$h$-topology for rigid spaces and an application to $p$-adic Simpson correspondence
Abstract
In this paper, we study the $h$-topology for rigid spaces. Along the way, we establish several foundational results on morphisms of rigid spaces: we prove generic flatness and openness of the flat locus in the rigid setting, and we show that (for affinoid rigid spaces) strict transforms become flat after a blowup. Moreover, we show that any fppf cover admits a quasi-finite refinement and prove a version of Zariski's main theorem for rigid spaces. As an application, we deduce that for a proper rigid space $X$ over $C$, the category of pro-étale vector bundles on $X$ is equivalent to the category of Higgs bundles on the $h$-site of $X$, thereby generalizing Heuer's results to the singular setting.
Summary
This paper delves into the study of the h-topology for rigid spaces, a concept analogous to the h-topology in algebraic geometry. The primary goal is to extend the p-adic Simpson correspondence, initially established by Heuer for smooth rigid spaces, to the more general case of possibly singular proper rigid spaces over a complete, algebraically closed non-archimedean extension of Qp. The authors establish several foundational results for morphisms of rigid spaces, including generic flatness, openness of the flat locus, and a platification by blowup result. They also prove that any fppf cover admits a quasi-finite refinement and provide a version of Zariski's main theorem for rigid spaces. The core methodology involves defining the h-topology for rigid spaces, proving key properties like generic flatness and openness of the flat locus, and establishing a platification by blowup result. These results are then used to show that h-covers are generated by finite surjective maps and eh-coverings. As a major application, the authors prove that for a proper rigid space X, the category of pro-étale vector bundles on X is equivalent to the category of Higgs bundles on the h-site of X. This generalizes Heuer's result to the singular setting. This generalization is significant because it provides a non-abelian categorical generalization of Scholze's Hodge-Tate decomposition for non-smooth rigid spaces, extending previous work by Heuer and Guo.
Key Insights
- •The paper establishes the h-topology on rigid spaces, providing a Grothendieck topology where coverings are finite families of quasicompact morphisms that are surjective.
- •They prove an analog of the classical algebraic geometry theorems of "the flat locus is open" and "generic flatness" in the rigid setting. Specifically, they show the flat locus is Zariski open and, for tft morphisms of affinoid rigid spaces, there exists a nowhere dense Zariski closed subset Z such that the morphism is flat over the complement of Z.
- •The paper demonstrates a "platification by blowup" result for rigid spaces, showing that for a tft morphism of affinoid rigid spaces, a strict transform becomes flat after a U-admissible blowup where U is the complement of a Zariski closed set where the original morphism is flat.
- •The authors prove a quasi-finite refinement result for fppf covers of rigid spaces and provide a version of Zariski's main theorem for rigid spaces, stating that a quasi-finite morphism can be locally factored into a finite morphism, an open immersion, and an étale morphism.
- •The paper provides a structure theorem for h-covers, showing that any h-cover can be refined by a sequence of morphisms including a finite surjective map, an open covering, an étale covering, and an eh-covering by a smooth rigid space.
- •The main application is the generalization of the p-adic Simpson correspondence to singular rigid spaces, proving the equivalence between pro-étale vector bundles and Higgs bundles on the h-site.
- •The paper builds on and generalizes Heuer's results, extending the p-adic Simpson correspondence to singular rigid spaces. This generalization is important because it provides a non-abelian categorical generalization of Scholze's Hodge-Tate decomposition for non-smooth rigid spaces.
Practical Implications
- •The results have implications for non-abelian Hodge theory in the p-adic setting, particularly for singular spaces. The equivalence between pro-étale vector bundles and Higgs bundles provides a powerful tool for studying these objects.
- •The paper opens up avenues for studying Simpson's correspondence for algebraic varieties over the complex numbers with klt singularities, potentially generalizing existing results.
- •The foundational results on morphisms of rigid spaces (flatness, blowups, quasi-finite maps) can be used in other areas of rigid analytic geometry and related fields.
- •The h-topology provides a new tool for studying rigid spaces and their properties, potentially leading to new insights and applications.
- •Future research directions include exploring the h-sheafified version of the Simpson gerbe constructed by Bhatt and Zhang, and investigating a version of singular non-abelian Hodge theory in positive characteristic.