Stratification of opers with apparent singularities and isomonodromic bubbling

Abstract

On a compact Riemann surface of genus $g \geq 2$, we study Simpson's stratification of the de Rham moduli space of rank-2 flat connections. Upon representing flat $\mathrm{SL}_2(\mathbb{C})$-connections by branched projective structures, or equivalently by opers with apparent singularities, the stratum containing a flat connection is determined by the minimal number of apparent singularities needed to represent it. We construct explicit holomorphic families in which two simple apparent singularities collide and disappear, so that the family crosses into a smaller stratum; this realizes Simpson's closure relation between strata near generic points. The analogous families of Higgs bundles are described by Baker--Akhiezer divisors on the spectral curve in which two points converge to a pair of points exchanged by the spectral involution. We show that, before the collision, the two apparent singularities are joined by twin paths bounding a bubble, so that these families arise from the isomonodromic bubbling surgery discussed by Calsamiglia, Deroin and Francaviglia.