Globally Holomorphic Polydifferentials

• Globally holomorphic polydifferentials are global sections of tensor powers of the canonical sheaf, fundamental for understanding algebraic curves and their symmetry groups.
• Their dimensionality and explicit bases, determined via the Riemann–Roch theorem and concrete formulas on curves like the Drinfeld curve, elucidate key geometric and arithmetic properties.
• The analysis of Galois module structures and automorphism actions provides detailed decompositions that inform applications in modular forms, deformation theory, and the study of singular spaces.

Globally holomorphic polydifferentials are global sections of tensor powers of the canonical sheaf of Kähler differentials on an algebraic or complex space. They constitute a fundamental object in the study of algebraic curves, their automorphism groups, Galois actions on cohomology, and dualities in both smooth and singular settings. The analysis of their structure, bases, and module decompositions reveals intricate relationships between geometry, arithmetic, and representation theory.

1. Fundamental Definitions and Framework

A globally holomorphic polydifferential of order $m$ on a smooth irreducible projective curve $X$ over a field $k$ is an element of the space $H^0(X, \Omega_X^{\otimes m})$, where $\Omega_X$ is the sheaf of Kähler differentials. More generally, on a reduced pure-dimensional complex space $X$ of dimension $n$, one studies $H^0(X, \Omega_X^p)$ for $0 \leq p \leq n$.

When a finite group $G$ acts on $X$, the spaces $H^0(X, \Omega_X^{\otimes m})$ are naturally $kG$-modules. The task of computing their module structure, particularly when $p= \text{char}(k)$ divides $|G|$ and the action is wild, is a central and unresolved problem except in specific cases involving additional structure or symmetry (Marchment et al., 15 Jan 2026, Bleher et al., 2021, Kalm, 2015).

2. Dimensionality and Basis Construction

For a smooth projective curve $X$ of genus $g$, the Riemann–Roch theorem gives

$$\dim_k H^0(X, \Omega_X^{\otimes m}) = \begin{cases} g, & m=1 \ (2m-1)(g-1), & m > 1 \end{cases}$$

provided $g \geq 2$ and $m \geq 1$ (Bleher et al., 2021, Marchment et al., 15 Jan 2026). For higher dimensions, analogous statements hold via Dolbeault cohomology if $X$ is a complex space.

In the explicit case of the Drinfeld curve $C$ given by $XY^q-X^qY-Z^{q+1}=0$ over $F$ of characteristic $p$, with $q=p^r$, an explicit $F$-basis is constructed: $$\omega_{i\,j} := x^i y^j \cdot \frac{(dx)^{\otimes m}}{x^{mq}}$$ where $x=X/Z$, $y=Y/Z$, and the indices $(i,j)$ vary so $0 \leq i, j$, $i+j \leq m(q-2)$. This basis is well-adapted to the $SL_2(\mathbb{F}_q)$-action and corresponds to exactly $\dim_F H^0(C, \Omega_C^{\otimes m})$ elements, yielding a complete and constructive description suitable for explicit computation (Marchment et al., 15 Jan 2026).

3. Galois Module Structure and Decomposition

The $kG$-structure of $H^0(X, \Omega_X^{\otimes m})$ is determined—in sharply characterized cases—by the $G$-divisor class of the associated divisor ($mK_X$ for canonical) modulo $G$-invariant principal divisors, and by detailed ramification data, including lower ramification filtrations and fundamental characters at ramified points (Bleher et al., 2021). For $G$ with cyclic Sylow $p$-subgroups, a hierarchical filtration of the module allows passage from inertia to global structure, and an explicit inductive algorithm provides a full decomposition into indecomposable summands.

Specializing to the Drinfeld curve ($G=SL_2(\mathbb{F}_q)$), one finds that the space $H^0(C, \Omega_C^{\otimes m})$ admits a decomposition into $G$-submodules $W_k$, each spanned by those $\omega_{i\,j}$ with $deg(\omega_{i\,j})\equiv k\mod (q+1)$. For general $q$, only a partial decomposition is possible; complete decomposition occurs in the prime field case $q=p$, wherein all indecomposable $\mathbb{F}[G]$-module summands and their multiplicities are described with explicit formulas (Marchment et al., 15 Jan 2026).

The following table summarizes the Galois module structure in representative cases for curves with group action:

Setting	Module Structure Description	Reference
$G$ with cyclic Sylow $p$-subgroups, $p>0$	Determined by $G$-divisor class and ramification data	(Bleher et al., 2021)
Drinfeld curve, $G = SL_2(\mathbb{F}_p)$	Full decomposition into indecomposable modules, explicit basis	(Marchment et al., 15 Jan 2026)
Arbitrary reduced complex space	Via fine sheaf resolutions of $\Omega^p_X$, Serre duality	(Kalm, 2015)

4. Actions of Automorphism Groups and Explicit Formulae

When $G$ acts on $X$, the group action extends naturally to $H^0(X, \Omega_X^{\otimes m})$. For the explicit bases above, representation theory is informed by how $G$ permutes and scales the basis elements. For example, for $G = SL_2(\mathbb{F}_q)$ acting on the Drinfeld curve,

$$\omega_{i\,j} \cdot \sigma = (\alpha x + \beta y)^i (\gamma x + \delta y)^j \cdot \frac{(dx)^{\otimes m}}{x^{mq}}$$

where $$\sigma = \begin{pmatrix} \alpha & \beta \ \gamma & \delta \end{pmatrix}\in SL_2(\mathbb{F}_q)$$. Each basis element is sent to a linear combination of elements of the same total degree modulo $q+1$ (Marchment et al., 15 Jan 2026).

In cases where a full decomposition is possible (e.g., $q=p$), the representation decomposes over blocks indexed by indecomposable modules $U_{a,b}$ and their Green correspondents $V_{a,b}$ for $G$, along with multiplicities given by explicit floor and ceiling combinatorial expressions.

5. Duality, Extension Criteria, and Singular Spaces

For a reduced pure $n$-dimensional complex space $X$, the structure of global polydifferentials is systematically understood via fine resolutions of $\Omega_X^p$ and dualizing sheaves. Fine sheaves of currents, denoted $A^{p,q}_X$ and $B^{n-p,q}_X$, extend the Dolbeault complex across singularities. The cohomology $H^q(X, \Omega_X^p)$ is computed via the cohomology of the complex $(A^{p,\bullet}_X, \partial)$ (Kalm, 2015).

Strongly holomorphic $p$-forms are characterized by the vanishing of residue obstructions: a meromorphic $p$-form extends holomorphically if and only if

$\nabla(\omega\wedge\varphi) = 0$

where $\nabla$ is the total differential from a free resolution and $\omega$ is the structure form. This provides effective, geometric extension criteria, especially significant in the presence of singularities.

Classical Serre duality is realized as a pairing between $H^a(X, \Omega_X^p\otimes F)$ and $H^{n-a}(X, \omega_X^{n-p}\otimes F^*)$ via integrations over $X$, and is built from these explicit resolutions.

6. Applications: Modular Forms, Deformation Theory, and Special Cases

Globally holomorphic polydifferentials play a critical role in various arithmetic and geometric contexts. For modular curves, knowledge of $H^0(X, \Omega_X^{\otimes m})$ as a Galois module yields information about cusp forms, Hecke algebras, and congruence relations among modular forms in characteristic $p$ (Bleher et al., 2021). In particular, the precise structure of the $k\mathrm{PSL}(2,\mathbb{F}_\ell)$-module of even-weight cusp forms in characteristic $p=3$ has been determined via these techniques.

Equivariant deformation spaces for curves with group action are similarly governed by the $G$-invariant part of $H^0(X, \Omega_X^{\otimes 2})$, with dimensions and structure explicitly evaluated in terms of ramification invariants. For hyperelliptic families and curves with explicit group action, computations of the indecomposable summands of the global polydifferentials are feasible and are supported by combinatorial formulae derived from the ramification filtration (Bleher et al., 2021, Marchment et al., 15 Jan 2026).

7. Explicit Examples and Structural Phenomena

In the Drinfeld curve case, all structural features—dimension, explicit basis, $G$-action, and module decomposition—are computable explicitly. For $m=1$, the canonical polydifferential representation aligns with classical results: the space is semisimple if and only if $p=q$, echoing Deligne–Lusztig theory. For small $m$ and $p$, hand computations verify the summand structure and dimensions predicted by the general theorems; e.g., for $p=3$, $m=2$, the dimension and module structure match the basis count and decomposition into $U_{a,b}$-summands (Marchment et al., 15 Jan 2026).

Globally, for complex spaces with only normal crossing singularities, every holomorphic $p$-form on the smooth part extends across singularities, and $\Omega_X^p$ remains reflexive and locally free. For spaces with isolated singularities, extension holds in degrees below $n-1$. Non-Cohen–Macaulay loci may obstruct extension and require higher Ext groups for duality, reflecting subtle geometric defects (Kalm, 2015).

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References:

(Marchment et al., 15 Jan 2026) The Galois Structure of the Spaces of polydifferentials on the Drinfeld Curve (Bleher et al., 2021) The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces (Kalm, 2015) The $\bar{\partial}$-equation, duality, and holomorphic forms on a reduced complex space