\documentclass[11pt]{amsart} \usepackage{amsmath,amssymb,amsthm} \usepackage{url} \setlength{\emergencystretch}{3em} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\Q}{\mathbb Q} \newcommand{\Z}{\mathbb Z} \newcommand{\C}{\mathbb C} \newcommand{\R}{\mathbb R} \newcommand{\PP}{\mathbb P} \newcommand{\End}{\operatorname{End}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Nm}{\operatorname{Nm}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\age}{\operatorname{age}} \newcommand{\fD}{\mathfrak D} \newcommand{\fd}{\mathfrak d} \newcommand{\cH}{\mathcal H} \newcommand{\cM}{\mathcal M} \title[Weil classes on sixfolds of Weil type for $\Q(\sqrt{-3})$]{Weil classes on every abelian sixfold of Weil type for $\Q(\sqrt{-3})$} \author{Ben Severn} \email{benzsevern@gmail.com} \subjclass[2020]{Primary 14C30; Secondary 14K20, 14D07, 14H30} \keywords{Hodge conjecture, Weil classes, abelian varieties of Weil type, complex reflection groups, nilpotent orbits, Brauer induction} \date{Draft v0.5, 16 October 2026} \begin{document} \begin{abstract} We prove that the Weil classes of every polarized complex abelian sixfold of Weil type for $K=\Q(\sqrt{-3})$, of every Hermitian discriminant, are algebraic. The same holds for every abelian eightfold of Weil type for $K$ of signature $(4,4)$ with split discriminant, and every power of such a sixfold whose special Mumford--Tate group is $SU_H$ satisfies the Hodge conjecture. The proof applies the transfer theorem of the companion paper \cite{SevernTransfer} to the Shephard--Todd group $G_{34}$ with twenty reflections as local monodromies, and passes from eightfolds to sixfolds by a compensation argument. Two general tools are proved on the way. If every local monodromy of a transfer datum is conjugate to its inverse, the lattice condition of the transfer theorem follows from Brauer's induction theorem. Dominance of the period map follows from exact data at a maximal degeneration, by the nilpotent orbit theorem. We also give an explicit higher-dimensional form of Schoen's compensation argument, which shows that one decided cell of $2n$-folds, $n\ge4$, decides every cell of sixfolds over the same imaginary quadratic field. No outside unrefereed manuscript is used. \end{abstract} \maketitle \section{Introduction} Let $A$ be a complex abelian variety of dimension $2n$ on which an imaginary quadratic field $K$ acts with both embeddings occurring $n$ times on the tangent space. The two-dimensional space $W_K(A)=\bigwedge^{2n}_KH^1(A,\Q)$ consists of Hodge classes, the Weil classes \cite{Weil}, \cite[\S5]{vanGeemen}. A compatible polarization gives a $K$-Hermitian form of signature $(n,n)$ whose discriminant in $\Q^\times/\Nm K^\times$ is an invariant of the rational polarized data. For $n=3$ the Weil classes are known to be algebraic for the split discriminant \cite{Schoen98, Markman}, and for $K=\Q(\sqrt{-3})$ and the class $-2$ \cite{SevernG32}. For $K=\Q(i)$ the split case was proved earlier by Koike, who showed that the Prym map from fourth cyclic \'etale covers of curves of genus four dominates the nine-dimensional Shimura variety of the family (discriminant $1$ in the normalization of that paper) \cite{Koike}. Outside discriminant $-1$, the algebraicity of Weil classes on sixfolds is described as open in \cite[Introduction]{Mostaed}. \begin{theorem}[Theorem~\ref{thm:qsqrt3}]\label{thm:main} Let $K=\Q(\sqrt{-3})$. \begin{enumerate} \item The Weil classes of every polarized abelian eightfold of Weil type for $K$ with signature $(4,4)$ and discriminant $[1]$ are algebraic. \item The Weil classes of every polarized abelian sixfold of Weil type for $K$, of signature $(3,3)$ and any discriminant, are algebraic. \item Every power of a sixfold $A$ as in (2) with $\operatorname{Hg}(A)=SU_H$ satisfies the Hodge conjecture. \end{enumerate} \end{theorem} The proof has four steps. The transfer theorem of \cite{SevernTransfer} (recalled as Theorem~\ref{thm:transfer}) decides a whole Weil cell from one family of Prym--Tyurin factors of Galois covers of $\PP^1$, given a lattice condition (L) on suppliers of Schoen cycles and dominance (D) of the period map. We take $G=G_{34}$ in its six-dimensional reflection representation and twenty reflections as local monodromies. This is a datum of type $(K,4)$ in the split cell. \begin{itemize} \item (L) holds by Theorem~\ref{thm:brauer}: since every local monodromy is an involution, Brauer's induction theorem writes the determinant character as an integer combination of admissible Schoen suppliers. An explicit witness is also given. \item (D) holds by Proposition~\ref{prop:boundary}: at a maximal degeneration of the Hurwitz space, the logarithms of the boundary monodromies span the unipotent radical of the stabilizer of a rational Lagrangian, which by the nilpotent orbit theorem forces full rank. All hypotheses are exact linear algebra over $\Q(\sqrt{-3})$, computed from the monodromy tuple. \item The transfer theorem then decides the split eightfold cell, for every member. \item Proposition~\ref{prop:compensation} passes to every sixfold cell, through products $A\times E\times\bar E$ with $E$ an elliptic curve with complex multiplication by $\mathcal O_K$; this is a higher-dimensional form of Schoen's compensation argument \cite[Proposition~10]{Schoen98}. \end{itemize} Theorem~\ref{thm:main}(2) contains the split sixfolds over $K$ \cite{Markman} and the cell $[-2]$ of \cite{SevernG32}; it is new for every other discriminant. The proof uses Schoen's theorem, Milne's theorem on Lefschetz classes, Brauer's induction theorem, the nilpotent orbit theorem and the Clemens--Schmid sequence. It does not use the unrefereed manuscripts that the companion paper uses conditionally. It does depend on the companion transfer paper \cite{SevernTransfer}, and through it on the cocycle formula of \cite{SevernG32}; the three papers are meant to be read and refereed together, and Section~\ref{sec:recall} states exactly what is imported. Section~\ref{sec:typeIII} draws a consequence for sixfolds of type III: the Hodge conjecture on all powers of the \emph{general member} of every effective PEL family whose rational quaternion algebra contains $\Q(\sqrt{-3})$. Unlike Theorem~\ref{thm:main}, which holds for every member, this consequence is limited to members whose Hodge group is the generic one of their family. It says nothing about powers of members with smaller Hodge group, about eightfold powers, or about other fields; Section~\ref{sec:scope} describes what the method needs for other fields. \section{Recollections}\label{sec:recall} We recall what we use from \cite{SevernTransfer}, with the same conventions. Throughout, $K$ is an imaginary quadratic field and $\vartheta\in K$ is nonzero with $\bar\vartheta=-\vartheta$. \subsection{Transfer data}\label{sec:supply} Fix a finite group $G$, conjugacy classes $\mathcal C_1,\dots,\mathcal C_r$, and the Hurwitz space $\cH$ of connected Galois $G$-covers $C\to\PP^1$ branched over $r$ ordered points with local monodromy in $\mathcal C_j$; its components are smooth and irreducible of dimension $r-3$ modulo $PGL_2$. By Chevalley--Weil, $H^1(C,\C)=\bigoplus_\tau\tau\otimes M_\tau$; put $m_\tau=\dim M_\tau$, and $c_\tau(g)=\rk(\tau(g)-1)$. \begin{proposition}[{\cite[Defect identity]{SevernTransfer}}]\label{prop:defect} If $\tau$ is irreducible and nontrivial, then $m_\tau=-2d+\sum_jc_\tau(g_j)$, $d=\dim\tau$; the Hodge numbers of $M_\tau$ are $\sum_j\age\tau(g_j^{\mp1})-d$. \end{proposition} \begin{definition}[{\cite[Definition of a Weil transfer datum]{SevernTransfer}}]\label{def:datum} A Weil transfer datum of type $(K,n)$ is $\fD=(G,\sigma;\mathcal C_1,\dots,\mathcal C_r)$ with $\sigma$ irreducible of dimension $d$, character field $K$ and Schur index one over $K$, a nonempty Nielsen class, $m_\sigma=2n$ and Hodge numbers $(n,n)$ on $M_\sigma$. \end{definition} For $C\in\cH$ the $\sigma$-isotypic factor $A_\sigma(C)\subseteq J(C)$ is a polarized abelian $2n$-fold of Weil type for $K$, with Hermitian form $H$ of signature $(n,n)$ \cite[\S2]{SevernTransfer}. On a component $S$ the discriminant class $\fd(S)$ is constant. A \emph{supplier} is a pair $(H,\chi)$ of a subgroup $H\le G$ and a linear character $\chi$ of $H$ of order $m\ge3$; $C'=C/\ker\chi\to C/H$ is a cyclic cover of a curve of genus $q$, with $s$ branch points and local exponents $\alpha$, and the primitive part of $J(C')$ has $E_\chi$-dimension $h=2q-2+s=\sum_\tau m_\tau\langle\tau|_H,\chi\rangle$. \begin{definition}[{\cite[admissible suppliers]{SevernTransfer}}]\label{def:admissible} A supplier is admissible if $h$ is even and positive, and $\alpha$ is simple in Schoen's sense (after a permutation, $\alpha_{2i}=-\alpha_{2i-1}$ for all $i$); for $s=0$ simplicity is automatic. \end{definition} \begin{lemma}[Schoen; {\cite[Theorem~2.0 and Corollary~3.1]{Schoen88}}]\label{lem:schoen} If $(H,\chi)$ is admissible, the Weil classes of the primitive part of $J(C/\ker\chi)$ are algebraic, at every $C\in\cH$. Exponents may share factors with $m$. \end{lemma} Let $U$ be the centralizer of $G$ in $\operatorname{Sp}(H^1(C,\Q))$ and $T$ its maximal abelian quotient. Then $X^*(T)=\Z^{\rm pairs}\oplus(\Z/2)^{\rm quat}$: one free coordinate for each pair $\{\tau,\bar\tau\}$ of non-self-dual $\tau$ with $m_\tau>0$, oriented by a chosen member, and one parity coordinate for each quaternionic $\tau$ with $m_\tau>0$. A supplier gives $a(\chi)=(\langle\tau|_H,\chi\rangle-\langle\bar\tau|_H,\chi\rangle)_{\rm pairs}\oplus(\langle\tau|_H,\chi\rangle\bmod2)_{\rm quat}$, and $e_\sigma$ is the coordinate $+1$ at the pair of $\sigma$. Condition (L) is that $e_\sigma$ lies in the group $L(\fD)$ generated by the $a(\chi)$ of admissible suppliers. \subsection{Dominance and the discriminant}\label{sec:disc} Let $\mathsf A=\Hom_G(\sigma,H^0(C,K_C))$ and $\mathsf B=\Hom_G(\bar\sigma,H^0(C,K_C))$, each of dimension $n$, and $\mu_C\colon\mathsf A\otimes\mathsf B\to H^0(\PP^1,K_{\PP^1}^{\otimes2}(B_C))$ the multiplication followed by contraction. \begin{lemma}[{\cite[Codifferential]{SevernTransfer}}]\label{lem:codiff} $\mu_C$ is, up to a nonzero normalization, the codifferential at $C$ of the period map of the family $A_\sigma$ into the $n^2$-dimensional domain of $SU(n,n)$. \end{lemma} \begin{proposition}[{\cite[Rank at one point]{SevernTransfer}}]\label{prop:dominance} If $\rk\mu_{C_1}=n^2$ at one point $C_1$ of a component $S$, the period map of $S$ has generic rank $n^2$. \end{proposition} Let $L'=\Hom_G(V_\sigma,\pi_*\Q|_{U_C}\otimes K)$, a local system of $K$-spaces of rank $d$ on $U_C=\PP^1-B_C$ with a flat positive definite Hermitian form, and $V=H^1_{\rm par}(U_C,L')$, of $K$-dimension $2n$, with the skew-Hermitian parabolic cup product $\mathcal S$ and the Hermitian form $h=\vartheta\mathcal S$. \begin{lemma}[Cocycle formula; {\cite[Cocycle formula]{SevernTransfer}, \cite[Lemma~6.2]{SevernG32}}]\label{lem:cocycle} With unitary monodromy $T_1,\dots,T_r$ along clockwise loops, $Q_i=T_1\cdots T_{i-1}$ and a parabolic cocycle given by $a_i$ with $c_i=(1-T_i)a_i$ and $F_i=\sum_{j0$, $m_\eta>0$, the coordinates of $X^*(T)$ (Subsection~\ref{sec:supply}). It is additive. By Frobenius reciprocity, $A(\operatorname{Ind}_H^G\chi)=a(\chi)$ for every subgroup $H$ and linear character $\chi$ of $H$, of any order, where the right side is given by the formula for $a_1(\chi)$. Since $\chi_\sigma$ is not real, $\sigma\not\cong\bar\sigma$, and $A(\sigma)=e_\sigma$. \emph{Step 2: Brauer induction.} By Brauer's induction theorem \cite[\S10.5, Theorem~20]{SerreLin}, \cite[(15.9)]{CurtisReiner}, $\sigma=\sum_in_i\operatorname{Ind}_{E_i}^G\lambda_i$ in $R_\C(G)$, with $n_i\in\Z$, $E_i\le G$ elementary and $\lambda_i$ linear. Hence $e_\sigma=\sum_in_ia(\lambda_i)$, an identity in $X^*(T)$ including the parity coordinates. \emph{Step 3: terms that vanish.} If $\lambda_i$ has order $1$ or $2$ it is real, so $\langle\tau|_{E_i},\lambda_i\rangle=\langle\bar\tau|_{E_i},\lambda_i\rangle$ and the free coordinates of $a(\lambda_i)$ vanish. For quaternionic $\eta$ the invariant alternating form on $\eta|_{E_i}$ is nondegenerate on the $\lambda_i$-isotypic part, because $\lambda_i=\lambda_i^{-1}$, and an alternating nondegenerate form on $\lambda_i\otimes\C^k$ with $\lambda_i$ one-dimensional forces $k$ even; so the parity coordinates vanish too. If $\lambda_i$ has order $m_i\ge3$, then $(E_i,\lambda_i)$ is a supplier with $h_i=\sum_\tau m_\tau\langle\tau|_{E_i},\lambda_i\rangle\ge0$; if $h_i=0$, every $\tau$ with $m_\tau>0$ has $\langle\tau|_{E_i},\lambda_i\rangle=0$, and $a(\lambda_i)=0$. Dropping these terms, $e_\sigma$ is an integer combination of the $a(\lambda_i)$ with $m_i\ge3$ and $h_i>0$. \emph{Step 4: the remaining suppliers are admissible.} Let $(H,\chi)$ be a supplier with $h>0$, and let $w\in G$ with $wg_jw^{-1}=g_j^{-1}$. Put $D=\langle g_j\rangle$. Since $w$ normalizes $D$, $HxD\mapsto HxwD$ is a well-defined bijection of $H\backslash G/D$, with inverse given by $w^{-1}$, and $(xw)g_j^k(xw)^{-1}=xg_j^{-k}x^{-1}$; so it preserves the least $\ell$ with $xg_j^\ell x^{-1}\in H$ and inverts the local monodromy. Hence the multiset of local exponents of $C'\to C/H$ over the $j$-th branch point is stable under $\alpha\mapsto-\alpha$. Pair each exponent with $\alpha\ne-\alpha$ with an occurrence of $-\alpha$. The only unpaired exponents are $\alpha=m/2$, for $m$ even. Their number is even: in the cyclic quotient of the orbifold fundamental group of $C/H$ the handle commutators vanish, so the exponents sum to $0$ modulo $m$, the paired ones contribute $0$, and $k\cdot m/2\equiv0\pmod m$ forces $k$ even. Hence $\alpha$ is simple, $s$ is even, and $h=2q-2+s$ is even. So $(H,\chi)$ is admissible (Definition~\ref{def:admissible}), and $e_\sigma\in L(\fD)$. \end{proof} \begin{remark} No coprimality between the exponents and $m$ is required, and unramified suppliers ($s=0$) with $h>0$ are admissible (Definition~\ref{def:admissible}, Lemma~\ref{lem:schoen}). The data of \cite{SevernG32} and the $(\Q(i),[-3])$ datum in $W(E_8)\cdot\mu_4$ of \cite{SevernTransfer} are not real: the reflections of $G_{32}$ have determinants $\omega^{\pm1}$, and the order-twelve local monodromies of the latter have character values $\pm i$. For $G_{32}$, (L) needed an explicit witness; for the $W(E_8)\cdot\mu_4$ datum it fails \cite{SevernTransfer}. The proof is an existence argument; Section~\ref{sec:g34} also records an explicit witness for its datum. \end{remark} \section{Rank at a maximal degeneration}\label{sec:boundary} A second way to certify (D) works at the boundary of the Hurwitz space and needs only the monodromy tuple. Dominance by degeneration has a precedent in \cite{ADFIO}, which proves that the Prym--Tyurin map from the Hurwitz space of $W(E_6)$-covers of $\PP^1$ branched over $24$ points to $\mathcal A_6$ is dominant by degenerating the branch points to a maximally degenerate point of $\overline{\mathcal M}_{0,24}$ and computing the rank of a $21\times21$ matrix there. Here the target is the domain of $SU(n,n)$ of a Weil family rather than $\mathcal A_g$, and the rank condition is an exact certificate over $K$. Fix $\mathbf g\in S$, let $V=H^1_{\rm par}(U_C,L')$ with the skew-Hermitian cup form $\mathcal S$ and $h=\vartheta\mathcal S$ of signature $(n,n)$ (Subsection~\ref{sec:disc}). Let $\mathcal P=\{\gamma_1,\dots,\gamma_{r-3}\}$ be a pants decomposition of the $r$-punctured sphere by simple closed curves. Three operators are attached to $\gamma_j$, in this order. \begin{enumerate} \item[(i)] The full Dehn twist $\tau_j$ along $\gamma_j$ is a braid. With the Artin convention, if $\gamma_j$ encloses the points $a,\dots,b$ and $c=g_a\cdots g_b$, it acts on tuples by $g_i\mapsto cg_ic^{-1}$ for $a\le i\le b$, and is $(\sigma_a\cdots\sigma_{b-1})^{b-a+1}$. \item[(ii)] The \emph{return power} $k_j\ge1$ is the least power with $\tau_j^{k_j}(\mathbf g)=\mathbf g$. Thus $\tau_j^{k_j}$ is the least power of the twist whose lift to the Hurwitz space, starting at the cover of $\mathbf g$, is closed; it acts on $V$ by a unitary $T_j$, computed by pulling back cocycles: for $u(xy)=u(x)+\rho(x)u(y)$ the pullback of $u$ under one twist is $u'_i=u(c)+P\,u_i-(Pg_iP^{-1})\,u(c)$, with $P$ the corresponding conjugator, and $T_j$ is its $k_j$-th iterate. \item[(iii)] Let $m_j$ be the order of the semisimple part of $T_j$. Then $T_j^{m_j}$ is unipotent, and $N_j=\log T_j^{m_j}$. \end{enumerate} If the boundary product $c$ is trivial in $G$, the twist fixes the tuple ($k_j=1$) but still acts nontrivially on $V$, as a Picard--Lefschetz transvection; it is not the identity on cohomology. The local monodromy of the period map around the boundary divisor $q_j=0$ is the loop of (ii), and its unipotent part has logarithm $N_j$ up to a positive rational factor. That the braid pullback in (ii) is the geometric Gauss--Manin monodromy, up to a root of unity, is Lemma~\ref{lem:gm}; the rest is Step~1 of the proof below. \begin{lemma}[Geometric monodromy]\label{lem:gm} Let $\sigma$ be absolutely irreducible, realized on $V_\sigma$ over $K$, and let $\mathbf g$ be a product-one tuple generating $G$. Let $a\colon\mathcal C\to B$ be an actual family of $G$-covers $p\colon\mathcal C\to\PP^1_B$ over a connected base $B$, \'etale over the interior of the Hurwitz space (for instance a rigidified chart, or the space of covers with a marked point over a tangential base point $\infty$), with the cover of $\mathbf g$ over a base point. Let $L'$ be the coefficient system of Section~\ref{sec:disc}, the $\sigma$-multiplicity of $p^\circ_*K$ on the complement of the branch locus. Then: \begin{enumerate} \item[(1)] $\Hom_{K[G]}(V_\sigma,R^1a_*K)\cong R^1\bar\pi_*(j_*L')$ naturally, as local systems of $K$-spaces on $B$; \item[(2)] the monodromy of this local system along a loop $\beta$ of $B$ is the braid pullback of parabolic cocycles by the braid of $\beta$, composed with transport of the coefficients by $\sigma(h_\beta)$, where $h_\beta\in G$ is the monodromy of the $G$-torsor along the section over $\infty$, and $\beta(\mathbf g)=h_\beta\mathbf gh_\beta^{-1}$; \item[(3)] if $\beta(\mathbf g)=\mathbf g$, then $h_\beta\in Z(G)$ and $\sigma(h_\beta)$ is a scalar root of unity of order dividing $|Z(G)|$; on the pointed family, $h_\beta=1$. \end{enumerate} \end{lemma} \begin{proof} (1) Near a branch point a normal tame cover has the model $z\mapsto z^e$, and locally constant functions upstairs are the inertia-invariant functions on the punctured preimage; so $p_*K=j_*p^\circ_*K$. The finite proper map $p$ has no higher direct images of the constant sheaf, so the Leray spectral sequence gives $R^1a_*K=R^1\bar\pi_*(p_*K)$. Taking the $\sigma$-multiplicity is exact in characteristic zero, by the averaging projector, and commutes with these direct images and with base change; this is the relative form of the identification used fibrewise in Section~\ref{sec:disc}. (2) This is \cite[Lemma~2.2 and Theorem~2.5]{DW1}, restated in \cite[Theorem~1.3]{DW2}, in their right-action convention: the monodromy is the product of the braid term and the coefficient term, where the coefficient element satisfies the displayed conjugation relation. On an actual family of $G$-covers the coefficient system comes from the finite $G$-torsor, so the coefficient element is the image of $h_\beta\in G$; this finiteness comes from the family, not from irreducibility alone. (3) If $\beta(\mathbf g)=\mathbf g$, then $h_\beta$ centralizes a generating tuple, so $h_\beta\in Z(G)$, and by Schur's lemma $\sigma(h_\beta)\in\End_{K[G]}(V_\sigma)=K$ is a root of unity of order dividing $|Z(G)|$. On the pointed family the marked point over $\infty$ returns to itself along a closed loop, so $h_\beta=1$. The pointed family and any rigidified chart are finite and \'etale over the interior of the Hurwitz component, so they have the same generic rank of the period map (Step~0 below). \end{proof} For $G_{34}$, $Z(G)=\mu_6$, and every $k_j$ of Section~\ref{sec:g34} is an exact return (checked independently; Appendix~\ref{app:g34}). So on the pointed family the geometric monodromy of $\tau_j^{k_j}$ is $T_j$, and on an unpointed realization it is $\zeta_jT_j$ with $\zeta_j^6=1$. In both cases, after raising to the power $6m_j$, the unipotent logarithm is a positive integer multiple of $N_j$. With one fixed orientation convention all these multiples carry the same sign, so (H1)--(H4) for the $N_j$ are (H1)--(H4) for the actual variation. For a pair of pants $\Pi$ of $\mathcal P$ let $P_1,P_2,P_3$ be the boundary products and $G_\Pi=\langle P_1,P_2,P_3\rangle$. \begin{proposition}[Rank at a maximal degeneration]\label{prop:boundary} Assume: \begin{enumerate} \item[(H0)] for every pair of pants $\Pi$, $c_\sigma(P_1)+c_\sigma(P_2)+c_\sigma(P_3)-2d+2\dim\sigma^{G_\Pi}=0$; \item[(H1)] $N_jN_k=0$ for all $j,k$; \item[(H2)] $I=\sum_j\operatorname{im}N_j$ is a $K$-subspace of dimension $n$, totally isotropic for $h$, and $N_j(I)=0$ for every $j$; \item[(H3)] the forms $\beta_j(x,y)=\mathcal S(x,N_jy)$, which are Hermitian and descend to $V/I$, are semidefinite of one common sign, and $\sum_j\beta_j$ is definite on $V/I$; \item[(H4)] the $N_j$ span a $\Q$-vector space of dimension $n^2$. \end{enumerate} Then the period map of $S$ has generic rank $n^2$, and (D) holds for $S$. \end{proposition} \begin{proof} \emph{Step 0 (ordering).} Let $\Sigma\le S_r$ be the group of permutations of the branch points preserving the class datum, that is with $\mathcal C_{\pi(j)}=\mathcal C_j$. It acts on the ordered Hurwitz space $\cH$, and the quotient $\cH/\Sigma$ is the Hurwitz space of covers whose branch points are unordered within each class. A braid whose permutation lies in $\Sigma$ carries $\mathbf g$ to a tuple $\mathbf g'$ in the same Nielsen class. Its ordered component $S'$ may differ from $S$, but $S$ and $S'$ map finitely and surjectively onto the same component $\bar S$ of $\cH/\Sigma$. The period map of the family $A_\sigma$ depends on the cover and the fixed representation, not on the labels of the branch points, so it descends to $\bar S$. After a level structure these maps are \'etale over the interior, and an \'etale finite map preserves generic rank. So the generic rank on $S$ equals that on $\bar S$ and on $S'$, and $\mathbf g$ may be replaced by $\mathbf g'$. For the datum of Section~\ref{sec:g34} all twenty points lie in one class, so $\Sigma=S_{20}$. \emph{Step 1 (the boundary point).} Pinching the curves of $\mathcal P$ gives a stable $r$-pointed rational curve with $r-2$ components and $r-3$ nodes, a point of the normal-crossing boundary of $\overline{\mathcal M}_{0,r}$ \cite[Theorem~2.7]{Knudsen}, with independent smoothing parameters $q_j$. The restrictions of the $G$-cover to the pairs of pants give covers of the components. At a node the two inertia elements are inverse of the same order $e$, and the local model $u=x^e$, $v=y^e$, $xy=t$, $uv=t^e$ \cite[\S4]{HarrisMumford} glues them, after sufficiently divisible powers of the $q_j$. The local Hurwitz cover over the punctured polydisc is a finite cover of the configuration space \cite[\S1.2]{FriedVolklein}. The loop of $q_j$ once around $0$ is the full Dehn twist $\tau_j$ (plumbing: one turn of $\arg q_j$ rotates the gluing annulus once relative to its boundary). The stabilizer of a chosen sheet of the local Hurwitz covering, under local monodromy over the punctured polydisc, is a finite-index subgroup $\Lambda\subseteq\Z^{r-3}$, which need not be a product of coordinate subgroups, but it contains $\bigoplus_jk_j\Z$ for the return powers $k_j$ of (ii). So the Kummer cover $w_j^{|Z(G)|k_jm_j}=q_j$, on an unpointed realization (or $w_j^{k_jm_j}=q_j$ on the pointed family), dominates the connected local Hurwitz cover, by maps \'etale over the interior; further divisible powers may be taken if a family chart needs them. By Lemma~\ref{lem:gm}, on it the local monodromy around $w_j=0$ is unipotent, with logarithm a positive integer multiple of $N_j$. On any further base change by positive powers the logarithms are positive integer multiples $c_jN_j$. With the orientation conventions fixed once (the clockwise loops of Lemma~\ref{lem:cocycle} and \cite[p.~463]{CKS}), all the logarithms carry the same sign $\varepsilon\in\{\pm1\}$. A consistent reversal of conventions changes $\varepsilon$ for every $j$ at once, and cannot be made curve by curve. For the datum of Section~\ref{sec:g34}, three of the seventeen curves have $k_j=2$, and every $m_j=1$ (Appendix~\ref{app:g34}). \emph{Step 2 (nilpotent orbit).} By the several-variable nilpotent orbit theorem \cite[(4.12)]{Schmid}, \cite[(1.15)]{CKS}, after the finite base change the lifted period map has the form $\tilde\Phi(z)=\exp\big(\sum_jz_jM_j\big)\cdot\psi(u)$, $M_j=\varepsilon c_jN_j$, $u_j=e^{2\pi iz_j}$, with $\psi$ holomorphic on the full polydisc, valued in the compact dual. Put $F_0=\psi(0)$. The real polarization of the underlying weight-one variation is, up to a positive constant and one global sign, $\Tr_{K/\Q}\mathcal S$; the $K$-action commutes with the monodromy and preserves the Hodge filtration. \emph{Step 3 (the limit is Hodge--Tate).} We give the deduction directly. Let $M=\sum_ja_jM_j$ with all $a_j>0$, an element of the open monodromy cone. By (H1), $M^2=0$. By (H2), $\operatorname{im}M\subseteq I$ and $M(I)=0$. By (H3), for $x\notin I$ we have $\mathcal S(x,\varepsilon Mx)=\sum_ja_jc_j\beta_j(x,x)\ne0$, because the $\beta_j$ are semidefinite of one sign with definite sum. So $V/I\to I$, $x\mapsto Mx$, is injective, hence an isomorphism since both sides have $K$-dimension $n$. Thus $\operatorname{im}M=\ker M=I$, and the monodromy weight filtration of $M$, centred at $1$, is $W_0=\operatorname{im}M=I$, $W_1=\ker M=I$, $W_2=V$. In particular $\operatorname{Gr}^W_1=0$. This filtration does not depend on the $a_j$; the cone theorem \cite[(4.4)]{CKS}, \cite[Theorem~(3.3)]{CK82} says the same in general, and here it is also visible directly. The role of the polarization is \cite[(4.5)]{CKS}: $(W,F_0)$ is a mixed Hodge structure polarized by $M$, with $F_0=\psi(0)$ the limiting filtration of Step~2. Since the variation has weight one, its Hodge numbers lie in $\{0,1\}$. So $\operatorname{Gr}^W_0$ has type $(0,0)$ and $\operatorname{Gr}^W_2$ has type $(1,1)$: the limit is Hodge--Tate. The sign convention matters only through (H3), where the Hermitian form is $\mathcal S(x,N_jy)$ and not $h(x,N_jy)$, which is skew-Hermitian (Remark~\ref{rem:boundary}). (H0) gives the same conclusion geometrically: by the Clemens--Schmid sequence \cite[(3.3)--(3.7), Theorem~9.12, \S11]{Clemens}, restricted to an arc $u_j=t^{a_j}$ with $a_j>0$, the weight-one part of the limit is $\bigoplus_\Pi H^1$ of the normalized components, whose $\sigma$-multiplicity is $\bigoplus_\Pi H^1_{\rm par}$ of the three-point local systems, of dimension the left side of (H0). \emph{Step 4 (the orbit has full rank).} Fix an embedding $\iota\colon K\to\C$ and write $V_\iota=V\otimes_{K,\iota}\C$, of dimension $2n$, and $I_\iota$. Deligne's bigrading \cite[(2.13)]{CKS} gives $V_\iota=I_\iota\oplus F^1_0$ with $\dim F^1_0=n$. The rational Lie algebra $\mathfrak n_I=\{X\in\mathfrak u(h):X(V)\subseteq I,\ X(I)=0\}$ has dimension $n^2$ over $\Q$; via $X\mapsto\mathcal SX$ it is the space of Hermitian forms on $V/I$. Its complexification maps isomorphically onto $\Hom_\C(V_\iota/I_\iota,I_\iota)$: the diagonal matrix units, and for $a