Logarithmic Non-Abelian Hodge Theory for curves in prime characteristic
Mark Andrea de Cataldo, Siqing Zhang
TL;DR
This work introduces a logarithmic, characteristic-$p$ analogue of non-abelian Hodge theory for curves, termed Log-$p$-NAHT, which relates de Rham moduli of logarithmic $G$-connections on a curve $C$ to Dolbeault moduli of logarithmic $G$-Higgs bundles on the Frobenius twist $C'$. A key novelty is that, unlike the no-pole case, the two moduli stacks are not étale locally isomorphic over the Hitchin base; instead they are linked by an Artin–Schreier-type Galois cover of a refined base $ rak cA$ parameterizing residues. The authors construct a global pseudo-torsor $\\mathscr{H}$ over $ rak cA$ and a twisted product morphism $oldsymbol n$ that realizes a Log-$p$-NAHT isomorphism over the image $ rak cA_{ ext{im}}$, with a semistable refinement when $(p,G)$ satisfies good-height hypotheses. In the $G=GL_r$ case, this yields a cohomological embedding of the Dolbeault intersection cohomology into the de Rham side for degree $pd$ and, when $ ext{gcd}(p d,r)=1$, an isomorphism of filtered cohomologies; the work further situates Log-$p$-NAHT within broader comparisons to complex-analytic Log-NAHT, higher-parahoric frameworks, and recent tame/logarithmic developments, while establishing weak Abelian fibration structures and substantial geometric and cohomological consequences in prime characteristic.
Abstract
For a curve C and a reductive group G in prime characteristic, we relate the de Rham moduli of logarithmic G-connections on C to the Dolbeault moduli of logarithmic G-Higgs bundles on the Frobenius twist of C. We name this result the Log-p-NAHT. It is a logarithmic version of Chen-Zhu's characteristic p Non Abelian Hodge Theorem (p-NAHT). In contrast to the no pole case, the two moduli stacks in the log case are not isomorphic etale locally over the Hitchin base. Instead, they differ by an Artin-Schreier type Galois cover of the base. In contrast to the case over the complex numbers, where some parabolic/parahoric data are needed to specify the boundary behavior of the tame harmonic metrics, no parabolic/parahoric data are needed in Log-p-NAHT. We also establish a semistable version of the Log-p-NAHT, and deduce several geometric and cohomological consequences. In particular, when G=GL_r, the Log-p-NAHT induces an embedding of the intersection cohomology of the degree d Dolbeault moduli to that of the degree pd de Rham moduli, and the embedding is an isomorphism when r is coprime to d and p>r.