Moduli Spaces of Higgs Harder–Narasimhan Strata

Definitions and mathematical statement

The supplied record does not specify the underlying variety, ground field, numerical invariants, ambient moduli stack, or refinement of the strata. Consequently, it does not determine a single fully specified moduli problem. The following definitions describe a customary curve setting only; they are not additional hypotheses attributed to the source.

If $X$ is a smooth projective connected curve over an algebraically closed field and $L$ is a line bundle on $X$, an $L$-twisted Higgs bundle is a pair $(E,\phi)$, where $E$ is a vector bundle and $\phi:E\to E\otimes L$ is a morphism. The usual Higgs-bundle setting takes $L=K_X$, the canonical bundle. For a nonzero vector bundle, its slope is $\mu(E)=\deg(E)/\operatorname{rk}(E)$. A Higgs bundle is slope semistable if $\mu(F)\leq\mu(E)$ for every nonzero proper $\phi$-invariant subbundle $F\subset E$. Ordinary slope semistability imposes the corresponding condition on all nonzero proper subbundles of $E$.

A Higgs Harder--Narasimhan filtration is a filtration by $\phi$-invariant subbundles whose induced Higgs quotients are semistable with strictly decreasing slopes. An ordinary Harder--Narasimhan filtration uses semistability of vector bundles without the invariance restriction. Its type records the ranks and degrees of the successive quotients. In a specified ambient moduli problem, a Harder--Narasimhan stratum consists of objects of a fixed such type. For Higgs bundles, stratifying by the ordinary Harder--Narasimhan type of the underlying bundle is distinct from stratifying a moduli stack of ordinary vector bundles; the record does not identify which interpretation is intended.

In the usual stack-theoretic sense, a coarse moduli space for a stack $\mathcal S$ is a morphism $q:\mathcal S\to M$ to an algebraic space that is universal for morphisms from $\mathcal S$ to algebraic spaces and induces a bijection between isomorphism classes over each algebraically closed extension field and the corresponding geometric points of $M$. Whether the source uses this sense or a quotient identifying a broader equivalence relation must be clarified.

Source question, with unresolved parameters retained. For the Higgs Harder--Narasimhan strata and ordinary Harder--Narasimhan strata intended by the source, can coarse moduli spaces be constructed? For the intended refined strata, can one construct comparably rich geometric structures, including compatible group actions and Hitchin fibrations?

In the customary rank-$r$, $L$-twisted curve setting, the characteristic coefficients define the Hitchin map \[ h(E,\phi)=(a_1(\phi),\ldots,a_r(\phi)) \in\bigoplus_{i=1}^{r}H^0(X,L^{\otimes i}). \] This describes the usual meaning of a Hitchin map, but the record does not specify its intended target on a refined stratum, descent to a coarse moduli space, or the properties required to call it a fibration. It likewise leaves unspecified the acting groups, compatibility conditions, refinement procedure, and the comparison expressed by "comparably rich." These omissions prevent a unique precise existence assertion.

Short English statement

The question asks for coarse moduli spaces of Higgs and ordinary Harder--Narasimhan strata, and for compatible group actions and Hitchin fibrations on refined strata. The ambient moduli problem and the requested refinements and compatibility properties are unspecified. The supplied dated curation labels the problem partially solved through results on refined strata, while retaining the question for unrefined strata in the stated generality.

Sources

• Supplied local problem registry: permanent UnsolvedMath ID $30004146$, title "Moduli Spaces of Higgs Harder--Narasimhan Strata," exact source problem code catalogue entry. The record gives proposed year $2019$, membership in the Oberwolfach Reports Open Problems collection, and no named proposer.
• Supplied report citation: Geometry and Physics of Higgs Bundles (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/23, pp. 1357--1417. https://doi.org/10.4171/owr/2019/23. The record describes these reports as documenting mathematical workshops and their open-problem discussions; no precise problem page or speaker attribution is supplied.
• Supplied literature source, labelled "Hamilton": https://arxiv.org/abs/1911.13194.
• Supplied literature source, labelled "Oxford thesis record": https://ora.ox.ac.uk/objects/uuid:6267baf5-3fde-405a-abc7-7618b0439e98.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially_solved and states that non-reductive GIT constructs refined Harder--Narasimhan and Hesselink--Harder--Narasimhan stratifications with quasi-projective coarse moduli spaces and projective completions, without fully settling the question for unrefined strata in the stated generality. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that its mathematical content was preserved while notation, typography, duplicate clauses, or omitted definitions were repaired. This supplied editorial label does not establish mathematical correctness or resolve the formulation gaps above.