Regular-singular connection

Definition

Let $U=X\setminus D$ be the complement of finitely many points in a smooth complex algebraic curve. A flat connection $(V,\nabla)$ on $U$ is regular singular along $D$ if, locally at every $x\in D$ , there is a locally free extension $\overline V$ across $x$ for which

\nabla:\overline V\longrightarrow \overline V\otimes\Omega_X^1(D)

has at most a logarithmic pole. Equivalently, its horizontal sections have moderate growth near each puncture.

For a principal $G$ -connection, regular singularity can be tested after a faithful linear representation of $G$ .

Riemann–Hilbert consequence

Regular-singular algebraic flat connections correspond analytically to local systems on $U$ . Thus ordinary monodromy records the connection on the punctured curve, although choosing a logarithmic extension across $D$ may require additional extension data.

References

Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics 163, Springer, 1970. DOI.