Quaternionic Systems of Hodge Bundles

• Quaternionic systems of Hodge bundles are quaternionic Higgs bundles with a Hodge decomposition compatible with both the Higgs field and the quaternionic involution, refining Simpson’s fixed-point description.
• They describe the R*-fixed geometry of the real moduli space of Higgs bundles, organizing real Bialynicki–Birula flows and enforcing parity constraints.
• The study employs moduli-theoretic methods and eigenvalue decompositions to reveal topological consequences and the distinct roles of real versus quaternionic structures.

Searching arXiv for the cited Higgs-bundle papers and related real/quaternionic systems of Hodge bundles literature. Quaternionic systems of Hodge bundles are quaternionic Higgs bundles equipped with a Hodge decomposition that is compatible with both the Higgs field and the quaternionic involution. In the setting of a compact Riemann surface $X$ endowed with an anti-holomorphic involution $\sigma:X\to X$, they arise as the quaternionic analogue of Simpson’s fixed-point description for the $\mathbb C^*$-action on Higgs-bundle moduli, but now on the real locus of the Dolbeault moduli space. Their principal role is to describe the $\mathbb R^*$-fixed geometry of $RM(r,d)=M(r,d)^\sigma$, to organize real Bialynicki–Birula flows, and to support connected-component results for the real locus of semistable Higgs-bundle moduli (Schaffhauser et al., 24 Jul 2025).

1. Moduli-theoretic setting and conceptual origin

Let $X$ be a compact Riemann surface of genus $\geqslant 2$ with anti-holomorphic involution $\sigma:X\to X$. The Dolbeault moduli space $M(r,d)$ of semistable Higgs bundles of rank $r$ and degree $\sigma:X\to X$0 carries the standard real structure

$\sigma:X\to X$1

Its fixed locus is denoted

$\sigma:X\to X$2

The Higgs-field scaling action $\sigma:X\to X$3 is compatible with this real structure through

$\sigma:X\to X$4

and therefore induces an $\sigma:X\to X$5-action on $\sigma:X\to X$6 (Schaffhauser et al., 24 Jul 2025).

The complex antecedent of this construction is Simpson’s fixed-point description: on the complex Higgs-bundle moduli space, the $\sigma:X\to X$7-fixed locus is described by systems of Hodge bundles. In the rank-three $\sigma:X\to X$8-Higgs setting, this mechanism appears explicitly via the $\sigma:X\to X$9-action

$\mathbb C^*$0

with moment map

$\mathbb C^*$1

whose fixed points are exactly variations of Hodge structure (VHS). For rank three, the fixed-point types are $\mathbb C^*$2-VHS, $\mathbb C^*$3-VHS, and $\mathbb C^*$4-VHS, and these are precisely the critical submanifolds of the Morse function (Zúñiga-Rojas, 2018).

A plausible implication is that quaternionic systems of Hodge bundles should be viewed not as an isolated construction, but as the $\mathbb C^*$5-compatible refinement of the same fixed-point formalism that governs complex Higgs moduli.

2. Quaternionic Higgs bundles and the definition of quaternionic systems of Hodge bundles

A real or quaternionic Higgs bundle is a triple $\mathbb C^*$6 in which $\mathbb C^*$7 is an anti-holomorphic map covering $\mathbb C^*$8, compatible with the Higgs field, and satisfying

$\mathbb C^*$9

The sign $\mathbb R^*$0 defines the real case, while the sign $\mathbb R^*$1 defines the quaternionic case. Equivalently, $\mathbb R^*$2 corresponds to a holomorphic isomorphism

$\mathbb R^*$3

such that

$\mathbb R^*$4

Thus the quaternionic condition is the sign choice $\mathbb R^*$5 in the square of the lift of the involution (Schaffhauser et al., 24 Jul 2025).

A quaternionic system of Hodge bundles is a quaternionic Higgs bundle $\mathbb R^*$6 for which $\mathbb R^*$7 admits a Hodge decomposition

$\mathbb R^*$8

satisfying

$\mathbb R^*$9

with every Hodge summand preserved by the quaternionic structure,

$RM(r,d)=M(r,d)^\sigma$0

and with each induced map

$RM(r,d)=M(r,d)^\sigma$1

a morphism of quaternionic bundles (Schaffhauser et al., 24 Jul 2025).

The paper treats real and quaternionic systems in parallel and states that both are $RM(r,d)=M(r,d)^\sigma$2-compatible refinements of Simpson’s Hodge decomposition for $RM(r,d)=M(r,d)^\sigma$3-fixed Higgs bundles. The only formal difference between a real system of Hodge bundles and a quaternionic one is the identity

$RM(r,d)=M(r,d)^\sigma$4

instead of

$RM(r,d)=M(r,d)^\sigma$5

3. Fixed points of the scaling action and the Hodge decomposition

The fixed-point statement in the real setting is sharpened by the identity

$RM(r,d)=M(r,d)^\sigma$6

Thus a point of the real locus fixed by the induced $RM(r,d)=M(r,d)^\sigma$7-action is automatically a $RM(r,d)=M(r,d)^\sigma$8-fixed point in the ambient complex moduli space, and hence lies in the domain of Simpson-type Hodge decomposition (Schaffhauser et al., 24 Jul 2025).

More precisely, if a stable real or quaternionic Higgs bundle $RM(r,d)=M(r,d)^\sigma$9 is fixed by some $X$0, then $X$1 carries a real or quaternionic system of Hodge bundles structure. The construction proceeds by taking an isomorphism

$X$2

decomposing $X$3 into generalized eigenspaces of $X$4, and assembling these eigenspaces into Hodge summands. The paper emphasizes that the Hodge grading is built from the $X$5-strings of eigenvalues, while conjugate eigenvalues are paired by $X$6, yielding a decomposition with

$X$7

(Schaffhauser et al., 24 Jul 2025).

This mechanism parallels the complex rank-three picture, where a system of Hodge bundles is a decomposition

$X$8

such that

$X$9

In that context, the fixed loci of the circle action on $\geqslant 2$0 are exhausted by the three VHS types

$\geqslant 2$1

with concrete Higgs-field block forms for each case (Zúñiga-Rojas, 2018). This suggests that quaternionic systems of Hodge bundles encode the same graded fixed-point phenomenon together with the extra involutive constraint imposed by $\geqslant 2$2.

4. Real versus quaternionic structures on stable points

For a stable Higgs bundle $\geqslant 2$3 satisfying

$\geqslant 2$4

the moduli-theoretic dichotomy is rigid: $\geqslant 2$5 admits either a real structure or a quaternionic structure, but not both, and this structure is unique up to isomorphism. Consequently,

$\geqslant 2$6

When $\geqslant 2$7, the paper states that the same decomposition holds for the whole real locus,

$\geqslant 2$8

where $\geqslant 2$9 is the quaternionic part (Schaffhauser et al., 24 Jul 2025).

The fixed-point analysis further separates two stable situations. In the geometrically stable case, the underlying Higgs bundle $\sigma:X\to X$0 is itself stable, and the decomposition takes the form

$\sigma:X\to X$1

with

$\sigma:X\to X$2

In the stable but not geometrically stable case, the underlying Higgs bundle is only polystable and splits as

$\sigma:X\to X$3

with

$\sigma:X\to X$4

and $\sigma:X\to X$5 exchanges the two factors. The eigenspace decomposition then comes in conjugate pairs: $\sigma:X\to X$6 and on each pair the involution acts by

$\sigma:X\to X$7

where the sign distinguishes the real and quaternionic cases (Schaffhauser et al., 24 Jul 2025).

These statements establish that quaternionic systems of Hodge bundles are not merely decorations on pre-existing Hodge data. They govern one of the two mutually exclusive liftings of the Galois-invariance condition to stable Higgs bundles.

5. Parity constraints and existence conditions

The quaternionic condition imposes arithmetic restrictions that do not appear in the ordinary complex theory. The paper records the following parity constraints for quaternionic vector bundles (Schaffhauser et al., 24 Jul 2025).

• When $\sigma:X\to X$8: a quaternionic vector bundle must have even rank.
• General parity condition:

$\sigma:X\to X$9

• When $M(r,d)$0 and $M(r,d)$1 is odd: there are no quaternionic bundles.
• When $M(r,d)$2: quaternionic bundles can occur in odd rank, subject to the parity condition above.
• For $M(r,d)$3: the real locus of the Picard variety may contain only real line bundles, only quaternionic line bundles, both, or be empty, depending on the parity of degree and genus; this determines whether $M(r,d)$4 contains real or quaternionic Higgs bundles.

The first constraint is explained fiberwise: on a real fiber, the identity

$M(r,d)$5

forces the fiber dimension to be even. The remaining conditions are global restrictions coupling rank, degree, and genus.

These parity statements are structurally important because the $M(r,d)$6-fixed locus of the real moduli space is not exhausted by real systems of Hodge bundles. Quaternionic fixed points are forced in certain topological regimes, especially when $M(r,d)$7 or when rank-degree parity excludes real alternatives.

6. Real Bialynicki–Birula flows and topological consequences

Quaternionic systems of Hodge bundles enter the topology of Higgs-bundle moduli through real Bialynicki–Birula flows. For a Galois-invariant Higgs bundle $M(r,d)$8, the Bialynicki–Birula flow

$M(r,d)$9

restricts to a real map

$r$0

and complete Bialynicki–Birula flows

$r$1

restrict to

$r$2

The paper uses this to flow points in the fiber

$r$3

down to the nilpotent cone

$r$4

The fixed points encountered along the flow are described by real or quaternionic systems of Hodge bundles (Schaffhauser et al., 24 Jul 2025).

The main topological theorem is that, when $r$5, the fibers $r$6 are connected, and therefore the connected components of $r$7 are indexed by $r$8. In particular, if $r$9 has $\sigma:X\to X$00 connected components, then

$\sigma:X\to X$01

so there are $\sigma:X\to X$02 connected components. When $\sigma:X\to X$03, the number of connected components depends on the parity of $\sigma:X\to X$04, $\sigma:X\to X$05, and $\sigma:X\to X$06 (Schaffhauser et al., 24 Jul 2025).

The quaternionic contribution is indirect but essential. The proof strategy flows a Galois-invariant point toward fixed points and then into the nilpotent cone; because some fixed points are quaternionic rather than real, the real Bialynicki–Birula stratification necessarily includes quaternionic systems of Hodge bundles among its fixed “nodes.”

7. Relation to classical systems of Hodge bundles and rank-three critical loci

The broader Higgs-bundle context is furnished by the theory of critical loci of the circle-action Morse function. For rank-three $\sigma:X\to X$07-Higgs bundles on a compact Riemann surface $\sigma:X\to X$08 of genus $\sigma:X\to X$09, a $\sigma:X\to X$10-Higgs bundle is a pair $\sigma:X\to X$11 with

$\sigma:X\to X$12

and the fixed points of the relevant action are exactly VHS loci. In rank three these are the three types

$\sigma:X\to X$13

which are exactly the critical submanifolds of the Morse function (Zúñiga-Rojas, 2018).

Two of these critical loci are identified with moduli spaces of $\sigma:X\to X$14-stable holomorphic triples. In the $\sigma:X\to X$15-case,

$\sigma:X\to X$16

and in the $\sigma:X\to X$17-case,

$\sigma:X\to X$18

where

$\sigma:X\to X$19

The remaining $\sigma:X\to X$20-locus has the explicit description

$\sigma:X\to X$21

Moreover, the embeddings

$\sigma:X\to X$22

extend to the critical loci and induce cohomological stabilization in specified degree ranges (Zúñiga-Rojas, 2018).

This complex rank-three theory does not itself define quaternionic systems of Hodge bundles, but it supplies the ambient template: systems of Hodge bundles are fixed points of scaling actions, critical loci can often be described by auxiliary moduli spaces, and the geometry of those loci controls global topology. A plausible implication is that quaternionic systems of Hodge bundles should be understood as the real-structure-compatible counterpart of that template, replacing purely complex fixed-point strata by fixed loci adapted to the involution

$\sigma:X\to X$23