Higgs-Grassmannian Schemes in Moduli Theory

• Higgs-Grassmannian schemes are geometric spaces that generalize classical Grassmannians by incorporating Higgs fields to parameterize quotient bundles.
• They play a pivotal role in moduli problems, enabling techniques in stability, reduction, and stratification across algebraic, arithmetic, and quantum field theory applications.
• They facilitate numerical and cohomological investigations through universal exact sequences and invariants, testing properties like ampleness, semistability, and Poincaré duality.

A Higgs-Grassmannian scheme generalizes the classical Grassmannian construction by incorporating the additional structure of Higgs fields, thus yielding a parameter space for quotient bundles equipped with compatible Higgs dynamics. Canonical examples arise in both algebraic geometry—as the moduli of Higgs bundles—and in the theory of loop groups and affine Grassmannians, with fundamental applications to stability conditions, reduction procedures, representation theory, and quantum field theory strata. The Higgs-Grassmannian framework crucially “feels” the Higgs field in defining positivity and semistability properties, and unifies local and global aspects of moduli theory via stratification and reduction, with technical constructions precise enough for arithmetic and geometric applications (Capasso, 29 Dec 2025, Chen, 2012, Fazzi et al., 2023).

1. Foundational Definitions and Constructions

Given a smooth projective variety $X$ over an algebraically closed field $K$ of characteristic zero, a rank $r$ Higgs bundle is a pair $(E, \varphi)$, where $E$ is a locally free $\mathcal{O}_X$-module and $\varphi: E \to E \otimes \Omega^1_X$ with $\varphi \wedge \varphi = 0$ (Capasso, 29 Dec 2025). For fixed $1 \leq s \leq r-1$, the s-th Higgs-Grassmannian functor $hGr_s(E)$ assigns to each $X$-scheme $T$ the set of isomorphism classes of surjections

$p_T^*E \xrightarrow{\epsilon} Q \to 0$

with $Q$ locally free of rank $s$ and the Higgs-compatibility condition requiring the induced map

$$\mathrm{Ker}(\epsilon) \xrightarrow{p_T^*\varphi} p_T^*E \otimes p_T^*\Omega^1_X \xrightarrow{\epsilon \otimes \mathrm{Id}} Q \otimes p_T^*\Omega^1_X$$

to vanish identically. The functor $hGr_s(E)$ is representable by a closed subscheme of the classical Grassmannian $Gr_s(E)$, termed the Higgs-Grassmannian scheme.

In the context of loop groups, let $k$ be an algebraically closed field, $F = k((t))$, $G$ a split reductive group, and $T \subseteq G$ a maximal torus. The affine Grassmannian $\mathcal{X} = G(F)/G(O)$ admits a stratification and quotient via the notion of $\xi$-stability, leading to the construction of $\mathcal{X}^\xi / T$ as an ind-$k$-scheme (Chen, 2012).

2. Universal Properties and Exact Sequences

On $Gr_s(E)$, one has the tautological sequence:

$$0 \to S_{r-s,E} \xrightarrow{\eta} p_s^*E \xrightarrow{\epsilon} Q_{s,E} \to 0,$$

where $S_{r-s,E}$ and $Q_{s,E}$ are the universal subbundle and quotient bundle, respectively. The locus $hGr_s(E)$ is carved out by the vanishing of the composed morphism

$$S_{r-s,E} \xrightarrow{\eta} p_s^*E \xrightarrow{p_s^*\varphi} p_s^*E \otimes p_s^*\Omega^1_X \xrightarrow{\epsilon \otimes \mathrm{Id}} Q_{s,E} \otimes p_s^*\Omega^1_X.$$

There is a universal property: for any scheme $Y \to X$ and quotient Higgs bundle $f^*E \xrightarrow{q} \mathcal{Q} \to 0,$ where $\varphi$ descends to $\mathcal{Q}$, there exists a unique morphism $g:Y \to hGr_s(E)$ such that $\mathcal{Q} \cong g^*Q_{s,E}$, and $\mathrm{Ker}(q)$ matches the pullback of the universal subbundle (Capasso, 29 Dec 2025).

3. Stability, Stratification, and Reduction Procedures

For $\xi \in X_*(T) \otimes \mathbb{R}$ generic, $\xi$-stability is defined via convex polytopes $Ec(x)$ comprising the cocharacters arising from the Tits–Iwasawa decomposition over varying Borels $B \supset T$. A point $x$ is $\xi$-stable iff $\xi \in Ec(x)$; semistability permits $\xi$ in the closure (Chen, 2012). In the case $G = SL_d$, $\xi$-stability translates into “slope inequalities” on lattice indices, closely paralleling the notion of slope-stability for vector bundles.

Analogous to Harder–Narasimhan reduction, the Arthur–Kottwitz procedure stratifies the affine Grassmannian minus its $\xi$-stable locus according to parabolic subgroups $P \supset T$, yielding infinite-dimensional affine fibrations over $\xi$-stable Levi factors. This mirrors the reduction and filtration structure of Higgs bundles or vector bundles.

4. Positivity and Semistability Criteria via Higgs–Grassmannians

Positivity conditions for Higgs bundles are detected on Higgs-Grassmannian schemes through numerical invariants. In particular, classes

$$\lambda_s(E) = c_1(\mathcal{O}_{Gr_1(Q_{s,E})}(1)) - \frac{1}{r}\pi_s^*c_1(E) \in N^1(Gr_1(Q_{s,E}))$$

and

$$\theta_s(E) = c_1(Q_{s,E}) - \frac{s}{r}\rho_s^*c_1(E) \in N^1(hGr_s(E)),$$

test ampleness and nefness. The main theorems establish that a Higgs bundle $E$ is curve-semistable precisely when $\theta_s(E)$ (equivalently, $\lambda_s(E)$) is nef for all $1 \leq s \leq r-1$ (Capasso, 29 Dec 2025). In particular, $E$ is H-numerically flat (H-nflat) if $E$ and $E^\vee$ are H-nef, and this is equivalent to semistable pullbacks with vanishing degree of $c_1$ on all curves.

5. Ind-Scheme Structure, Quotients, and Projectivity

The locus $hGr_s(E) \hookrightarrow Gr_s(E)$ is a closed subscheme and thus proper and projective over $X$. For affine Grassmannians equipped with $\xi$-stability, the quotient $\mathcal{X}^\xi/T$ is constructed as an ind-scheme via exhaustion by finite-type projective varieties, compatible with geometric invariant theory (GIT): the $\xi$-stable locus coincides with a Mumford-stable locus for an appropriate subtorus action (Chen, 2012). This structure ensures that the quotient fulfills the valuative criterion of properness, enabling arithmetic applications.

For double affine Grassmannians, the ind-scheme stratification into smooth symplectic leaves and transverse slices encodes the RG flow hierarchy in higher-dimensional field theories (Fazzi et al., 2023).

6. Applications and Examples

Higgs-Grassmannian schemes apply to moduli problems for Higgs bundles: they parametrize quotient Higgs bundles with compatibility conditions, enabling the study of positivity notions such as H-ampleness and H-nefness, with concrete implications in curve-semistability and numerical flatness. Examples demonstrate that the Higgs positivity conditions do not coincide with classical ones; e.g., rank-2 Higgs bundles may be H-ample even if the underlying bundle is not ample in the usual sense (Capasso, 29 Dec 2025).

The local theory of $\xi$-stability on affine Grassmannians, initiated by Chaudouard–Laumon and extended by Chen, provides a formal-disc analogue to global $\xi$-stability on moduli spaces of Higgs bundles. The quotient $\mathcal{X}^\xi/T$ serves as a local parameter space for $\xi$-stable Higgs lattices and is instrumental in studying purity phenomena of affine Springer fibers—a route to the Goresky–Kottwitz–MacPherson purity conjecture (Chen, 2012).

In quantum field theory, the Higgs-Grassmannian schemes stratify the double affine Grassmannian, thereby encoding hierarchy and RG flows in 6d $(1,0)$ SCFTs, with the partial order on coweights corresponding to possible RG trajectories (Fazzi et al., 2023).

7. Cohomological and Arithmetic Properties

For $G = SL_d$, each finite-level quotient $X_n^\xi/T$ is homologically smooth and satisfies Poincaré duality, with the unstable locus having codimension at least $(d-1)n$. The generating function for the Poincaré series is explicitly

$$P_{X^\xi/T}(t) = (1-t^2)^{-(d-1)} \prod_{i=1}^{d-1} \frac{1}{1-t^{2i}},$$

with vanishing odd cohomology and Weil weights governed by Frobenius action (Chen, 2012). These properties provide tools for arithmetic and enumerative geometry computations, and underpin conjectures such as the cohomological purity of affine Springer fibers.

A plausible implication is that cohomological invariants and purity results obtained for Higgs-Grassmannian schemes can inform analogous results in moduli spaces of Higgs bundles, representations of affine Kac–Moody algebras, and quantum field theory stratification phenomena. However, conjectures such as vanishing discriminants for curve-semistable Higgs bundles remain open in general (Capasso, 29 Dec 2025).