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.

Disclosure

“emain positive, while the Higgs field must have a zero. These conditions are e ≥ 2 and degZ(θ) = 2g − 2 − 2e > 0. Acknowledgements. The author initially aimed to prove the foliation conjecture over compact curves with the assistance of OpenAI’s ChatGPT (versions 5.5 and 5.6 Plus), but did not receive a valuable approach even after hundreds of chats. Finally, AI instead suggested to examine whether the foliation assertion might be false, given its strength. This change of perspective led”

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Suggesting mathematical examples or conjectures

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2026-08-23T05:02:37.690325Z

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