A theory of dormant opers on pointed stable curves 9782856299562

Table of contents :
Chapter 0. Introduction
0.1. Review of opers on complex algebraic curves
0.2. What about opers in positive characteristic?
0.3. Counting problem of dormant opers
0.4. Comparison with arguments due to Mochizuki, Joshi, and Pauly
0.5. Structure and main theorems of the manuscript
0.6. Acknowledgements
0.7. Notation and Conventions
Chapter 1. Logarithmic connections over log schemes
1.1. The adjoint representation and the Maurer-Cartan form
1.2. Principal G-bundles and their associated Lie algebroids
1.3. Logarithmic connections and curvature
1.4. Explicit descriptions of gauge transformations
1.5. The moduli stack of pointed stable curves
1.6. Local pointed curves and monodromy
Chapter 2. Opers on a family of pointed stable curves
2.1. (g, )-opers on a log curve
2.2. Local description and automorphisms of a (g, )-oper
2.3. The case of g= sl2
2.4. The G-bundle EG and the vector bundle Vg
2.5. q1[origin=c]-10 -normality and a torsor structure on the sheaf of (g, )-opers
2.6. (g, )-opers on pointed stable curves
2.7. The adjoint quotient of g
2.8. Radii of (g, )-opers
2.9. The moduli space of (g, )-opers of prescribed radii
2.10. The universal moduli stack
Chapter 3. Opers in positive characteristic
3.1. Frobenius twists and relative Frobenius morphisms
3.2. Lie algebroids in positive characteristic
3.3. p-curvature on an -flat G-bundle
3.4. Dormant/p-nilpotent (g, )-opers and their moduli spaces
3.5. The Hitchin-Mochizuki morphism
3.6. Varying the parameter
3.7. The finiteness of the Hitchin-Mochizuki morphism
3.8. The universal moduli stacks of dormant/p-nilpotent (g, )-opers
Chapter 4. Flat vector bundles and differential operators
4.1. Logarithmic -connections on vector bundles
4.2. Logarithmic differential operators with parameter
4.3. Comparison of log connections
4.4. (GLn, )-opers and comparison with (sln, )-opers
4.5. (n, , )-projective connections and (GLn, , )-opers
4.6. (n, )-theta characteristics
4.7. Radii of (GLn, , )-opers and (n, , )-projective connections
4.8. Change of (n, )-theta characteristic
4.9. A canonical PGLn-bundle underlying (sln, )-opers
4.10. Dormant (GLn, )-opers
4.11. Comparison of the moduli functors
4.12. Hypergeometric differential operators
Chapter 5. Duality of opers
5.1. (so2l+1, )-opers and (sp2m, )-opers
5.2. (GO2l+1, , )-opers and (GSp2m, , )-opers
5.3. Dual opers
5.4. Isomorphisms of moduli spaces induced by duality
Chapter 6. Local deformation of opers
6.1. Deformation spaces of a curve and an -flat G-bundle
6.2. Cohomology of complexes associated to a (g, )-oper
6.3. Infinitesimal deformations of a (g, )-oper
6.4. Flat vector bundles in positive characteristic
6.5. Infinitesimal deformations of a dormant (g, )-oper
Chapter 7. The pseudo-fusion ring for dormant opers
7.1. Semi-graphs and clutching data
7.2. Gluing -flat G-bundles
7.3. Factorization of (g, )-opers
7.4. Pseudo-fusion rules and pseudo-fusion rings
7.5. The ring structure of the fusion ring
7.6. Factorization property of a fusion rule
7.7. The pseudo-fusion ring associated to dormant opers
7.8. The pseudo-fusion rule for sl2
Chapter 8. Generic étaleness of the moduli space of dormant opers
8.1. Formally local description of a flat bundle
8.2. Deformation of a flat bundle
8.3. Deformation of a dormant GLn-oper
8.4. Dormant opers on a 3-pointed projective line
8.5. The generic étaleness of the moduli stack of dormant opers
Chapter 9. Comparison with Quot schemes
9.1. Quot schemes
9.2. Comparison with the moduli space of dormant (GLn, )-opers
9.3. The relation between the Quot schemes
9.4. Counting maximal subbundles
9.5. Joshi's conjecture
Bibliography
Index

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