Irregular-singular connection

Let $X$ be a smooth complex curve, let $D\subset X$ be a finite set, and write $U=X\setminus D$ . A meromorphic flat connection on $X$ , with poles along $D$ , has an irregular singularity at $x\in D$ if it is not regular singular there.

Formal description

After a finite ramified cover and a formal gauge transformation, an irregular connection decomposes into pieces containing nonconstant exponential factors. These factors, together with formal monodromy, describe its formal type.

Analytic classification

Formal type does not determine the analytic connection. Sectorial fundamental solutions differ by Stokes data, which record the missing analytic information.

References

Philip Boalch, “Geometry and braiding of Stokes data; fission and wild character varieties,” Annals of Mathematics 179 (2014), 301–365. arXiv.