Simpson Filtration and Oper Stratum Conjecture

TL;DR

This paper establishes the unique minimal and maximal strata in the oper stratification of the de Rham moduli space, characterizing their dimensions and properties within the space of flat bundles.

Contribution

It proves the uniqueness and dimension of the minimal and maximal strata in the oper stratification of the de Rham moduli space.

Findings

The closed oper stratum is the unique minimal stratum.

The open dense stratum consists of irreducible flat bundles with stable underlying bundles.

Dimensions of the minimal and maximal strata are explicitly determined.

Abstract

In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum.