How do Simpson strata closure relations behave in Hodge moduli spaces?

What Simpson's nestedness condition actually asks

Simpson's scaling action on the Hodge family sends an object to a limiting system of Hodge bundles, and the connected components of the scaling fixed locus partition the Dolbeault and de Rham moduli spaces into strata. Simpson conjectured that the closure of each stratum is a union of whole strata and that the two closure orders agree. The first assertion is the nestedness condition: if two strata intersect, then the closure of one contains the other.

The paper separates two questions that are easy to conflate. Knowing which limiting types can occur under specialization does not determine which points of a permitted stratum lie in the closure. The authors therefore treat the numerical order as a necessary condition on closure relations, not as a description of the closure itself.

The Heinloth weight strictly increases across components

For a degree-zero system of Hodge bundles E = ⊕ Ep with arrows lowering the index by one, the Heinloth weight is δ(E) = Σ p·deg(Ep), using the paper's normalization. The paper proves that this weight is constant on each connected component Pα of the full scaling fixed locus, with value δα, and that for every rank n≥1 the closure of a stratum S⋆α is contained in S⋆α together with the union of strata S⋆β whose weight satisfies δβ > δα, for ⋆ in {Hod, Dol, dR}.

This gives a partial order on fixed component labels: β ⪯δ α when β = α or δβ > δα. The closure bound refines the numerical bound from upper semicontinuity by excluding specialization between distinct components of the same weight. The authors emphasize that numerical comparability alone does not determine which permitted strata are met, nor whether an intersection contains the whole target stratum.

Why the weight is a maximum over Griffiths filtrations

The proof characterizes the Heinloth weight as the maximum of the weighted degrees of Griffiths transverse filtrations, with equality precisely for Simpson filtrations. A generic Simpson filtration extends through a proper flag Quot scheme, and saturating its special steps compares the two weights. If the weights are equal, the saturation lengths and the contribution of subsequent destabilizing modifications vanish, so the graded objects stay in the same connected fixed component.

These modifications happen in an auxiliary Rees family, in the spirit of Langer's interpretation of Simpson's procedure via Langton's semistable reduction, while the original family stays fixed. The maximum property concerns weighted degrees rather than filtration polygons, and the authors note it does not identify the Simpson decomposition with the ordinary Shatz stratification, since these already differ in rank three.

Stable full chains refine the numerical condition

For stable full chains, whose graded pieces are line bundles L1,…,Ln with nonzero arrows Li → Li+1 ⊗ KX, the paper defines prefix degrees sk = deg(L1 ⊕ … ⊕ Lk) for 1 ≤ k < n. These are exactly the degrees of the nonzero proper steps of the associated full flag, and the vector s = (s1,…,sn−1) indexes the corresponding connected fixed components. Stability of a degree-zero full chain is equivalent to positivity of every proper prefix degree.

The stable full chain components are parametrized by labels s in Zn−1>0 with Cs ≤ 2h1, where C is the Cartan matrix of type An−1 and h = g−1, and the component is isomorphic to Pics1(X) × ∏ Symdi(s)(X). The labels satisfy 1 ≤ sk ≤ hk(n−k). For these labels the paper proves S⋆s ⊆ ⋃t≥s S⋆t, with t ≥ s meaning coordinatewise comparison, and dim S⋆s − dim S⋆t = (t1 − s1) + (tn−1 − sn−1).

A zero dimension gap drives the counterexamples

For a full chain, positivity of the proper prefix degrees makes saturation of a specialized flag already a Simpson filtration, so an actual specialization satisfies t − s = ℓ ≥ 0, where ℓ is the vector of saturation lengths. The dimension formula then shows that only the two endpoint entries t1 − s1 and tn−1 − sn−1 affect the dimension.

If t ≠ s while t1 = s1 and tn−1 = sn−1, the weight increases while the dimension stays unchanged. Two distinct irreducible strata of the same dimension cannot contain one another in their closures, so any such specialization violates the closure condition in Simpson's nestedness conjecture. This is the mechanism the authors exploit.

Counterexamples exist in every rank n≥4

The paper constructs, for every g ≥ 2 and n ≥ 4, distinct connected components Ps and Pt of the full fixed locus, both consisting entirely of stable full chains, such that the intersection S⋆s ∩ S⋆t is nonempty and is a proper subset of S⋆t, for ⋆ in {Hod, Dol, dR}. Consequently Simpson's nestedness conjecture fails for the Dolbeault and de Rham moduli spaces in every rank n ≥ 4, and the same closure condition fails in the Hodge moduli space.

In rank four the authors choose arrow divisor degree vectors (0,2,0) and (1,0,1), whose labels differ only at the second coordinate. The Dolbeault incidence comes from a scaling equivariant projective line, and the Hodge incidence follows from the same Dolbeault curve. For flat bundles, smoothing one nonsaturated step of a transverse flag when the incoming and outgoing arrow divisors share a simple common point produces an actual family with the prescribed generic and special Simpson types. In these examples the Heinloth weights satisfy δt = δs + 1, consistent with the strict numerical closure condition.

The numerical order records direction, not containment

The counterexamples show that the strict increase of the Heinloth weight records the direction of specialization without forcing the whole target stratum into the source closure. The numerical condition is necessary, not sufficient: it constrains which strata can be met, but it does not decide which points of a permitted stratum belong to the closure.

The authors also discuss Simpson filtrations from the viewpoint of Θ stratifications, using the maximum property of the weight. On the locus δ = 0, where the underlying vector bundle is semistable, the HN ray is the unique rational ray minimizing Q independently of ε and of the base point x0. The numerical order is the lexicographic order of (δ, −m), and separating the coarse fixed components of the centers refines the Simpson decomposition. At strictly semistable centers, extension data can change the optimal filtration even when the coarse point is fixed: for g ≥ 3 there is a family of rank-three semistable Higgs bundles with split central member E0, all representing one coarse fixed point of weight one, with m(E0) = 1/2 and m(Eu) = 2/3 for u ≠ 0.

Where the result applies and where it stops

The closure theorem holds in every rank n ≥ 1 for the full Hodge family over a smooth complex projective curve of genus g ≥ 2, and the counterexamples require n ≥ 4. The refined prefix-degree comparison is stated for stable full chains, where all graded pieces are line bundles; for higher rank graded pieces, positive block prefix degrees do not control invariant subbundles inside a graded piece, so the same stability argument is no longer sufficient.

The authors note that the relation to Θ stratifications and the deformation theory of stratum intersections will receive a more complete treatment in a forthcoming update, so the Θ-stratification discussion in this paper is presented as an informal discussion rather than a finished account. The most decision-relevant point for readers is the split between the numerical order and the actual closure: the strict increase of the Heinloth weight is a reliable necessary condition on specialization, but the nestedness condition fails in every rank n ≥ 4, so any argument that treats numerical comparability as equivalent to containment of whole strata is not supported by this evidence.

Sources used in this answer