[Paper Review] Irreducible connections admit generic oper structures

D. Arinkin|arXiv (Cornell University)|Feb 29, 2016

Algebraic Geometry and Number Theory6 references7 citations

TL;DR

This paper proves that every meromorphic $G$-connection on a smooth projective curve over an algebraically closed field of characteristic zero admits a gauge-equivalent connection with a possibly degenerate oper structure; in particular, every irreducible meromorphic $G$-connection admits a generic (non-degenerate) oper structure. The result is established via degeneration of the moduli space of connections to Higgs bundles and relies on local analysis at formal disks and gauge equivalence arguments.

ABSTRACT

Let G be a connected reductive group and X be a smooth curve over an algebraically closed field of characteristic zero. We show that every meromorphic G-connection on X admits a possibly degenerate oper structure; in particular, every irreducible meromorphic G-connection admits a generic oper structure.

Motivation & Objective

• To establish that every meromorphic $G$-connection on a smooth projective curve admits a gauge-equivalent connection with a possibly degenerate oper structure.
• To show that irreducible meromorphic $G$-connections admit generic (non-degenerate) oper structures.
• To extend the classical result for $\mathrm{GL}(n)$ to arbitrary connected reductive groups $G$.
• To bridge global connections on curves with local oper structures on formal disks via approximation and gauge equivalence.
• To provide a global proof of the local oper structure existence for irreducible connections, using global irreducibility and moduli space techniques.

Proposed method

• Use of the Lie algebra $\mathfrak{g}^{(-1)} = \{x \in \mathfrak{g} : [x,\mathfrak{n}] \subset \mathfrak{b}\}$ to define the space of possibly degenerate opers.
• Reduction of the global problem to local analysis at formal disks using completion at points of the curve.
• Application of gauge transformations via $G(\mathbb{F})$-action to relate arbitrary connections to oper forms.
• Leveraging the Iwasawa decomposition in the local setting to analyze stability under gauge equivalence.
• Using the openness of gauge equivalence classes in the formal setting to construct approximating connections.
• Applying the moduli space degeneration from connections to Higgs bundles to prove the main theorem.

Experimental results

Research questions

• RQ1 Does every meromorphic $G$-connection on a smooth projective curve admit a gauge-equivalent connection in the space of possibly degenerate opers?
• RQ2 Can the oper structure be chosen to be non-degenerate (i.e., generic) for irreducible $G$-connections?
• RQ3 Is the existence of generic oper structures on irreducible connections a consequence of global geometric properties of the moduli space?
• RQ4 How does the local theory of opers on formal disks relate to the global theory on curves?
• RQ5 Can the classical Frobenius normal form construction for $\mathrm{GL}(n)$ be generalized to arbitrary reductive groups?

Key findings

• Every meromorphic $G$-connection on a smooth projective curve over an algebraically closed field of characteristic zero is gauge-equivalent to a possibly degenerate oper.
• Every irreducible meromorphic $G$-connection admits a gauge-equivalent generic oper structure.
• The result extends the classical Frobenius normal form for $\mathrm{GL}(n)$ to arbitrary connected reductive groups.
• The proof relies on the openness of gauge equivalence classes in the formal disk setting and approximation techniques.
• The global existence of generic oper structures for irreducible connections follows from the existence of irreducible local connections and the approximation lemma.
• The main theorem implies the local version (Theorem B) of Frenkel and Zhu, establishing that irreducible connections on formal disks admit generic oper structures.