Twisted calculus

Curvature formula for direct images of twisted relative canonical bundles endowed with a singular metric

In this note, we obtain various formulas for the curvature of the $L^2$ metric on the direct image of the relative canonical bundle twisted by a holomorphic line bundle endowed with a positively curved metric with analytic singularities, generalizing some of Berndtsson's seminal results in the smooth case. When the twist is assumed to be relatively big, we further provide a very explicit lower bound for the curvature of the $L^2$ metric.

多变量中的扭曲微积分

Twisted calculus in several variables

本文引入多变量扭曲微分算子的新概念，建立形式框架，证明扭曲连接模与扭曲导数作用模的等价性，并将Le Stum和Quirós的汇合定理推广到多变量情形。

Twisted calculus on affinoid algebras

We introduce the notion of a twisted differential operator of given radius relative to an endomorphism $$σ$$ of an affinoid algebra A. We show that this notion is essentially independent of the choice of the endomorphism $$σ$$. As a particular case, we obtain an explicit equivalence between modules endowed with a usual integrable connection (i.e. differential systems) and modules endowed with a $$σ$$-connection of the same radius (i.e. q-difference systems). Moreover, this equivalence preserves cohomology and in particular solutions.

多变量扭曲微分算子

Twisted differential operators in several variables

该研究引入新概念构建多变量扭曲微分算子的一般形式体系，研究Huber环上扭曲坐标以构造并比较相关算子环，证明两类模的等价性，契合$p$-进霍奇上同调与棱柱上同调的最新进展。

Twisted differential operators of negative level and prismatic crystals

We introduce twisted differential calculus of negative level and prove a descent theorem: Frobenius pullback provides an equivalence between finitely presented modules endowed with a topologically quasi-nilpotent twisted connection of level minus one and those of level zero. We explain how this is related to the existence of a Cartier operator on prismatic crystals. For the sake of readability, we limit ourselves to the case of dimension one.

Twisted differential operators and $q$-crystals

We discuss the notion of a q-PD-envelope considered by Bhatt and Scholze in their recent theory of q-crystalline cohomology and explain the relation with our notion of a divided polynomial twisted algebra. Together with an interpretation of crystals on the q-crystalline site, that we call q-crystals, as modules endowed with some kind of stratification, it allows us to associate a module on the ring of twisted differential operators to any q-crystal. For simplicity, we explain here only the one dimensional case.