arXiv 1707.007522017-07math.AGmath.NT

刚性连接和$F$ -同晶

Rigid connections and $F$-isocrystals

Hélène Esnault, Michael Groechenig

备注：: final version, accepted in Acta Mathematica

摘要：

光滑射影复簇$X$上的不可约可积联络$(E,\nabla)$称为刚性联络，如果它产生相应模空间$\mathcal{M}_{dR}(X)$的孤立点。根据Simpson的动机猜想，不可约刚性平坦联络是几何起源的，也就是说，它是作为定义在$X$的开稠密子簇上的光滑投射簇族的Gaussian-Manin联络的子簇而产生的。在这篇文章中，我们研究模$p$减少不可约刚性连接，并建立结果证实辛普森的预测。特别是，对于大的$p$，我们证明了$p$ -曲率模$p$减少不可约刚性平坦连接是幂零的，并建立在这个结果，我们构建了一个$F$ -isocrystallization {不可约}刚性平坦连接。更精确地说，我们证明了在有限型环$R$上存在$X$和$(E,\nabla)$的光滑模型$X_R$和$(E_R,\nabla_R)$，使得对于有限域$k$的每个Witt环$W(k)$和每个同态$R \to W(k)$，$\widehat{X}_{W(k)}$上基变化$(\widehat{E}_{W(k)},\widehat{\nabla}_{W(k)})$的$p$ -adic完成表示$F$ -同晶.随后我们证明了$p$ -曲率为零的{不可约}刚性平坦联络是酉的。这使我们能够证明新的情况下的Grothendieck-Katz $p$ -曲率猜想。我们还证明了存在一个完整的同伴对应$F$ -isocrystals源于不可约cohomologically刚性连接。

英文摘要：

An irreducible integrable connection $(E,\nabla)$ on a smooth projective complex variety $X$ is called rigid if it gives rise to an isolated point of the corresponding moduli space $\mathcal{M}_{dR}(X)$. According to Simpson's motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gauß-Manin connection of a family of smooth projective varieties defined on an open dense subvariety of $X$. In this article we study mod $p$ reductions of irreducible rigid connections and establish results which confirm Simpson's prediction. In particular, for large $p$, we prove that $p$-curvatures of mod $p$ reductions of irreducible rigid flat connections are nilpotent, and building on this result, we construct an $F$-isocrystalline realization for {irreducible} rigid flat connections. More precisely, we prove that there exist smooth models $X_R$ and $(E_R,\nabla_R)$ of $X$ and $(E,\nabla)$, over a finite type ring $R$, such that for every Witt ring $W(k)$ of a finite field $k$ and every homomorphism $R \to W(k)$, the $p$-adic completion of the base change $(\widehat{E}_{W(k)},\widehat{\nabla}_{W(k)})$ on $\widehat{X}_{W(k)}$ represents an $F$-isocrystal. Subsequently we show that {irreducible} rigid flat connections with vanishing $p$-curvatures are unitary. This allows us to prove new cases of the Grothendieck--Katz $p$-curvature conjecture. We also prove the existence of a complete companion correspondence for $F$-isocrystals stemming from irreducible cohomologically rigid connections.

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