A $p$-adic Simpson correspondence for singular rigid-analytic varieties

Technical Summary: A p-Adic Simpson Correspondence for Singular Rigid-Analytic Varieties

Problem Statement
The paper addresses the extension of the $p$-adic Simpson correspondence to the setting of singular rigid-analytic varieties. Previous foundational work established this correspondence for smooth projective curves (Faltings [Fal05]), higher-dimensional smooth algebraic varieties (Abbes, Gros, and Tsuji [AGT16]), and arbitrary smooth proper rigid-analytic varieties over a complete, algebraically closed non-archimedean field $C$ (Heuer [Heu23]). The central challenge addressed here is the removal of the smoothness hypothesis. In the singular case, the standard sheaf of differentials $\Omega^1_X$ is insufficient to define Higgs bundles in a way that captures the necessary cohomological properties, and the étale topology does not admit a basis of smooth objects, complicating descent arguments.

Methodology
The authors employ the $\acute{e}h$-topology (introduced for rigid-analytic varieties by Guo [Guo19]) to resolve singularities locally. The $\acute{e}h$-topology is generated by étale coverings, blowup coverings, and universal homeomorphisms. A key structural property utilized is that the $\acute{e}h$-site of any rigid-analytic variety admits a basis of smooth rigid-analytic varieties via resolution of singularities.

The methodology proceeds through three main stages:

1. Descent on Smooth Varieties: The authors first establish that for a smooth rigid-analytic variety $X$, the category of Higgs bundles on the $\acute{e}h$-site ($X_{\acute{e}h}$) is equivalent to the category of Higgs bundles on the étale site ($X_{\acute{e}t}$). This relies on proving that vector bundles and Higgs bundles satisfy $\acute{e}h$-descent and that the sheaf of differentials $\Omega^1_{\acute{e}h}$ coincides with the pullback of the étale differentials on smooth objects.
2. Simplicial Machinery: To handle singular varieties, the authors construct smooth proper $\acute{e}h$-hypercovers. They develop the theory of simplicial rigid-analytic varieties, defining simplicial Higgs bundles and establishing a $p$-adic Simpson correspondence for smooth proper simplicial varieties. This involves lifting the simplicial variety to $B^+_{dR}/\xi^2$ and utilizing an exponential map.
3. Descent to the Singular Case: By utilizing the local smoothness of the $\acute{e}h$-topology, the authors construct a 1-truncated smooth proper $\acute{e}h$-hypercover for an arbitrary proper rigid-analytic variety $X$. They then use the equivalence established in the simplicial smooth case and descent properties (specifically that vector bundles and Higgs bundles form $\acute{e}h$-sheaves) to descend the correspondence to the singular variety $X$.

Key Contributions and Results

• Main Theorem (Theorem 4.4): For a proper rigid-analytic variety $X$ over $C$, the category of pro-étale vector bundles on $X$ ($\text{Vect}(X_{pro\acute{e}t})$) is equivalent to the category of Higgs bundles on the $\acute{e}h$-site of $X$ ($\text{Higgs}(X_{\acute{e}h})$).

  • A Higgs bundle on $X_{\acute{e}h}$ is defined as a finite locally free $\mathcal{O}_{X_{\acute{e}h}}$-module $E$ equipped with a morphism $\theta: E \to E \otimes \Omega^1_{X_{\acute{e}h}}(-1)$ satisfying $\theta \wedge \theta = 0$.
  • Here, $\Omega^1_{X_{\acute{e}h}}$ is the sheaf associated to the presheaf sending a smooth rigid-analytic variety $Y$ to $\Gamma(Y, \Omega^1_Y)$. This sheaf effectively replaces the standard differential sheaf with the first Deligne–Du Bois complex in the singular context.

• Cohomological Comparison (Theorem 4.7): The correspondence induces a natural isomorphism between the pro-étale cohomology of a vector bundle $E$ and the $\acute{e}h$-cohomology of the corresponding Higgs bundle $(E, \theta)$:
  $$R\Gamma_{pro\acute{e}t}(X, E) \simeq R\Gamma_{\acute{e}h}(X, (E, \theta))$$
  The authors prove that these cohomology groups are concentrated in cohomological degrees $[0, 2n]$, where $n$ is the dimension of $X$.

• Geometric Sen Operator (Proposition 4.8): The paper constructs a unique Higgs field $\eta_E$ for any $v$-vector bundle on the diamond associated to a rigid-analytic variety $X$. This field vanishes if and only if the bundle descends from the $\acute{e}h$-site, generalizing the geometric Sen operator to the singular setting.

• Descent Properties: The paper rigorously establishes that the presheaves of vector bundles and Higgs bundles satisfy $\acute{e}h$-descent (Theorem 2.5, Corollary 2.17). It also clarifies the behavior of differential sheaves, noting that while $\Omega^1_{\acute{e}h}$ is not a vector bundle (its rank may vary), it behaves well enough for the construction of the correspondence.

Significance and Claims
The authors claim that this work generalizes the results of Faltings, Abbes-Gros-Tsuji, and Heuer to the case of arbitrary proper rigid-analytic varieties, removing the smoothness constraint.

• Non-Abelian Hodge Theory: The result is presented as a non-abelian categorical generalization of Scholze's primitive comparison theorem and the generalized Hodge decomposition for singular varieties (Guo [Guo19]).
• Analogy with Complex Geometry: The paper notes that while Simpson's correspondence for singular complex algebraic varieties is largely open (with partial results for klt singularities), the methods developed here, utilizing the $\acute{e}h$-topology and Deligne–Du Bois complexes, suggest a plausible path to generalizing these results to the complex algebraic setting as well.
• Technical Foundation: The work provides the necessary descent machinery and simplicial framework to handle singularities in $p$-adic non-abelian Hodge theory, specifically addressing the subtleties of differential sheaves on singular spaces where the rank is not constant.

The paper does not propose new experimental applications or future conjectures beyond the immediate generalization of the correspondence and the potential extension to complex algebraic varieties mentioned in Remark 1.3.