Tangent discontinuity in the oper stratification of de Rham moduli spaces

Abstract

Let $X$ be a smooth complex projective curve of genus $g$, and let $\mathcal{M}_{\mathrm{dR}}(X,r)$ be the moduli space of flat bundles of rank $r$. Over the stable locus, Simpson showed the oper stratification with Lagrangian fibers and asked whether these fibers are closed and fit together into a smooth foliation. This question is often referred to as the foliation conjecture. In this paper, we give a counterexample to this conjecture in rank two on every curve of genus $g\geq4$. The main idea is to show that the tangent planes are discontinuous along a holomorphic curve crossing two adjacent strata, hence the foliation assertion fails.