In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of Homological Mirror conjecture and geometric Strominger-Yau-Zaslow conjecture. In particular, we glue from "flat pieces" an analytic K3 surface. As a byproduct of our approach we obtain an action of an arithmetic subgroup of the group $SO(1,18)$ by piecewise-linear transformations on the 2-dimensional sphere $S^2$ equipped with naturally defined singular affine structure.
We study integral dlt models of a proper C((t))-variety X along a toric stratum of the special fiber. We prove that the associated Berkovich retraction - from the non-archimedean analytification of X onto the dual complex of the model - is an affinoid torus fibration around the simplex corresponding to the toric stratum, which extends results from Nicaise-Xu-Yu. This allows us to construct new types of non-archimedean retractions for maximally degenerate families of quartic K3 surfaces and quintic 3-folds, by gluing several non-archimedean SYZ fibrations, each one toric along a codimension one…
Motivated by mirror symmetry and the enumeration of holomorphic disks, we construct the theory of Gromov-Witten invariants in the setting of non-archimedean analytic geometry. We build on our previous works on derived non-archimedean geometry and non-archimedean quantum K-invariants, as well as recent developments of rigid analytic motives and virtual fundamental classes in derived geometry. Our approach gives also a new perspective for the classical algebraic case.
We propose a derived version of non-archimedean analytic geometry. Intuitively, a derived non-archimedean analytic space consists of an ordinary non-archimedean analytic space equipped with a sheaf of derived rings. Such a naive definition turns out to be insufficient. In this paper, we resort to the theory of pregeometries and structured topoi introduced by Jacob Lurie. We prove the following three fundamental properties of derived non-archimedean analytic spaces: (1) The category of ordinary non-archimedean analytic spaces embeds fully faithfully into the $\infty$-category of derived non-arc…
I describe an algebro-geometric theory of skeleta, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine manifolds with singularities. Skeleta are spaces equipped with a structure sheaf of topological semirings, and are locally modelled on the spectra of the same. The primary result of this paper is that the topological space underlying a non-Archimedean analytic space may locally be recovered from the sheaf of `pointwise valuations' of its analytic functions.
Let $U$ be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in $U$ uniquely determine a commutative associative algebra equipped with a compatible multilinear form. This proves a variant of the Frobenius structure conjecture by Gross-Hacking-Keel in mirror symmetry, and the spectrum of this algebra is supposed to give the hypothetical mirror family. Although the statement of our theorem involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constan…
Gromov's compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov's compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct …
We construct non-archimedean SYZ fibrations for maximally degenerate Calabi-Yau varieties, and we show that they are affinoid torus fibrations away from a codimension two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt-models along one-dimensional strata.
We develop the theory of pinchings for non-archimedean analytic spaces. In particular, we show that although pinchings of affinoid spaces do not have to be affinoid, pinchings of Hausdorff analytic spaces always exist in the category of analytic spaces.
For a complex toric variety $X$ the logarithmic absolute value induces a natural retraction of $X$ onto the set of its non-negative points and this retraction can be identified with a quotient of $X(\mathbb{C})$ by its big real torus. We prove an analogous result in the non-Archimedean world: The Kajiwara-Payne tropicalization map is a non-Archimedean analytic stack quotient of $X^{an}$ by its big affinoid torus. Along the way, we provide foundations for a geometric theory of non-Archimedean analytic stacks, particularly focussing on analytic groupoids and their quotients, the process of analy…
We investigate the topological properties of Berkovich analytifications over hybrid fields, that is a field equipped with the maximum of its native norm and the trivial norm. We prove that the analytification of the affine line or of a smooth projective curve over a countable Archimedean hybrid field is contractible, and show that it can be non-contractible when the field is uncountable. Further, we prove that the analytification of affine space over a non-Archimedean hybrid field or over a discrete valuation ring is contractible. As an application, we show that the Berkovich affine line over …
Counts of holomorphic disks are at the heart of the SYZ approach to mirror symmetry. In the non archimedean framework, these counts are expressed as counts of analytic cylinders. In simple cases, such as cluster varieties, these counts can be extracted from a combinatorial algorithm encoded by a scattering diagram. The relevant scattering diagram encodes information about infinitesimal analytic cylinders. In this paper, we give a formula relating counts of a restricted class of cylinders on affine log Calabi Yau surfaces to counts on a single blowup of a toric surface, using non archimedean Gr…
We suggest an analog of the Bass-Quillen conjecture for smooth affinoid algebras over a complete non-archimedean field. We prove this in the rank-1 case, i.e. for the Picard group. For complete discretely valued fields and regular affinoid algebras that admit a regular model (automatic if the residue characteristic is zero) we prove a similar statement for the Grothendieck group of vector bundles.
In this paper we study two types of descent in the category of Berkovich analytic spaces: flat descent and descent with respect to an extension of the ground field. Quite surprisingly, the deepest results in this direction seem to be of the second type, including the descent of properties of being a good analytic space and being a morphism without boundary.
We prove non-Archimedean analogs of results of Noguchi and Winkelmann showing algebraic degeneracy of rigid analytic maps to projective varieties omitting an effective divisor with sufficiently many irreducible components relative to the rank of the group they generate in the Neron-Severi group of the variety.
We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of Berkovich non-archimedean analytic spaces. The motivation for our construction is Tony Yue Yu's non-archimedean enumerative geometry in Gromov-Witten theory. The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson-Thomas invariants. In this paper we give the moduli construction over a non-archimedean field $\kk$. We use the machinery of formal schemes, that is, we d…
Log Calabi-Yau mirror symmetry and non-archimedean disks
Sean Keel, Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257, USA；Logan White, Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257, USA；Tony Yue YU, Department of Mathematics, M/C 253-37, Caltech, 1200 E.\ California Blvd., Pasadena, CA 91125, USA
We study the tropicalizations of analytic subvarieties of normal toric varieties over complete non-archimedean valuation fields. We show that a Zariski closed analytic subvariety of a normal toric variety is algebraic if its tropicalization is a finite union of polyhedra. Previously, the converse direction was known by the theorem of Bieri and Groves. Over the field of complex numbers, Madani, L. Nisse, and M. Nisse proved similar results for analytic subvarieties of tori.
Let $K$ be a complete algebraically closed non-archimedean valued field of characteristic zero, and let $X$ be a finite type scheme over $K$. We say $X$ is $K$-analytically Borel hyperbolic if, for every finite type reduced scheme $S$ over $K$, every rigid analytic morphism from the rigid analytification $S^{\mathrm{an}}$ of $S$ to the rigid analytification $X^{\mathrm{an}}$ of $X$ is algebraic. Using the Viehweg-Zuo construction and the $K$-analytic big Picard theorem of Cherry-Ru, we show that, for $N \geq 3$ and $g \geq 2$, the fine moduli space $\mathcal{M}^{[N]}_{g,K}$ over $K$ of genus $…
Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semi distance dCK that he introduced for analytic spaces defined over a non-Archimedean metrized field k. We prove various characterizations of smooth projective varieties for which dCK is an actual distance. Secondly, we explore several notions of hyperbolicity for a smooth algebraic curve X defined over k. We prove a non-Archimedean analogue of the equivalence between having negative Euler characteristic and the normality of certain families of analytic maps taking values in X.
We introduce a variant of homotopy K-theory for Tate rings, which we call analytic K-theory. It is homotopy invariant with respect to the analytic affine line viewed as an ind-object of closed disks of increasing radii. Under a certain regularity assumption we prove an analytic analog of the Bass fundamental theorem and we compare analytic K-theory with continuous K-theory, which is defined in terms models. Along the way we also prove some results about the algebraic K-theory of Tate rings.
Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let $\mathscr{X}$ be a $K$-analytic space (in the sense of Huber). In this work, we pursue a non-Archimedean characterization of Campana's notion of specialness. We say $\mathscr{X}$ is $K$-analytically special if there exists a connected, finite type algebraic group $G/K$, a dense open subset $\mathscr{U}\subset G^{\text{an}}$ with $\text{codim}(G^{\text{an}}\setminus \mathscr{U}) \geq 2$, and an analytic morphism $\mathscr{U} \to \mathscr{X}$ which is Zariski dense. With this definition, we…
We associate a weight function to pairs consisting of a smooth and proper variety X over a complete discretely valued field and a differential form on X of maximal degree. This weight function is a real-valued function on the non-archimedean analytification of X. It is piecewise affine on the skeleton of any regular model with strict normal crossings of X, and strictly ascending as one moves away from the skeleton. We apply these properties to the study of the Kontsevich-Soibelman skeleton of such a pair, and we prove that this skeleton is connected when X has geometric genus one. This result …
Let $k$ be a discretely valued non-Archimedean field. We give an explicit description of analytic functions whose norm is bounded by a given real number $r$ on tubes of reduced $k$-analytic spaces associated to special formal schemes (those include $k$-affinoid spaces as well as open polydiscs). As an application we study the connectedness of these tubes. This generalizes (in the discretely valued case) a result of Siegfried Bosch. We use as a main tool a result of A.J. de Jong relating formal and analytic functions on special formal schemes and a generalization of de Jong's result which is pr…
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 …
Let $K$ be a complete non-trivially valued non-Archimedean field. Given an algebraic group over $K$ on which every regular function is constant, any rigid analytic function is shown to be constant too. It follows that an algebraic group over $K$ is affine if and only if the associated $K$-analytic space is Stein; that is, rigid analytic embeddings of it in an affine space may always be chosen to be given by algebraic functions. Arguably curiously, the corresponding statement over the complex numbers is false.
We show an example of a non-archimedean version of the Calabi-Yau theorem in complex geometry. Precisely, we consider totally degenerate abelian varieties and certain probability measures on their associated analytic spaces in the sense of Berkovich.
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…
It is now a classical result that an algebraic space locally of finite type over $\mathbf{C}$ is analytifiable if and only if it is locally separated. In this paper we study non-archimedean analytifications of algebraic spaces. We construct a quotient for any etale non-archimedean analytic equivalence relation whose diagonal is a closed immersion, and deduce that any separated algebraic space locally of finite type over any non-archimedean field $k$ is analytifiable in both the category of rigid spaces and the category of analytic spaces over $k$. Also, though local separatedness remains a nec…
We study the topology of the punctured disc defined over a non-archimedean field of characteristic zero. Chapter two includes a new proof of the so-called p-adic Riemann existence theorem. This release completes the study of breaks and break decompositions of the monodromy representation of a sheaf around the origin of the punctured disc.
We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of Toën--Vaquié--Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we can define a topology on certain subcategories of the of the category of affine schemes with respect to this category. By examining this topology for the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in analytic geometry. This gives a functor of points approach to non-Archimedean analytic geo…
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 …
Let $V$ be a complete discrete valuation ring with fraction field $F$ of characteristic zero and with residue field $\mathbb{F}$. We introduce analytic cyclic homology of complete torsion-free bornological algebras over $V$. We prove that it is homotopy invariant, stable, invariant under certain nilpotent extensions, and satisfies excision. We use these properties to compute it for tensor products with dagger completions of Leavitt path algebras. If $R$ is a smooth commutative $V$-algebra of relative dimension $1$, then we identify its analytic cyclic homology with Berthelot's rigid cohomology…
We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic informations of the underlying classical moduli space. The main tool in our construction is the representability theorem in derived analytic geometry, which has been established in our previous work. The representability theorem provides us sufficient and necessary conditions for an analytic moduli functor to possess the structure of a derived analytic stack. In order to verify the conditions of the representability theorem, we prove several general results in the context…
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…
We define a new type of valuation of a ring that combines the notion of Krull valuation with that of multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic space. After defining natural generalized valuation spectra and computing the spectrum of Z and Z[X], we define analytic spectra and sheaves of analytic functions on them.
Let $u$ be a nowhere vanishing holomorphic function on the Drinfeld space $Ω^{r}$ of dimension $r-1$, where $r \geq 2$. The logarithm $\log_{q}\lvert u \rvert$ of its absolute value may be regarded as an affine function on the attached Bruhat-Tits building $\mathcal{BT}^{r}$. Generalizing a construction of van der Put in case $r=2$, we relate the group $\mathcal{O}(Ω^{r})^{*}$ of such $u$ with the group $\mathbf{H}(\mathcal{BT}^{r}, \mathbb{Z})$ of integer-valued harmonic 1-cochains on $\mathcal{BT}^{r}$. This also gives rise to a natural $\mathbb{Z}$-structure on the first ($\ell$-adic or de …
We prove a version of Montel's theorem for analytic functions over a non-archimedean complete valued field. We propose a definition of normal family in this context, and give applications of our results to the dynamics of non-archimedean entire functions.
A non-Archimedean analog of the classical Big Picard Theorem, which says that a holomorphic map from the punctured disc to a Riemann surface of hyperbolic type extends accross the puncture, is proven using Berkovich's theory of non-Archimedean analytic spaces.
Peterzil and Starchenko have proved the following surprising generalization of Chow's theorem: A closed analytic subset of a complex algebraic variety that is definable in an o-minimal structure, is in fact an algebraic subset. In this paper, we prove a non-archimedean analogue of this result.
This paper provides an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker, Payne and Rabinoff in the case of curves, we explain a result by Cueto, Häbich and the author comparing the tropical Grassmannian of planes to the analytic Grassmannian. We also give an overview of the general higher-dimensional theory developed by Gubler, Rabinoff and the author. In particular, we explain the construction of generalized skeleta in which are polyhedral substructures of Berkovich…
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…
