Core Concepts

This research paper proves that the mod-p reductions of rigid integrable G-connections on smooth projective varieties have nilpotent p-curvatures, extending previous results for regular connections and providing further evidence for Simpson's motivicity conjecture.

Abstract

**Bibliographic Information:**Huang, P., Qin, Y., & Sun, H. (2024). Rigid G-connections and nilpotency of p-curvatures. arXiv preprint arXiv:2410.09929v1.**Research Objective:**This paper aims to generalize Esnault and Groechenig's result on the nilpotency of p-curvatures for rigid connections to the case of rigid integrable G-connections. This generalization provides further support for Simpson's conjecture, which posits that rigid integrable connections are motivic.**Methodology:**The authors employ techniques from nonabelian Hodge theory, including the Corlette-Simpson correspondence, Ogus-Vologodsky correspondence for principal bundles, and the spectral data morphism for G-Higgs bundles. They build upon previous work by Esnault and Groechenig, adapting their strategy for proving the nilpotency of p-curvatures in the case of regular connections.**Key Findings:**The paper's main result is the proof that for a rigid integrable G-connection on a smooth projective complex variety, there exists a model over a finite-type scheme over Z such that the restrictions to all closed points have nilpotent p-curvatures. This result is achieved by establishing a correspondence between rigid integrable G-connections and nilpotent G-Higgs bundles, leveraging the fact that the latter objects have nilpotent p-curvatures.**Main Conclusions:**The authors successfully extend the nilpotency of p-curvatures result to rigid integrable G-connections, strengthening the evidence for Simpson's motivicity conjecture. This work contributes significantly to the understanding of the arithmetic properties of complex local systems and their connections to geometry.**Significance:**This research has significant implications for the study of nonabelian Hodge theory and its applications in algebraic geometry and number theory. The findings provide further support for Simpson's motivicity conjecture, a fundamental open problem in the field.**Limitations and Future Research:**The paper focuses on smooth projective varieties. Exploring the nilpotency of p-curvatures for rigid G-connections on more general varieties could be a potential direction for future research. Additionally, investigating the implications of this result for specific types of G-connections and their associated geometric structures could lead to further insights.

To Another Language

from source content

arxiv.org

Stats

Quotes

"Motivated by Simpson’s conjecture on the motivicity of rigid irreducible connections, Esnault and Groechenig demonstrated that the mod-p reductions of such connections on smooth projective varieties have nilpotent p-curvatures. In this paper, we extend their result to integrable G-connections."
"Simpson addressed this transcendental nature in [Sim90], where he posed a question about which integrable connections defined over Q have associated monodromy representations also defined over Q. He conjectured that such integrable connections should originate from geometry"
"In this paper, we aim to generalize Esnault–Groechenig’s result on nilpotency of p-curvatures [EG20, Theorem 1.4] to rigid integrable G-connections."

Key Insights Distilled From

by Pengfei Huan... at **arxiv.org** 10-15-2024

Deeper Inquiries

This paper cleverly leverages the interplay between the Corlette-Simpson correspondence, the Ogus-Vologodsky correspondence, and the spectral data morphism to establish the nilpotency of p-curvatures for rigid integrable G-connections. These techniques hold promise for investigating other arithmetic properties of G-connections. Here are some potential avenues:
Integrality Properties: The paper already mentions the work of Klevdal and Patrikis [KP22] on the integrality of cohomologically rigid G-connections. One could explore if the methods used here, particularly the interplay between characteristic 0 and characteristic p geometries, could provide alternative proofs or stronger integrality results. For instance, can we establish integrality properties for specific classes of rigid G-connections with nilpotent p-curvatures?
Crystalline Nature: The nilpotency of p-curvatures hints at a deeper crystalline nature of these connections. It would be interesting to investigate if rigid integrable G-connections with nilpotent p-curvatures can be realized within the framework of crystalline cohomology or other p-adic cohomology theories. This could provide a more refined understanding of their arithmetic properties.
Special Values of L-functions: The motivicity of objects often has profound implications for the special values of associated L-functions. While Simpson's motivicity conjecture remains open in general, the results of this paper provide evidence in its direction. One could explore if the nilpotency of p-curvatures for rigid G-connections can be used to deduce information about the special values of L-functions associated with these connections, potentially leading to new cases of conjectures like the Bloch-Kato conjecture.
Moduli Spaces and Deformation Theory: The paper utilizes the deformation theory of G-Higgs bundles and integrable G-connections. Further study of the deformation theory of these objects, particularly in the context of nilpotent p-curvatures, could unveil finer arithmetic information. For example, one could investigate the structure of the formal neighborhoods of rigid points in the moduli spaces and explore potential connections with p-adic Hodge theory.

The paper focuses on smooth projective varieties. The situation could become significantly more intricate when singularities are introduced. It is plausible that the nilpotency of p-curvatures might not hold in general for rigid integrable G-connections on singular varieties.
Here's why:
Monodromy and Singularities: The relationship between the monodromy of a connection and the singularities of the underlying variety is well-established. Singularities can give rise to more complicated monodromy representations, which in turn might lead to non-nilpotent p-curvatures.
Logarithmic Connections: A natural framework for studying connections on singular varieties is that of logarithmic connections. It would be interesting to investigate if there are examples of rigid logarithmic G-connections on varieties with specific types of singularities where the nilpotency of p-curvatures fails.
Counter-Examples: Constructing explicit counter-examples would be a challenging but potentially fruitful direction. One could start by considering varieties with simple singularities, such as nodal curves or surfaces with isolated singularities, and attempt to construct rigid integrable G-connections with non-nilpotent p-curvatures.

This research has significant implications for the study of representations of fundamental groups, particularly in bridging the gap between characteristic 0 and characteristic p geometries.
Here's how:
Motivic Perspective: Simpson's motivicity conjecture lies at the heart of understanding the arithmetic nature of representations of fundamental groups. This paper provides further evidence for this conjecture by establishing a key property (nilpotency of p-curvatures) expected of motivic objects. This strengthens the link between representations arising from geometry and those satisfying specific arithmetic conditions.
New Constraints on Representations: The nilpotency of p-curvatures imposes a strong constraint on the possible representations of fundamental groups that can arise from rigid G-connections. This could potentially lead to new restrictions on the possible images of such representations, providing insights into the structure of these representations.
p-adic Methods: The techniques employed in this paper, particularly the use of the Ogus-Vologodsky correspondence, highlight the power of p-adic methods in studying representations of fundamental groups. This could inspire further applications of p-adic techniques to investigate the geometry and topology of complex algebraic varieties.
Anabelian Geometry: This research aligns with the spirit of anabelian geometry, which seeks to recover geometric information about a variety from its fundamental group. The results suggest that arithmetic properties of representations, such as the nilpotency of p-curvatures, could encode subtle geometric information about the variety.

0