A theory of dormant opers on pointed stable curves -- a proof of Joshi's conjecture

Abstract: This manuscript presents a detailed and original account of the theory of opers defined on pointed stable curves in arbitrary characteristic and their moduli. In particular, it includes the development of the study of dormant opers, which are opers of a certain sort in positive characteristic. The theory of dormant opers (or more generally, opers in positive characteristic) on pointed stable curves, which has proved to be rather rich and deep, was born in the work of S. Mochizuki, who developed the theory for $\mathfrak{sl}_2$-opers and used it to establish $p$-adic Teichm\"{u}ller theory. Some parts of Mochizuki's work were later extended in the case of proper smooth curves by K. Joshi, C. Pauly, and other mathematicians. This manuscript represents an advance in the theory of opers that takes the subject beyond the work of Mochizuki, Joshi, and Pauly. In particular, we provide general unified formulations and the basics of principal bundles and connections defined on families of pointed stable curves. The notion of an oper is accordingly introduced in the context of logarithmic algebraic geometry. Some of the results can be regarded as generalizations of results obtained in the fundamental work on the geometric Langlands program developed by A. Beilinson and V. Drinfeld. We also describe various properties and assertions about (dormant) opers, such as duality, comparison with differential operators, and compactification of the moduli space. Our goal is to give an explicit formula, conjectured by Joshi, for the generic number of dormant $\mathfrak{sl}_n$-opers. We do so by obtaining a detailed understanding of the moduli space of dormant opers and computing the Gromov-Witten invariants for Quot-schemes in characteristic zero. This formula reveals an interaction between studies in $p$-adic Teichm\"{u}ller theory and certain areas of mathematics, including Gromov-Witten theory.

• The paper extends the theory of dormant opers on stable curves and proves Joshi's conjecture on the generic number of dormant $\mathfrak{sl}_n$-opers.
• It defines and examines the moduli space of dormant opers, establishing geometric properties and finite results related to Gromov-Witten invariants.
• Dormant opers are crucial for understanding algebraic structures in positive characteristic, and the results provide a foundation for further research in algebraic geometry.

Dormant Opers on Pointed Stable Curves: Theoretical Insights and Advances

The paper presented by Yasuhiro Wakabayashi constitutes a comprehensive exploration of the theory of opers defined over pointed stable curves, with a particular focus on those in positive characteristic, named dormant opers. Dormant opers, originating from the foundational work of S. Mochizuki, have been further developed to connect various domains within mathematics, including p-adic Teichmüller theory and Gromov-Witten theory. The research endeavors to generalize and extend these concepts, offering new formulations within logarithmic algebraic geometry.

Key Contributions

1. Advance in the Theory of Dormant Opers:
   • The manuscript extends the theory of opers on stable curves by providing unified formulations related to principal bundles and connections on pointed stable curves.
   • By integrating results and methodologies from various mathematical frameworks, the paper presents generalizations of the geometric Langlands program outcomes, initially developed by Beilinson and Drinfeld.
2. Joshi's Conjecture on Dormant opers:
   • A critical component is the proof of Joshi's conjecture regarding the generic number of dormant $\mathfrak{sl}_n$-opers. This involves a detailed study of the moduli space of dormant opers and employs techniques involving Gromov-Witten invariants.
3. Structure of Moduli Spaces:
   • The paper defines and examines the moduli space of dormant opers, providing explicit results around their geometric properties such as compactification, dimension, and connection with differential operators.
   • The finiteness and étaleness aspects of these spaces are thoroughly established, underscoring the deep intertwining of dormant opers with various characteristic polynomial invariants.
4. Numerical Results and Finiteness Theorems:
   • Strong numerical results are presented on the aforementioned conjecture, ensuring connections with Gromov-Witten theory and providing explicit calculations for specific moduli spaces.
   • The effort also includes a demonstration of the finiteness of certain Hitchin-Mochizuki morphisms within this context.

Theoretical and Practical Implications

• Logarithmic connections, and by extension dormant opers, play a pivotal role in understanding algebraic structures in characteristic $p > 0$, offering insights into the interplay of algebraic and geometric properties in such settings.
• By proving aspects of Joshi's conjecture and other structural theorems, the research opens avenues for further exploration into similar conjectures in the field of opers and their applications in broader mathematical contexts.
• The developed theories and results have practical implications for understanding vector bundles, principal bundles, and the associated connections, which are foundational elements in many theoretical physics models, particularly within string theory and related fields.

Future Directions

The implications of this work extend beyond the immediate results, suggesting various future research directions. This includes potential explorations into higher-rank structures, dormancy conditions in more generalized settings, and further computational relations within Gromov-Witten theory.

In conclusion, Wakabayashi's paper contributes significant theoretical advancements in the study of dormant opers on pointed stable curves, offering both a rigorous proof of conjectural aspects and a path forward for further mathematical exploration. The interplay of dormant opers with algebraic and geometric frameworks highlights the richness of this area of study, providing a foundation for future research and applications in algebraic geometry.