard separated handles for the last assertion. The winding arc and its endpoint disk can be placed in the last-handle and puncture region. The other handle loops lie outside. Carrying all these choices by the homeomorphism used in Equation (18) preserves the assertion. A change of their based access paths only conjugates those loops; because their \(t\)-multipliers are one, zero cocycle displacement on them remains zero after that change. ◻

Proof. We exhibit an irreducible group of twists and a rank-one nilpotent in the Lie algebra of its connected closure. There are three cases.

No active punctures. Here \(\eta\ne1\), since \(t\) is nontrivial. The surface relation is \((\eta-1)z(b_h)=0\). Thus \(z(b_h)=0\), and a coboundary removes \(z(a_h)\). The remaining coordinates are precisely the ordinary first cohomology coordinates of the first \(h-1\) handles. The twist character is trivial on this subsurface. Twists along a symplectic handle basis and along connecting band sums give rank-one transvections whose directions span with connected nonorthogonality graph. Their positive powers preserving \(\phi\) have the same logarithm directions. They act irreducibly already on the identity component. When \(h=1\) the character space is zero, as asserted.

At least three active punctures. Start with consecutive arcs inside the ordered puncture disk. Their common fixed equations make all \(s_i\) equal. Adjacent directions pair nontrivially: in a common three-puncture disk this follows by substituting the vector of one pair twist into the functional of its neighbor in Equation (16); the factors that occur are nonzero multiples of \(t_j\), \(1-t_i\), and \(1-t_k\).

For each handle generator add the winding arc of Lemma 16 from \(1\) to \(2\). Its direction is linked to the standard \(2,3\) direction by the same calculation in their three-puncture neighborhood. In the common fixed space, use a coboundary to make all \(s_i=0\). The winding equations then force \(z(x)=0\) for every handle generator. Thus the common fixed space is zero and the direction graph is connected. Lemma 14 gives irreducibility.

The twist about \(b_h\) supplies a rank-one nilpotent. In compatible generators it replaces \(a_h\) by \(a_hb_h\) and fixes the other generators, so its cocycle displacement is supported on \(a_h\) and is \(\eta z(b_h)\). The coordinate \(z(b_h)\) can be chosen freely and the single surface relation compensated at an active meridian. The image vector is not a coboundary, since it vanishes on all active meridians. This is a nonzero square-zero rank-one operator. A positive power of the twist preserves \(\phi\); a further power removes any finite deck scalar. Adjoin this twist to the irreducible disk-and-winding subgroup. The enlarged group is still generated, up to scalars, by rank-one changes of the identity. Lemma 15 now gives connected irreducibility for its closure.

Two active punctures. Write \(t_1=\lambda\), \(t_2=\lambda^{-1}\). The cocycle relation is \[ (\eta-1)z(b_h)+(1-\lambda)(s_1-s_2)=0. \tag{19}\] If \(\eta=1\), remove the common value \(s_1=s_2\) by a coboundary. All the handle coordinates are then ordinary symplectic coordinates, and their twists and band sums prove the assertion exactly as in the first case.

Suppose \(\eta\ne1\) and impose the gauge \(s_2=0\). The handle coordinates are arbitrary and Equation (19) determines \(s_1\). Take the standard two-puncture arc and the winding arcs corresponding to \(a_h\) and to \(a_j,b_j\) for \(j<h\). The standard fixed equation is \(s_1=0\); the others are \(s_1=z(x)\), after reversing \(x\) if necessary. Reversing a trivial-multiplier generator merely changes its sign, and for \(a_h\) it multiplies its coordinate by the nonzero scalar \(-\eta^{-1}\); these changes do not affect the argument. The common kernel is zero: first \(s_1=z(b_h)=0\), then \(z(a_h)=0\) and all remaining handle coordinates vanish.

We verify connectedness of the direction graph. Before the gauge, the standard arc direction has zero handle coordinates and \(s_1=s_2=-\lambda/(1-\lambda)\ne0\). Adding the coboundary with parameter \(\lambda/(1-\lambda)\) puts it in the gauge \(s_2=0\) and gives \[z(a_h)=\frac{(1-\eta)\lambda}{1-\lambda}\ne0, \qquad z(b_h)=s_1=0,\] with every other handle coordinate zero. The winding \(a_h\) functional is therefore nonzero on this direction, with either orientation of \(a_h\). The fixed-hyperplane description and Hermitian symmetry give the reciprocal nonzero pairing, so the \(a_h\)-winding direction has \(s_1\ne0\). That direction has zero coordinates on all \(a_j,b_j\) for \(j<h\), by the last assertion of Lemma 16; subtracting its gauge coboundary does not alter these coordinates, whose multipliers are one. Hence every remaining winding functional is nonzero on this direction. The graph is connected. All these pair twists have exceptional eigenvalue \(\lambda\lambda^{-1}=1\), so their nontrivial changes are rank-one nilpotents. Lemmas 14 and 15 finish this case.

In all cases the connected twist closure is irreducible and its Lie algebra contains a rank-one nilpotent. It lies in the projected Hodge group at every \(s\in\mathcal T_\phi^{\mathrm{Hg}}\) by Lemma 11. The latter is reductive and irreducible, so Lemma 13 applied to it gives the stated standard special linear or symplectic image.

The same constructions apply after every scalar conjugation: the active set is unchanged, a multiplier equal to one remains one, and every displayed nonzero factor remains nonzero. Thus every conjugate space has a nontrivial unipotent preserving its polarization Hermitian form. A definite Hermitian space admits no such unipotent isometry. Polarization is definite on a pure character space, so both Hodge types occur on every conjugate.

For the trivial character use the ordinary handle transvections on \(H^1(B)\), again taking powers in the finite-index stabilizer. They give the full symplectic group by the same spanning and connectedness argument and Lemma 13; the polarization rules out the larger special linear alternative in dimension greater than two. ◻

Proof. Choose a cyclotomic number field \(E\) containing all character values. The projector for \(\chi\) and the monodromy representation on its image are defined over \(E\). The determinant of an integral monodromy operator on this image is an algebraic integer: its eigenvalues are among those of the operator on integral first cohomology. Applying the same observation to the inverse shows that the determinant is an algebraic unit. At every embedding of \(E\) the operator preserves the nondegenerate Hermitian form obtained from the polarization, so the absolute value of its determinant is one. An algebraic integer all of whose conjugates have absolute value one is a root of unity. For example, the monic polynomials whose roots are the conjugates of its positive powers have bounded integral coefficients; their finite number forces two powers of a conjugate, and hence of the original integer, to coincide. The roots of unity in the fixed field \(E\) form a finite group. This finite root set is Zariski closed, so the determinant of the closure \(M_\chi\) has the same finite-image property. This proves the first assertion.

The matrices at any other embedding are obtained by applying that embedding to the matrices over \(E\). A finite set of matrices therefore remains finite after every embedding. Their direct sum is the scalar extension of the rational orbit representation, proving the second assertion. For the last assertion, change the sign of each negative definite Hermitian form and take their direct sum. It gives an invariant positive definite real form on the rational orbit representation, after restricting from its complexification. This representation also has an invariant lattice, its intersection with integral first cohomology. The image of each lattice basis vector has fixed length and hence finitely many possibilities. The monodromy image is finite. ◻

Proof. Let \(V_\mathbb Q\) be the rational character-orbit summand. The orbit contains the inverse character, so the polarization restricts nondegenerately to \(V_\mathbb Q\). It is a polarized subvariation of first cohomology. By Lemma 18 its monodromy is finite. We pass to an algebraic quotient of the full marked family. Since \(r\ge4\), the ordered moduli space of the marked points on \(\mathbb P^1\) is a smooth quasi-projective configuration space modulo projective coordinates; one may fix three labels. Include the labelled base point and the chosen point above it. The finite-index stabilizer of \(\phi\) gives a finite topological cover of this configuration space, which is algebraic and finite étale by the Riemann existence theorem. The base-point section splits the fundamental-group sequence of the universal punctured curve: in the homotopy sequence of this surface bundle, the section kills the boundary map into the fiber fundamental group and splits the projection on fundamental groups. Since the base subgroup preserves \(\phi\), sending its loops to the identity and the fiber loops by \(\phi\) defines a homomorphism from the total fundamental group to \(P\). Its finite topological cover is algebraic by the same theorem. Normalize the universal compact curve in this cover. Analytically near a branch section, a small simply connected base neighborhood trivializes the topological data, and the normalization is a disjoint union of charts \((u,s)\mapsto(u^e,s)\) with the fixed inertia order. The normalization is finite over the universal compact curve, so these charts make it a smooth proper family of compact covers. Thus this is a quotient of the full marked variation, not a restricted family. Indeed its monodromy on first cohomology is the action of the chosen lifts in \(\Gamma\): the lifted base-point section fixes the sheet during transport. On pulling back to \(\mathcal T_\phi\), it is the original marked family, since the topological covering epimorphism and the chosen sheet agree and the complex structure is uniquely obtained by lifting the base charts and filling the same power charts. In particular the pulled-back orbit variation is exactly \(V_\mathbb Q\) above.

A finite-index kernel in \(\Gamma\) kills the finite monodromy of \(V_\mathbb Q\). The corresponding finite topological cover of the same configuration quotient is algebraic and finite étale by Riemann existence; the smooth proper family is simply pulled back to it. The period map of \(V_\mathbb Q\), now with a constant marking, is a single-valued holomorphic map from this cover to a Siegel period domain. A Cayley transform realizes the domain as a bounded domain in a finite-dimensional complex vector space.

Each coordinate of the period map is consequently a bounded holomorphic function. Take a smooth projective compactification with normal-crossing boundary. The removable-singularity theorem, in local polydisks along the boundary, extends each coordinate holomorphically over the compactification. Such a function is constant on a connected projective variety. The period map of \(V_\mathbb Q\) is therefore constant.

We show that a mixed character would contradict this constancy. Pass to the cyclic quotient \[\pi:Y=C/\ker\chi\longrightarrow\mathbb P^1\] and let \(D_\chi\) consist of its \(r\) active branch points. The \(\chi\) and \(\chi^{-1}\) spaces in first cohomology are unchanged by this quotient. The map forgetting inactive and auxiliary markings is locally a submersion onto the configuration space of \(D_\chi\): the active points can be moved independently while the additional points are kept away from them. It thus suffices to show that the character period on the cyclic quotient has nonzero differential whenever it is mixed.

Here is a deformation calculation which also covers genus one for \(Y\). If a branch chart is \(x=u^e\), an invariant vector field on \(Y\) has the form \(u b(u^e)\partial/\partial u\); its pushforward is \(e x b(x)\partial/\partial x\). Consequently \[ (\pi_*T_Y)^{P/\ker\chi}=T_{\mathbb P^1}(-D_\chi),\qquad H^1(Y,T_Y)^{P/\ker\chi} =H^1(\mathbb P^1,T_{\mathbb P^1}(-D_\chi)). \tag{21}\] The second equality uses finiteness of \(\pi\) and exactness of invariants in characteristic zero. The tangent map for varying the branch points is this identification. Indeed, on a Čech cover, coordinate changes preserving the branch sections lift to the cyclic cover; differentiation gives precisely the inverse of the displayed pushforward on invariant vector fields. This describes the full tangent space of ordered branch configurations modulo projective coordinate changes. Since \(r\ge4\), there are no infinitesimal automorphisms of the pointed base. This argument does not require \(H^0(Y,T_Y)=0\).

If the \(\chi\)-space is mixed, there are nonzero forms \[\alpha\in H^0(Y,K_Y)_\chi, \qquad \beta\in H^0(Y,K_Y)_{\chi^{-1}}.\] Their product \(\alpha\beta\) is a nonzero invariant quadratic differential. It pairs nontrivially with some element of \(H^1(Y,T_Y)^{P/\ker\chi}\) by Serre duality and averaging over the finite group. Equivalently, invariant quadratic differentials descend to \(H^0(\mathbb P^1,K_{\mathbb P^1}^{\otimes2}(D_\chi))\): in the chart \(x=u^e\), invariance permits at most a simple pole at \(x=0\), and its pullback is holomorphic. This is the dual of (21).

For clarity, the period differential here follows directly from the same local deformation. If the transition functions of the deformed curve differ to first order by the vector-field cocycle \(\theta_{ij}\), Cartan’s formula for the closed holomorphic form \(\alpha\) shows that its first-order changes differ on overlaps by \(\mathcal L_{\theta_{ij}}\alpha =d(\theta_{ij}\mathbin{\lrcorner}\alpha)\). Its variation modulo holomorphic forms is therefore the class of \(\theta_{ij}\mathbin{\lrcorner}\alpha\) in \(H^1(Y,\mathcal O_Y)\), up to an immaterial common sign. Serre duality pairs this class with \(\beta\) by evaluating \(\theta\) on \(\alpha\beta\). The chosen tangent direction makes that value nonzero. This calculation also applies in genus one. Thus the period differential is nonzero. The contradiction proves purity, separately at every scalar translate.

Finally write \(F=\mathbb Q(\chi(P))\). The rational orbit summand is an \(F\)-vector space whose scalar embedding spaces are exactly these character spaces. Since each is pure, the Hodge circle acts by scalars on each of them. It therefore centralizes the entire rational algebra \(\operatorname{End}_F(V_\mathbb Q)\). The definition of the rational Hodge group at each fiber forces it to centralize the same algebra. Its action on \(V_\mathbb Q\) lies in the scalar torus \(\operatorname{Res}_{F/\mathbb Q}\mathbb G_m\), as claimed. ◻

Proof. We first check that the arc family acts irreducibly. When an actual rank-one unipotent is available, its logarithm yields the nilpotent needed to prove connected irreducibility; the remaining pseudoreflection case requires a separate proof of connected irreducibility. In that case an adjoint-eigenspace projection then produces a rank-one nilpotent in the Lie algebra.

Consecutive two-puncture arcs give an irreducible family on \(W_\chi\). To check the hypotheses of Lemma 14, use the cocycle coordinates \(z_i=z(g_i)\) and put \(s_i=z_i/(1-t_i)\). Their common fixed hyperplanes say that all \(s_i\) are equal; subtracting the resulting coboundary kills the cocycle. Consecutive arc directions have a nonzero mutual pairing by the formula in that lemma. Thus the directions span and their nonorthogonality graph is connected. The full group \(M_\chi\) acts irreducibly.

Its identity component is reductive in this representation. Indeed, the unipotent radical is characteristic in the identity component and hence normal in \(M_\chi\). Its nonzero fixed-vector space is then \(M_\chi\)-stable, and irreducibility makes that space all of \(W_\chi\). The radical acts trivially. The isotypical summands for \(M_\chi^\circ\) are permuted transitively by \(M_\chi\).

First suppose that the Lie algebra already contains a rank-one nilpotent \(N\). On an isotypical summand \(S\otimes A\) it acts as \(N_S\otimes1_A\). A summand on which it is nonzero must therefore have \(\dim A=1\) and \(\dim S\ge2\). Transitivity gives these same properties on every isotypical summand. An operator differing from a scalar by rank one cannot permute two summands of dimension at least two: its displacement on either of the summands would have rank at least two. Every arc pseudoreflection therefore fixes each summand. The irreducibility of the arc family forces there to be just one, proving connected irreducibility in this case.

There are two immediate sources of such an \(N\). If \(t_it_j=1\) for some active pair, its twist is a nontrivial rank-one unipotent by Lemma 14. Its logarithm belongs to \(\operatorname{Lie}(M_\chi^\circ)\). If all active pair products are \(-1\), then all the \(t_i\) are equal to the same primitive fourth root of unity. Equation (20) implies \(4\mid r\); since \(r\ge5\), we have \(r\ge8\). Choose a disk containing four active points. If \(g\) is the product of their consecutive meridians, then \(t(g)=1\). The boundary twist conjugates each inside meridian by \(g\), and its cocycle displacement is \[z_i\longmapsto z_i+(1-t_i)z(g)\quad(i\text{ inside}), \qquad z_j\longmapsto z_j\quad(j\text{ outside}).\] This is a rank-one unipotent. Its image is nonzero modulo coboundaries because there are active points outside the disk. Its functional is nonzero: one can prescribe a nonzero value of \(z(g)\) on the inside and satisfy the global cocycle relation using an outside entry. Its square-zero displacement follows also from \(t(g)=1\).

It remains to treat the case with no pair product equal to \(1\) and with some pair product \(\lambda\ne\pm1\). All pair twists are then semisimple pseudoreflections. Suppose first that the isotypical summands for \(M_\chi^\circ\) have dimension at least two. The rank argument just given makes every pair twist fix every summand, so there is just one summand \(S\otimes A\). If \(\dim S=1\), the connected group acts by scalars; its finite determinant then makes the connected group trivial, contrary to infinitude. Thus \(\dim S>1\).

An operator normalizing this isotypical action has the form \(B\otimes D\), with \(B\in\operatorname{GL}(S)\) and \(D\in\operatorname{GL}(A)\). To see this, choose an intertwiner \(B\) for the induced automorphism of the irreducible representation on \(S\); what remains centralizes that representation and is \(1\otimes D\) by Schur’s lemma. A nontrivial semisimple pseudoreflection cannot have this form when both factors have dimension at least two. Indeed the factors must themselves be semisimple (a nontrivial unipotent Jordan part would survive on the tensor product). Their eigenvalue products would then form a rectangular grid with just one exceptional entry. A two-by-two subrectangle containing that entry contradicts the equality of the products of its two diagonal pairs. Hence \(\dim A=1\), and the connected action is irreducible.

The only remaining reducible case has distinct one-dimensional isotypical summands. The Hermitian form pairs each such line nondegenerately with exactly one other line: for characters \(a,b\) of the connected group, a nonzero pairing requires \(a\overline b=1\) on the finite-index monodromy subgroup dense in that group. Distinct characters preclude two choices of \(b\). Hermitian symmetry makes this a pairing involution on the set of lines. Each pair twist preserves the involution and permutes the lines by either the identity or one transposition. Indeed, a permutation cycle of length \(\ell\) contributes rank at least \(\ell-1\) to the difference from a scalar identity, and disjoint cycles contribute independently; here that rank is one. The permutation group generated by the twists is transitive, by irreducibility. If the pairing involution had no fixed point, a transposition commuting with it would have an invariant two-element support, so could only exchange the members of one of its pairs. Such transpositions cannot act transitively on at least three lines. Thus there is a fixed point, and transitivity makes every line self-paired. Distinct lines are orthogonal; transitivity also makes their signs equal. The Hermitian form is definite.

The decomposition into distinct one-dimensional connected isotypicals and the pseudoreflection action are algebraic properties over \(k\). They persist at every scalar translate. The same argument therefore makes the Hermitian form definite on every translated character space. By Lemma 18 the rational orbit monodromy is finite, a contradiction. This proves connected irreducibility in the remaining case.

Use now a pair twist with eigenvalue ratio \(\lambda\ne\pm1\). After a scalar normalization it is \(R=\operatorname{diag}(\lambda,1,\ldots,1)\) for a decomposition \(W_\chi=L\oplus U\), \(\dim L=1\). Conjugation by \(R\) preserves \(\mathfrak m=\operatorname{Lie}(M_\chi^\circ)\) and has the three distinct eigenvalues \(1,\lambda,\lambda^{-1}\) on \[\operatorname{End}(L)\oplus\operatorname{End}(U),\qquad \operatorname{Hom}(U,L),\qquad \operatorname{Hom}(L,U),\] respectively. Polynomial projection onto these eigenspaces preserves \(\mathfrak m\). Connected irreducibility implies that some off-diagonal component is nonzero, since otherwise \(L\) and \(U\) would be invariant. A nonzero element of either off-diagonal space has rank one and square zero. It supplies the asserted nilpotent. ◻

Proof. For \(n\ge3\), Lemmas 20 and 13 give either the full standard special linear or the full standard symplectic derived algebra of connected monodromy. Its determinant is finite, so its connected scalar center is trivial. In the symplectic case all of \(M_\chi\) normalizes the connected symplectic group and hence preserves its unique invariant alternating form up to scalar.

In dimension at least four a semisimple pseudoreflection with ratio different from \(1\) cannot be such a similitude. Normalize its fixed hyperplane \(U\) to have eigenvalue \(1\). The alternating form does not vanish identically on \(U\), since \(\dim U=n-1>n/2\); thus the similitude multiplier is \(1\). Its exceptional eigenline must then be orthogonal to \(U\) and to itself, and hence be in the radical, an impossibility. It follows that every active pair product is \(1\). With at least three active points this forces all \(t_i=-1\). The sphere meridians generate the deck group, so \(\chi\) has order two. Conversely an order-two character space has a nondegenerate alternating polarization and monodromy contained in its symplectic group. Its pair twists are nontrivial unipotents, so the symplectic alternative occurs. This proves the stated connected monodromy groups.

The algebraic image of \(L_k\) on \(W_\chi\) is closed and is the Zariski closure of the character action, hence is \(M_\chi\). The image of \(L_k^\circ\) is \(M_\chi^\circ\), by the identity-component property of algebraic group images. By Lemma 11, \(L^\circ\subset J\), so connected character monodromy is contained in the image of \(J_k\) on \(W_\chi\). In the special linear case its action already contains the full traceless algebra. In the order-two case the Hodge group also preserves the restricted alternating polarization. Hence the derived Hodge projections are exactly the groups asserted in (i). Finite monodromy gives the toric conclusion by Lemma 19.

A connected reductive subgroup of \(\operatorname{GL}_2\) either acts through a torus or, when its derived image is nontrivial, has the full standard \(\operatorname{SL}_2\) as its derived image. Indeed a reducible two-dimensional representation of a reductive group is a sum of characters, while the only nonzero semisimple subalgebra of \(\mathfrak{sl}_2\) is \(\mathfrak{sl}_2\) itself. Dimension one is toric, and dimension zero is omitted. This proves (ii). Finally each displayed derived image is simple, and a surjective map from a semisimple Lie algebra to a simple one is supported by exactly one simple ideal. ◻

Proof. Pass to the cyclic quotient on which the character is faithful, of order \(m\), and write \(v_i=a_i/m\), \(0<a_i<m\). Choose a coordinate with infinity away from the branch points. The quotient has an equation \[y^m=\prod_i(x-b_i)^{a_i},\qquad \sum_i a_i=mS.\] The eigendifferentials for the inverse of the character of \(y\) are \(f(x)\,dx/y\). At a branch point with \(d_i=\gcd(m,a_i)\), a uniformizer has \(x-b_i=u^{m/d_i}\) and \(y\) equal to a unit times \(u^{a_i/d_i}\). The differential \(dx/y\) is holomorphic there, whereas a pole of \(f\) would destroy holomorphicity. Thus \(f\) is a polynomial. At infinity holomorphicity requires \(\deg f\le S-2\), giving dimension \(S-1\). Replacing \(y\) by \(\prod_i(x-b_i)/y\) computes the opposite eigenspace with exponents \(m-a_i\) and dimension \(3-S\). The choice of whether \(t\) labels this character or its inverse exchanges the two dimensions everywhere and does not affect the assertion. ◻

Proof. Write \(\mathfrak f\) for the shared ideal. The standard representations of a simple special linear or symplectic algebra determine their dimension and are preserved, up to duality, by isomorphisms of these algebras. The coincidence \(\mathfrak{sl}_2=\mathfrak{sp}_2\) has the same standard two-dimensional representation. Invert one character when necessary, so that an \(\mathfrak f\)-linear isomorphism \[F:W_\chi\longrightarrow W_{\chi'}\] exists. At this stage \(F\) need not respect the Hodge circle.

The projective actions of the entire group \(\Gamma\) are intertwined by \(F\). Indeed, \(\Gamma\) preserves the character spaces and normalizes \(J\) by Lemma 11. The kernel of either character projection on \(\operatorname{Lie}(J^{\mathrm{der}})_k\) is preserved, because the character space is preserved. Its unique complementary supporting ideal \(\mathfrak f\) is therefore preserved as well. Each lift induces the same automorphism of this ideal on both spaces. The two implementations of this automorphism differ, after conjugation by \(F\), by a scalar, by irreducibility and Schur’s lemma.

A two-puncture twist is projectively trivial if at least one endpoint is inactive: fill that inactive puncture in the rank-one local system, after which the twist is supported in a disk with at most one puncture. It is projectively nontrivial when both endpoints are active. For the latter assertion use \(r=n+2\ge4\) and the nonzero rank-one formula of Lemma 14; the global cocycle relation can be satisfied using an active point outside the disk, so its functional is nonzero as well as its direction. The graph of nontrivial pair twists is therefore the complete graph on the active set. Projective equivalence shows that the active sets of \(\chi\) and \(\chi'\) coincide. Label them identically and write \(t_i,t_i'\) for their multipliers.

Suppose first that \(n\ge3\). The pair twist has one exceptional eigenvalue and a fixed hyperplane. Its ratio is distinguished projectively by the multiplicities, so \[ t_i't_j'=t_it_j\qquad(i\ne j). \tag{22}\] If the ratio is \(1\), the operator is nontrivial unipotent and the same conclusion holds. These twists also intrinsically identify their image directions: for a semisimple twist this is its exceptional eigenline, and for a unipotent twist it is the image of its normalized displacement.

Choose three active points in an embedded disk, with consecutive meridians \(g_i,g_j,g_\ell\). Cocycles supported on these entries and with value zero on their product form a two-dimensional subspace \(U\subset W_\chi\). There are active points outside the disk, so passage to the cocycle quotient is injective on this space. The two consecutive pair directions are independent and span \(U\). Thus \(F\) identifies this space with its primed counterpart. The twist enclosing the three points acts on \(U\) by the scalar \(t_it_jt_\ell\) and on \(W_\chi/U\) by \(1\). Indeed its displacement on a cocycle is supported in the disk and has value zero on the product; on a cocycle with that product value zero, conjugation by the product multiplies every inside value by its multiplier. The quotient has dimension \(n-2>0\). Its identity action fixes the projective scalar in the comparison, even when the full operator is a nontrivial unipotent extension of the identity. We obtain \[t_i't_j't_\ell'=t_it_jt_\ell.\] Equation (22) makes every ratio \(t_i'/t_i\) equal to a common sign; the triple equality makes that sign \(1\). All local multipliers are equal. Since sphere meridians generate the deck group, the characters coincide after the permitted inversion. This proves the first part of (i). The symplectic assertion follows from Proposition 21: its character has order two, so complex conjugation acts internally and exchanges its two Hodge parts. Both parts have dimension \(n/2\) at every translate.

Now suppose \(n=2\), so the common active set has four points. The eigenvalue ratio of a projective two-by-two operator is defined up to inversion. Hence \[ t_i't_j'\in\{t_it_j,(t_it_j)^{-1}\}. \tag{23}\] Put \(A=t_1t_2\), \(B=t_1t_3\), and \(D=t_1t_4\). The Klein four permutations on the four indices give the following triples of pair products: \[\begin{array}{c|c} \pi&(t_{\pi(1)}t_{\pi(2)}, t_{\pi(1)}t_{\pi(3)},t_{\pi(1)}t_{\pi(4)})\\ \hline 1&(A,B,D)\\ (12)(34)&(A,B^{-1},D^{-1})\\ (13)(24)&(A^{-1},B,D^{-1})\\ (14)(23)&(A^{-1},B^{-1},D) \end{array}\] Uniformly inverting the four entries supplies the other four inversion patterns. Choose \(\pi\) in this Klein four group and \(\epsilon\in\{1,-1\}\) realizing the three choices in (23) involving index \(1\). Since \(t_1^2=ABD\), recovery of the four entries from these three products leaves only a common sign. Thus \[ t_i'=\delta\,t_{\pi(i)}^\epsilon\quad(1\le i\le4), \qquad \delta\in\{1,-1\}. \tag{24}\] The choices \(\delta,\epsilon,\pi\) are made once and are unchanged at every scalar translate of this identity.

This classifies only the projective twist data. It remains to use the shared ideal to align pure and mixed behavior and to make one fixed choice between the original model and its polarized dual for every scalar translate.

Mixedness agrees between the two character spaces at each translate. To justify this without any assumption about the pure orientations, split the infinitesimal Hodge-circle action into its central scalar part and its component in the shared simple ideal. On a standard two-dimensional module, the ideal component is zero precisely when the circle is scalar, that is, precisely when the character space is pure. The same ideal component acts on both equivalent modules, and this observation remains valid after every scalar translation.

We compare the two pure orientations. Permuting the four multipliers has no effect on their signatures; uniformly inverting them reverses the two pure signatures by Lemma 22. Consider therefore the additional common sign \(\delta=-1\). At a fixed translate write the four entries before this sign change as \(\exp(2\pi i v_i)\), \(0<v_i<1\), and put \(S=\sum_i v_i\). No \(v_i\) equals \(1/2\), since otherwise the sign change would produce an inactive multiplier. If \(a\) entries exceed \(1/2\), the sum after the change is \[ S'=S+2-a. \tag{25}\] For \(S=1\) one has \(a\in\{0,1\}\); agreement of purity excludes \(S'=2\), so \(a=0\) and \(S'=3\). For \(S=3\) one has \(a\in\{3,4\}\); the same argument gives \(a=4\) and \(S'=1\). For \(S=2\), agreement of mixedness forces \(a=2\) and \(S'=2\). Thus the common sign reverses pure orientation wherever the spaces are pure. Combining it with \(\epsilon\), the relative orientation of the two pure spaces is the same fixed sign \(\delta\epsilon\) at every pure scalar translate.

The standard \(\mathfrak{sl}_2\) module is self-dual. According to this fixed sign choose either the original model or its polarized dual, whose circle weights are the negatives of the original weights. Its pure type now agrees with that of \(W_{\chi'}\) at every translate. Choose an intertwiner over \(k\) of the resulting standard ideal modules. At a mixed translate the two circle weights are \(+1,-1\); their sum is zero, so the central scalar shift is zero on both modules. The intertwiner respects the common ideal action and hence the whole circle. At a pure translate both circles are the same scalar, so it respects them there as well. The same reasoning applies to every scalar conjugate of the chosen intertwiner. This proves (ii). Notice that polarized duality reverses the weights at every translate; no commutation between an arbitrary scalar automorphism and complex conjugation is being assumed. ◻

Proof. The intertwiner line is one-dimensional by Schur’s lemma and is stable under the Hodge group, since its derived group is normal. If the Hodge circle acts on this line by \(z^a\), the intertwiner shifts the spectrum of one space by \(a\). Both spectra are \(\{-1,1\}\), so \(\{-1+a,1+a\}=\{-1,1\}\) forces \(a=0\). Scalar conjugation gives the identical argument for the conjugated intertwiner and spaces. It is therefore an admissible Hodge isomorphism, and hence algebraic in the sense of Companion results 3(ii).

An invariant nondegenerate alternating form also spans a Hodge-stable line. Its scalar weight must be zero: nondegeneracy pairs the spectrum \(\{-1,1\}\) with its reflection about half that weight, so the midpoint is zero. The inverse tensor consequently has type \((1,1)\), and so do all its conjugates. It is the required pair tensor. ◻

Proof of Proposition 12 and Theorem 1. Fix \(s\in\mathcal T_\phi^{\mathrm{Hg}}\) and put \(J=J_s\). Set \(A=\operatorname{Jac}(C_s)\) and choose an Abel map \(a:C_s\to A\). Its pullback \(a^*:H^1(A,\mathbb Q)\to H_\mathbb Q\) is an isomorphism of rational Hodge structures. Its inverse is an algebraic degree-one operation by Companion results 3(ii). Transport the admissible character projectors \(e_\chi\) through this isomorphism. Their images are then actual summands of \(H^1(A,k)\), as required by the tensor criterion; we use the same notation for the identified curve and Jacobian spaces.

Collect the rational Galois orbits of character spaces on which \(J^{\mathrm{der}}\) acts trivially into a rational Hodge summand \(H_0\). Its Hodge group is a torus: its projections on the orbits are tori, and the group on their direct sum is a subgroup of their product. Companion results 3(iv) realize \(H_0\) as the first cohomology of a CM isogeny factor. For \(h\ge1\), every remaining nonzero character space has standard special linear or symplectic derived projection by Proposition 17. For \(h=0\) the same assertion, with the toric alternatives accounted for, is Proposition 21.

Each remaining character space is supported by exactly one simple ideal of \(\operatorname{Lie}(J^{\mathrm{der}})_k\). Indeed its image is a simple standard special linear or symplectic algebra. In a semisimple algebra mapping onto a simple one, exactly one simple ideal has nonzero image; the commuting images of two nonzero ideals could not both be that simple algebra. The supporting ideal maps isomorphically. If two character spaces share it, their standard representations are equivalent or dual. The only coincidence between the two group types is \(\mathfrak{sl}_2=\mathfrak{sp}_2\), with the same standard module.

In positive genus, Lemma 24 makes these comparisons admissible Hodge isomorphisms, using the opposite character for a dual comparison. On the sphere, Proposition 23 gives the same conclusion, including the simultaneous alignment of pure scalar translates in dimension two. Group the spaces with the same supporting ideal into one block. Treat a two-dimensional block as linear. A higher-dimensional symplectic block has its alternating pair tensor: it comes from Lemma 24 in positive genus and from the polarization on an order-two character on the sphere. A linear block uses the polarization pairing of a character and its opposite. Since finite character values are roots of unity, the opposite space is its coefficient-conjugate space \(\overline W\).

The simple-ideal decomposition gives simultaneous independent standard actions on these blocks. More explicitly, each ideal acts on precisely the copies in its own block, with the chosen standard or dual identifications, and acts trivially on the other blocks and on \(H_0\). The connected image of \(J^{\mathrm{der}}_k\) consequently has the Lie algebra of the product of these standard groups. The connected groups have the same image, so this is the product required in part (c) of Companion results 4. No independence of the central torus is needed.

A representative of every block is the transported image of an actual character space on \(C_s\). Corollary 10 supplies its determinant singleton: a nontrivial character uses the faithful cyclic quotient \(C_s/\ker\chi\), and the invariant character uses the polarization. The curve–Jacobian degree-one maps and all the block identifications are algebraic operations by Companion results 3(ii). They transfer these determinant lines to the chosen Jacobian slots. Thus every linear determinant singleton is CM-supplied; there are no orthogonal blocks. The data are saturated.

Companion results 4 now prove the assertion for every power after adjoining any CM \(M\). This is a pointwise application after saturation, so \(M\) does not enter the definition of \(\mathcal T_\phi^{\mathrm{Hg}}\). Finally, the algebraic curve–Jacobian correspondences identify their first cohomology, and a point and the fundamental class supply the other two curve degrees. External products of these maps and Companion results 5 transfer the tensor conclusion to every \(C_s^a\times M^b\). ◻

Proof. Each curve point has at most one zero coordinate. Since \(j<m\), the products are never all zero. A product of evaluations of linear polynomials is the evaluation of their product, so the formula gives a morphism to \(X_{P_j}\). Its defining sections give \[\mu_j^*\mathcal O(1) =\mathcal O_C(1)\boxtimes\cdots\boxtimes\mathcal O_C(1),\] which is ample. A positive-dimensional projective fiber would contain a curve on which this line bundle has degree zero. Thus the proper morphism is finite.

Homogenization \(f(z)\mapsto v^j f(u/v)\) identifies polynomials of degree at most \(j\) with binary forms of degree \(j\). On the bases the map multiplies \(j\) binary linear forms. Every degree-\(j\) form factors into such forms, including the factor \(v\) representing a zero at infinity. Choose roots of the factors’ evaluation vectors; a phase change on one factor then realizes any prescribed root vector above the product. This proves surjectivity. Where the factors are distinct and no evaluation vanishes, the only identifications are their permutations and phase tuples of total product one. The indicated finite quotient therefore maps finitely and birationally to the smooth target. Normality of the target makes this map an isomorphism.

By Lemma 6, target cohomology is the invariant cohomology of the product. In residual character \(b\), product kernel invariance forces character \(b\) in every factor. Connectedness of \(C\) kills nontrivial characters in \(H^0,H^2\), leaving \(V_b^{\otimes j}\). A geometric interchange of two degree-one factors is minus ordinary tensor interchange. The permutation invariants are exactly the alternating tensors. ◻

Proof. The invariant assertion follows from the finite quotient formula. The pullback of a base hyperplane is \(n\) times the root hyperplane. For \(d\geq1\), Lemma 27 makes all degrees outside \(d\) ambient and phase-invariant. For \(d=0\), degree zero is the regular representation of the finite torsor.

For \(d\geq1\), compute the rank at an evaluation plane. The curve \(C\to\mathbb P^1\) is branched at the zeros of its \(m\) coordinate forms. Pass to the effective cyclic quotient for \(b\). Exactly the \(s=s(b)\) points in its support have nontrivial inertia character. Its character summand of the finite pushforward of the constant sheaf is a rank-one local system off those points, extended by zero at them. Its compactly supported Euler characteristic is \(2-s\), the rank-one Euler characteristic of the punctured sphere. The connected cyclic cover has no nontrivial character in degrees zero or two. Hence \(\dim V_b=s-2\). The effective quotient makes this valid for nonfaithful characters and nonunit entries as well.

Lemma 29 gives the binomial rank there. The phase idempotent cuts out a local system on the connected \(\mathcal U_{d,m}\), so the rank is constant. In dimension zero every character has rank one in the regular representation, agreeing with the formula since a nonzero sum-zero tuple has \(s\geq2\).

For \(s\geq d+2\), any \(d+1\) selected coordinates detect \(P\). Its projection is injective with coordinate-general image, and the selected root coordinates cannot all vanish. Forgetting the others defines \(p_S\). Its pullback of \(\mathcal O(1)\) is \(\mathcal O_{X_P}(1)\), so the ample-line-bundle argument proves finiteness. It is surjective by taking roots of the remaining coordinate values of the determined base point.

The fibers are exactly the orbits of \(K_S=\ker(G_{m,n}\to G_{s,n})\). For \(d\geq1\), the induced finite map \(X_P/K_S\to X_{P_S}\) is birational and its target is smooth, so it is an isomorphism. For \(d=0\), it is directly the quotient of finite torsors. The character \(b\) is inflated from \(G_{s,n}\), so Lemma 6 proves the eigenspace assertion. Universal coordinate projections and constant phase actions give the asserted family compatibility. ◻

Proof. First target \(X=X_{P_S}\). Put \(h=h_S\), \(F=F_n^{s-2}\), and \(Y=Y_S\). Thus \(\dim F=d+h\), \(\dim X=d\), and \(\dim Y=h\). The graph embedding \[X\times Y\longrightarrow(F\times Y)\times X,\qquad (x,y)\longmapsto(f_{P_S}(x,y),y,x)\] defines \(\Gamma_S\in\operatorname{CH}^{d+h}(F\times Y\times X)_\mathbb Q\). The source dimension is \(d+2h\), so the shift is \(-2h\). On a Künneth tensor its action is \[\alpha\otimes\beta\longmapsto (\operatorname{pr}_X)_*\bigl(f_{P_S}^*\alpha\smile\operatorname{pr}_Y^*\beta\bigr).\] For a character \(b\) with support \(S\), put \(b_S=b|_S\) and \(L_{b_S}=H^{s-2}(F,\overline{\mathbb Q})_{b_S}\). The invariant Poincaré pairing on \(Y\) pairs only opposite characters and is perfect on each opposite pair, by its full nondegeneracy and the direct character decomposition. This also holds for the sum pairing on a finite torsor. It identifies \(H^h(Y,\overline{\mathbb Q})_{-b_S}(h)\) with the dual of \(H^h(Y,\overline{\mathbb Q})_{b_S}\). Theorem 31 therefore makes the displayed graph action an isomorphism of Hodge-graded spaces \[L_{b_S}\otimes H^h(Y,\overline{\mathbb Q})_{-b_S}(h) \xrightarrow{\;\sim\;} H^d(X,\overline{\mathbb Q})_{b_S}.\] Composing with \(p_S^*\) fills \(W_{P,b}\).

Both the graph cycle and the transpose graph for \(p_S^*\) have rational coefficients. Apply them to the entire source degree, without character projectors. After scalar extension their images contain all the character pieces just described. Hyperplane cycles from a point supply the invariant cohomology, including every degree outside the middle. The sum of these finite rational maps is thus surjective after extension to \(\overline{\mathbb Q}\); its rational cokernel is already zero. No algebraic Künneth projector is required. ◻

Proof. For \(b\ne0\), put \(S=S(b)\) and \(s=|S|\). If \(W_{P,b}\ne0\), then \(s\geq d+2\), and \(\dim Y_S=s-d-2\). If \(c=1\), \(m=d+2\), so \(s=m\) is the only possibility and the source dimension is zero. If \(c=2\), \(m=d+3\), so \(s=m\) gives dimension one and \(s=m-1\) gives dimension zero. The full-support source is \(X_{P^\perp}\), connected by Lemma 28.

For a missing-one support, zero extension identifies \[(\operatorname{pr}_S P)^\perp=\{q|_S:q\in P^\perp,\ q_i=0\}.\] This is the coordinate-general line cut from the dual two-plane by its \(i\)th coordinate hyperplane. Its root torsor is precisely the reduced fiber \(\{y_i=0\}\) on \(C\) after forgetting that coordinate. The phase group has size \(n^{m-2}\). Distinct boundary fibers are disjoint because a root-curve point has at most one zero coordinate. ◻

Proof. The ratios are row-basis independent and their denominators are units. The Plücker relations give \[z_i-1=\frac{\Delta_{12}\Delta_{3i}}{\Delta_{1i}\Delta_{23}}, \qquad z_i-z_j=-\frac{\Delta_{12}\Delta_{13}\Delta_{ij}} {\Delta_{23}\Delta_{1i}\Delta_{1j}},\] so the values lie in \(M_m\).

Let \(A_0(z)\) have columns \[(0,1)^t,\ (1,0)^t,\ (1,-1)^t,\ (1,-z_4)^t,\ldots,(1,-z_m)^t.\] Set \(t_1=1\) and \[t_2=\frac{\Delta_{23}}{\Delta_{13}},\quad t_3=\frac{\Delta_{23}}{\Delta_{12}},\quad t_i=\frac{\Delta_{1i}\Delta_{23}}{\Delta_{12}\Delta_{13}} \quad(i\geq4).\] All are nonzero regular functions of \(Q\). With \(B=(a_1,a_2)\), left multiplication by \[H=\begin{pmatrix}0&t_2\\1&0\end{pmatrix}B^{-1}\] gives, by Cramer’s rule, \[Ha_1=(0,1)^t,\quad Ha_2=(t_2,0)^t,\quad Ha_3=(t_3,-t_3)^t,\quad Ha_i=(t_i,-t_i z_i)^t .\] Hence \(Q=\operatorname{row}(A_0(z)\operatorname{diag}(t_i))\). Conversely, substituting this form into the ratios recovers \(z,t_2,\ldots,t_m\). These are inverse regular maps for the product isomorphism. The effective coefficient torus acts freely and transitively on the second factor, giving the quotient claim. ◻

Proof. The map is the \(n\)-power map on a torus times the identity. It is finite locally free of degree \(n^{m-1}\) and étale because \(n h_i^{n-1}\) is a unit. Its source is irreducible. The coordinate scalings satisfy the root equations by the preceding formulas for \(P,Q\); their inverses and phase equivariance are immediate.

On the normalized base \([s:t]\), the coordinate forms are \[t,\ s,\ s-t,\ s-z_4t,\ldots,s-z_m t.\] Their ordered zeros are the stated branch points. On \(t\ne0\), put \(v=s/t\), \(w_i=y_i/y_1\). The function field is \[w_2^n=v,\qquad w_3^n=v-1,\qquad w_i^n=v-z_i\quad(i\geq4).\] Valuations at the distinct finite points show that these \(m-1\) functions are independent in \(\mathbb C(v)^*/\mathbb C(v)^{*n}\). Kummer theory gives degree \(n^{m-1}\). Equivalently, adjoin the roots successively and use an Eisenstein test at the new branch point, where the earlier functions are units with local roots. Thus the cover is connected and its phase group is its full deck group.

At a finite point indexed by \(i\geq2\), a positive meridian multiplies \(w_i\) by \(\zeta_n\) and fixes the other local roots, giving \(\bar e_i\). At infinity all right sides have simple poles, so a meridian multiplies every \(w_i\), \(i\geq2\), by \(\zeta_n^{-1}\). This is \(\bar e_1=-\sum_{i\geq2}\bar e_i\). The entries have order \(n\), sum to zero, and generate the group.

For \(*\in\mathbb P^1\setminus D_z\), positive meridians generate the punctured-sphere fundamental group with their product as the single relation. The displayed epimorphism is precisely abelianization modulo \(n\): \[\pi_1(\mathbb P^1\setminus D_z,*)\to H_1(\mathbb P^1\setminus D_z,\mathbb Z/n) \simeq(\mathbb Z/n)^m/\langle(1,\ldots,1)\rangle .\] A pure orientation-preserving mapping class conjugates each positive peripheral meridian, so it fixes this epimorphism. The standard classification of connected covers by epimorphisms shows that the ordered tuple determines the single fixed cover type here. For each complex structure on the pointed base, pullback charts off the punctures and the local ramification power charts give the unique complex structure on its compact covering surface. The root equations realize it for every configuration in \(M_m\). ◻

Proof. We use the planar Schoenflies theorem in its annulus form: the region between two nested Jordan curves is an annulus. It gives the following disk extension fact. If \(D_0,D_1\) are closed disks in the interior of a closed disk \(U\), then an orientation-preserving homeomorphism \(D_0\to D_1\) extends to a homeomorphism of \(U\) which is the identity on \(\partial U\). Indeed, the two complementary regions in \(U\) are annuli. In annulus coordinates, the prescribed maps on their two boundary circles extend across the annulus: orientation-preserving circle homeomorphisms are isotopic, as is seen by taking increasing lifts to \(\mathbb R\) and linearly interpolating the lifts. Gluing this annulus map to the given disk map proves the extension fact.

Let \(g\) be the given sphere homeomorphism. Choose \(p\in S^2\), a point \(q\) different from \(p\) and \(g(p)\), and a small closed disk \(D\) about \(p\) for which both \(D\) and \(g(D)\) avoid \(q\). Their compact union lies in the interior of a closed disk \(U\subset S^2\setminus\{q\}\). The disk extension fact gives a homeomorphism \(f\) of \(U\), fixed on its boundary, which agrees with \(g\) on \(D\). Extend \(f\) by the identity outside \(U\). The Alexander trick isotopes \(f\) to the identity, supported in \(U\): in unit-disk coordinates a boundary-fixed map \(F\) is deformed by \[F_t(x)= \begin{cases} tF(x/t),&|x|\leq t,\\ x,&|x|\geq t, \end{cases} \qquad F_0=\operatorname{id},\quad 0<t\leq1.\] Now \(f^{-1}g\) fixes \(D\) pointwise, so it is a boundary-fixed homeomorphism of the complementary closed disk. A second Alexander trick isotopes it to the identity. Composing the two isotopies proves the assertion. ◻

Proof. Away from \(D_z\), \(q_z\) is finite étale with free transitive phase fibers. This proves the torsor assertion except for connectedness. The total \(\mathcal Y_0\) is irreducible: it is smooth over the irreducible \(M_m\), and every fiber is connected. Indeed its irreducible components, being disjoint open components of a smooth variety, would have nonempty open images on the base; two such images meet, contradicting connectedness of that fiber. Removing the boundary leaves a nonempty irreducible open. Thus \(B_{m,n}\) is connected.

Fix \((z_0,y_0)\in B_{m,n}\), put \(b_0=q_{z_0}(y_0)\), and identify the fixed labelled topological sphere with the normalized base at \((z_0,b_0)\). Also fix a \(G_{m,n}\)-equivariant identification of its topological cover with the root cover, taking the fixed upstairs point to \(y_0\). The based epimorphisms agree because both take the ordered positive meridians to \(\bar e_i\). A change of the chosen lift conjugates the epimorphism and hence leaves it unchanged for this abelian group.

Consider any prescribed marked complex sphere. A complex curve of genus zero is \(\mathbb P^1\), and its first three branch labels give a unique Möbius normalization to \(\infty,0,1\). The marking is then represented by an orientation-preserving self-homeomorphism \(g\) of the reference sphere which carries all reference labels to the normalized target labels. It fixes the first three labels at the endpoint. By Lemma 37, choose an unmarked isotopy \(g_t\) from the identity to \(g\). For each \(t\), let \(a_t\) be the unique Möbius transformation taking \(g_t(p_1),g_t(p_2),g_t(p_3)\) to \(\infty,0,1\). The formulas for the transformation of a distinct triple show that \(a_t\) is continuous, and \(a_0=a_1=\operatorname{id}\). Thus \(a_tg_t\) keeps the first three images fixed. Its remaining branch images and its auxiliary-point image give a path \(\gamma\) in \(V_m\), starting at \((z_0,b_0)\), whose endpoint marking is the prescribed \(g\), not only a marking of the same unmarked fiber.

Lift \(\gamma\) uniquely to \(B_{m,n}\) from \((z_0,y_0)\); write \(y_t\) for its tautological upstairs point. Naturality of oriented labelled meridians under \(a_tg_t\) and the abelianization description in Proposition 36 give equality of the exact based epimorphisms at every \(t\). The covering-space lifting criterion therefore gives a unique \(G_{m,n}\)-equivariant lift of the base marking which sends the fixed upstairs point to \(y_t\). Covering homotopy lifting and uniqueness make these lifts continuous along \(\gamma\). They extend over the compactification through the fixed local ramification power charts, giving a continuous trivialization of the compact root family pulled back to the path interval. They therefore induce exactly its cohomological parallel transport along the projection of \(\gamma\) to \(M_m\).

At the endpoint, this lifted covering identification is holomorphic for the complex structure of the prescribed marked cover: off the branch points both complex structures are pulled back from the normalized base, and the same statement holds in the local power charts. Hence it identifies the endpoint Hodge filtrations. This proves the asserted tensor consequence. No loop in \(V_m\) is required to lift to a loop in \(B_{m,n}\); its endpoint lift may differ by a deck phase, which is precisely retained by the transported point \(y_t\).

Finally \(V_m\to M_m\) is smooth and surjective, with fiber the punctured line. Its composite with the finite étale torsor is smooth and surjective. The product description of \(\widetilde\mathcal D_{2,m,n}\) gives the displayed fiber product; it is connected and its projection is the torus projection. ◻

Proof. The tensor variation \(\mathbb V^{\otimes 2a}(a)\), with the standard tensor/Tate polarization rescaled by a positive integer if needed, is integral, polarized, and of weight zero. Denote its polarization by \(Q_a\). Multiplying a denominator makes a rational tensor integral. For an integral flat tensor \(u\) on the universal cover, let \(H_u\subset S\) be the set where some determination of \(u\) has type \((0,0)\). Cattani–Deligne–Kaplan, Corollary 1.3 on p. 2 of the preprint [4], says that \(H_u\) is algebraic. Its proof writes \(H_u\), for \(K=Q_a(u,u)\), as a union of images of connected components of the algebraic variety \(S^{(K)}\) finite over \(S\) in their Theorem 1.1. There are finitely many such components, and finite morphisms are closed, so \(H_u\) is closed.

If \(u\) is not Hodge throughout the universal cover, \(H_u\) is proper. On a simply connected analytic chart, its determinations form a countable set of integral tensors. Each has a closed analytic Hodge locus: for a real weight-zero tensor, type \((0,0)\) is equivalent to membership in the holomorphic subbundle \(F^0\). None contains an open subset, since analytic continuation on the connected universal cover and deck equivariance would then make \(u\) globally Hodge. If \(H_u=S\), the chart would be covered by countably many nowhere-dense closed analytic subsets, contrary to Baire’s theorem. Hence \(H_u\) is proper.

There are countably many integral tensors in all powers. Remove \(H_u\) for the ones which are not globally Hodge. A rational Hodge tensor at a remaining point, after clearing a denominator, must be globally Hodge on the universal cover, equivalently Hodge after parallel transport along every path from that point. ◻

Proof. Write \(\pi:\mathcal Y_0\to M_m\). Its integral polarized variation is \(\mathbb V=R^1\pi_*\mathbb Z\). Apply Lemma 40. For a remaining \(z\), choose an auxiliary point and one of its lifts, hence a point of \(B_{m,n}\), and fix the initial based identification. A rational tensor Hodge in this marked fiber corresponds to a rational tensor in the root fiber. By the lemma, it remains Hodge under transport along every path in \(M_m\). For any prescribed marked fiber, Lemma 38 supplies a path in \(V_m\) and its based lift whose endpoint filtration is that of the prescribed cover. Applying root-family globality to the projected path shows that the tensor is Hodge in that marked fiber. The fiber was arbitrary, so this marked parameter belongs to \(\mathcal T_\phi^{\mathrm{Hg}}\). Theorem 1 applies.

Surjectivity of \(\lambda\) makes inverse images of the proper algebraic exceptional subsets proper. At a complex point above \(z\), roots of its torus coordinates exist, and Proposition  36 identifies its dual curve with \(X_{Q_0(z)}\). The Hodge conclusion is invariant under this algebraic isomorphism. ◻