On Liu morphisms in non-Archimedean geometry
Mingchen Xia
本文定义了Berkovich分析空间间的Liu态射与准Liu态射，并证明其行为与概形中的仿射态射和准仿射态射一致。
我们定义了Berkovich分析空间之间的Liu态射和准Liu态射。我们证明Liu态射和准Liu态射的行为与概形中的仿射态射和准仿射态射完全一致。
We define Liu morphisms and quasi-Liu morphisms between Berkovich analytic spaces. We show that Liu morphisms and quasi-Liu morphisms behave exactly as affine morphisms and quasi-affine morphisms of schemes.
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An algebraic characterisation of non-Archimedean Stein spaces
本文通过引入Liu代数和非阿基米德Stein代数，给出了非阿基米德Stein空间的代数刻画，并解决了两个关于仿射性和Stein空间的猜想。
Let $ϕ: C' \to C$ be a finite morphism between smooth, projective, irreducible curves defined over a non-archimedean valued, algebraically closed field $k$. This morphism induces a morphism between the analytifications of the curves. We will construct a compatible pair of deformation retractions of $C'^{an}$ and $C^{an}$ whose images $Υ_{C'^{an}}$ and $Υ_{C^{an}}$ are closed subspaces of $C'^{an}$ and $C^{an}$ which are homeomorphic to finite metric graphs. We refer to such closed subspaces as skeleta. In addition, the subspaces $C'^{an}$ and $C^{an}$ are such that their complements in the two…
In 2010, Hrushovski--Loeser showed that the Berkovich analytification of a quasi-projective variety over a non-Archimedean valued field admits a deformation retraction onto a finite simplicial complex. In this article, we adapt the tools and methods developed by Hrushovski--Loeser to study if such deformation retractions can be obtained to be compatible with respect to a given morphism. Amongst other results, we show that compatible deformation retractions exist over a constructible partition of the base and prove the general statement in the case of a morphism of relative dimension 1 where th…
Let $k$ be a non-archimedean complete valued field and $X$ be a $k$-analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: 1) for every complete valued extension $k'$ of $k$, every coherent sheaf on $X \times_{k} k'$ is acyclic; 2) $X$ is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex) and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, $X$ is compact); 3) $X$ admits a suitable exhaustion by compact analytic domains considered by Liu in his …
The goal of the current text is to study non-archimedean analytic derived de Rham cohomology by means of formal completions. Our approach is inspired by the deformation to the normal cone provided in \cite{Gaitsgory_Study_II}. More specifically, given a morphism $f \colon X \to Y$ of (derived) $k$-analytic spaces we construct the \emph{non-archimedean deformation to the normal cone} associated to $f$. The latter can be thought as an $\mathbf A^1_k$-parametrized deformation whose fiber at $1 \in \mathbf A^1_k$ coincides with the formal completion of $f$ and the fiber at $0 \in \mathbf A^1_k$ wi…
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Non-Archimedean entire curves in projective varieties dominating an elliptic curve
本文针对允许支配态射到椭圆曲线的无群射影曲面，证明其 K-解析 Brody 双曲性，从而建立不规则度为一情形的非阿基米德 Green–Griffiths–Lang 猜想。
Non-archimedean integrals as limits of complex integrals
本文利用同时具有阿基米德与非阿基米德范数的复数非标准模型，证明两类形式双复形间存在与积分相容的自然态射，从而说明非阿基米德积分可作为复积分族渐近中的极限出现。
We present several results on the compactness of the space of morphisms between analytic spaces in the sense of Berkovich. We show that under certain conditions on the source, every sequence of analytic maps having an affinoid target has a subsequence that converges pointwise to a continuous map. We also study the class of continuous maps that arise in this way. Locally, they turn analytic after a certain base change. We give some applications of these results to the dynamics of an endomorphism f of the projective space. We define the Fatou set as the normality locus of the family of the itera…
In this paper we study skeleta of residually tame coverings of a marked curve over a non-archimedean field. We first generalize a result by Liu and Lorenzini by proving a simultaneous semistable reduction theorem for residually tame coverings. We then use this to construct a functor from the category of residually tame coverings of a marked curve $(X,D)$ to the category of tame coverings of a metrized complex $Σ$ associated to $(X,D)$. We enhance the latter category by adding a set of gluing data to every covering and we show that this yields an equivalence of categories. Using this equivalenc…
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Let $k$ be a complete, nontrivially valued non-archimedean field. Given a finite morphism of quasi-smooth $k$-analytic curves that admit finite triangulations, we provide upper bounds for the number of connected components of the ramification locus in terms of topological invariants of the source curve such as its topological genus, the number of points in the boundary and the number of open ends.
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Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field $K$ of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is $K$-analytically Brody hyperbolic in equal characteristic zero. These two results are predicted by the Green-Griffiths-Lang conjecture on…
We construct quantum K-invariants in non-archimedean analytic geometry. Contrary to the classical approach in algebraic geometry via perfect obstruction theory, we build on our previous works on the foundations of derived non-archimedean geometry, the representability theorem and Gromov compactness. We obtain a list of natural geometric relations between the stacks of stable maps, directly at the derived level, with respect to elementary operations on graphs, namely, products, cutting edges, forgetting tails and contracting edges. They imply immediately the corresponding properties of quantum …
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Let $X$ be a smooth geometrically connected projective curve of genus two over a complete non-archimedean field $K$. For discretely valued $K$, the first main theorem in \cite{liu} gives a set of criteria on the Igusa invariants of the curve that determine the minimal Berkovich skeleton of $X$ together with its edge lengths and vertex weights. In this paper we use the theory of Berkovich spaces to give a new proof of this theorem that works for arbitrary complete non-archimedean fields. We furthermore interpret the final result in terms of tropical moduli spaces and tropical Igusa invariants. …
We associate to an analytic subvariety of a torus a tropical variety. In the first part, we generalize the results from tropical algebraic geometry to this non-archimedean analytic situation. The periodic case is applied to a totally degenerate abelian variety A. We obtain a dimensionality bound for the range of an algebraic morphism from a smooth variety Y to A in terms of the singularities of a strictly semistable model of Y. The main result is that Chambert-Loir's canonical measures on a closed subvariety of A induce piecewise Haar measures on the associated tropical variety and we give an …
Rational motives on pro-algebraic stacks
该研究定义了前代数栈的有理动机∞-范畴，证明其对特定态射类有6函子形式体系且在前概形上符合Liu-Zheng定义，为不可表示代数栈和非有限型态射的动机理论提供新途径。
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本文研究abelian 变种上的半同构向量束的模空间及其热带化，通过非阿基米德统一化视角，建立本质骨架与热带模空间的联系，并构造从解析基本群字符空间到模空间的覆盖映射。
In this article, we apply the approach of relative algebraic geometry towards analytic geometry to the category of bornological and Ind-Banach spaces (non-Archimedean or not). We are able to recast the theory of Grosse-Klönne dagger affinoid domains with their weak G-topology in this new language. We prove an abstract recognition principle for the generators of their standard topology (the morphisms appearing in the covers). We end with a sketch of an emerging theory of dagger affinoid spaces over the integers, or any Banach ring, where we can see the Archimedean and non-Archimedean worlds com…
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Proper pushforwards on analytic adic spaces
本文为解析仿射空间的部分固有态射构造固有推前映射，推广了van der Put的相关理论，还针对Kiehl意义下光滑且部分固有的态射构造了迹映射与对偶配对。
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本文利用形式模型与相对Riemann-Zariski空间，为刚性解析空间的结构态射构造了Huber万有紧化的新方法。
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$G$-gerbes on perfectoid spaces
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Let $X$ be an analytic space over a non-Archimedean, complete field $k$ and let $(f_1,..., f_n)$ be a family of invertible functions on $X$. Let $ϕ$ the morphism $X\to G_m^n$ induced by the $f_i$'s, and let $t$ be the map $X\to (R^*_+)^n$ induced by the norms of the $f_i$'s. Let us recall two results. 1) The compact set $t(X)$ is a polytope of the $R$-vector space $(R^*_+)^n$ (we use the multiplicative notation) ; this is due to Berkovich in the locally algebraic case, and has been extended to the general case by the author. 2) If moreover $X$ is Hausdorff and $n$-dimensional, then the pre-ima…
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