Complex-structure dependence of cyclotomic Hitchin loci

Definitions and mathematical statement

Let $S$ be a closed marked oriented surface of genus $g\geq2$, and fix a Hitchin component $\operatorname{Hit}_n$ of the $\operatorname{SL}(n,{\mathbb{R}})$ character variety, with the necessary lift/section data fixed. For a complex structure $J$, let $K_J$ be its canonical bundle. In Hitchin-section coordinates, \[ s_J:\bigoplus_{k=2}^nH^0(S,K_J^k)\longrightarrow\operatorname{Hit}_n, \] Higgs-field scaling by an $n$th root of unity acts on $q_k$ by multiplication by that root to the $k$th power. Fixedness means equality up to Higgs-bundle isomorphism, not literal equality of a nonzero field with its scalar multiple. The fixed Betti locus is \[ \operatorname{Cyc}_n(J)=s_J(0,\ldots,0,H^0(S,K_J^n)). \] Oscar Garcia-Prada asks whether this cyclotomic space is independent of the complex structure. The aid interprets independence as equality of subsets in the same marked Betti component, rather than noncanonical abstract isomorphism of equal-dimensional vector spaces.

It reports that for $n=2$ this locus is the whole Hitchin component, hence is independent of $J$. For every $n\geq3$, let $\operatorname{Fuch}_n$ denote the principal Fuchsian locus in the fixed component and $\rho_J$ the principal Fuchsian point uniformizing $J$, with the chosen lift convention. The aid's distinguishing identity is \[ \operatorname{Cyc}_n(J)\cap\operatorname{Fuch}_n=\{\rho_J\}. \] Consequently $\operatorname{Cyc}_n(J)=\operatorname{Cyc}_n(J')$ forces $J=J'$ in marked Teichmüller space. This establishes dependence for that interpretation; it does not say all different fibers are disjoint in every rank.

For $n=3$, the aid invokes the Labourie--Loftin correspondence to state more: the cubic-differential fibers over Teichmüller space form a pairwise-disjoint partition of $\operatorname{Hit}_3$. For $n\geq4$, consider instead the smooth universal cyclic map from the bundle with fibers $H^0(K_J^n)$ over Teichmüller space. Its real domain and target dimensions are \[ (4n+4)(g-1),\qquad2(n^2-1)(g-1), \] with positive gap \[ 2(n-3)(n+1)(g-1). \] The aid therefore asserts that its image has measure zero in smooth local coordinates and empty interior. This does not imply that the image is nondense or prove disjointness of different higher-rank fibers.

The family is mapping-class covariant: transporting $J$, representation marking and any lift/component data together transports $\operatorname{Cyc}_n(J)$ to the corresponding locus for the transported $J$. Covariance is not constancy of the marked subset. Exact Hitchin-section and nonabelian-Hodge normalization conventions require source review.

Short English statement

Under the marked-Betti interpretation, the supplied aid says the cyclotomic locus is independent of complex structure only in rank two. In higher rank its unique principal Fuchsian point recovers the complex structure; rank three has a partition theorem, while higher ranks have a dimension obstruction to filling the component.

Sources

• Local UnsolvedMath registry, permanent ID $20003190$, source code catalogue entry; canonical aim-topology-notes.json, notes[277].
• 2007 AIM workshop Representations of Surface Groups, Question 5.1, attributed to Oscar Garcia-Prada: https://aimath.org/WWN/surfacegroups/surfacegroups.pdf. The supplied verification reports page 3 and restoration only of line-break hyphenation and superscripts. Original remarks and literature field are empty.
• The aid invokes Hitchin-section coordinates and the Labourie--Loftin correspondence without authoritative complete bibliographic entries.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor proved, novelty confidence low. Its Fuchsian-intersection and dimension-gap claims are machine-generated aid, not peer reviewed, and require independent verification before citation.