The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as well. A classical example of the latter is a complete rank one valued fields with the Tate algebra of strictly convergent power series. The algebraic case was treated in our previous papers. Here we are going to carry over the research to the general analytic settings. Also considered are the local rings of analytic function germs induced by a given separated Weierstrass system, which turn out to be excellent and regular. Several results are established as, for instance, piecewise continuity of definable functions, several versions of the Łojasiewicz inequality, Hölder continuity of definable functions continuous on closed bounded subsets of the affine space and curve selection for definable sets. Likewise as before, at the center of our approach is the closedness theorem to the effect that every projection with closed bounded fiber is a definably closed map. It enables application of resolution of singularities and of transformation to a normal crossing by blowing up (here applied to the local rings of analytic function germs) in much the same way as over locally compact ground fields. We rely on elimination of valued field quantifiers, term structure of definable functions and b-minimal cell decomposition, due to Cluckers--Lipshitz--Robinson, as well as on relative quantifier elimination for ordered abelian groups, due to Cluckers--Halupczok. Besides, other two ingredients of the proof of the closedness theorem are existence of the limit (after finite partitioning of the domain) for a definable function of one variable and fiber shrinking, being a relaxed version of curve selection.
We overview some of the foundations of the so-called henselian rigid geometry, and show that henselian rigid geometry has many aspects, useful in applications, that one cannot expect in the usual rigid geometry. This is done by announcing a few characteristic results, one of which is an analogue of Zariski Main Theorem.
The main purpose of the paper is to establish a closedness theorem over Henselian valued fields $K$ of equicharacteristic zero (not necessarily algebraically closed) with separated analytic structure. It says that every projection with a projective fiber is a definably closed map. This remains valid also for valued fields with analytic structure induced by a strictly convergent Weierstrass systems, including the classical, complete rank one valued fields with the Tate algebra of strictly convergent power series. As application, we prove two theorems on existence of the limit and on piecewise c…
Let $K$ be a Henselian, non-trivially valued field with separated analytic structure. We prove the existence of definable retractions onto an arbitrary closed definable subset of $K^{n}$. Hence directly follow definable non-Archimedean versions of the extension theorems by Tietze--Urysohn and Dugundji. This generalizes our previous paper dealing with complete non-Archimedean fields with separated power series and remains true for Henselian valued fields with strictly convergent analytic structure, because every such a structure can be extended in a definitional way to a separated analytic stru…
This paper develops algebraic geometry over Henselian real valued (i.e. of rank 1) fields $K$, being a sequel to our paper about that over Henselian discretely valued fields. Several results are given including: a certain concept of fiber shrinking (a relaxed version of curve selection) for definable sets, the canonical projection $K^{n} \times K\mathbb{P}^{m} \to K^{n}$ and blow-ups of the $K$-points of smooth $K$-varieties are definably closed maps, a descent property for blow-ups, a version of the Lojasiewicz inequality for continuous rational functions and the theorem on extending continuo…
We develop geometry of affine algebraic varieties in $K^{n}$ over Henselian rank one valued fields $K$ of equicharacteristic zero. Several results are provided including: the projection $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$ and blow-ups of the $K$-rational points of smooth $K$-varieties are definably closed maps, a descent property for blow-ups, curve selection for definable sets, a general version of the Łojasiewicz inequality for continuous definable functions on subsets locally closed in the $K$-topology and extending continuous hereditarily rational functions, established for the real…
We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem that the projections $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$ are definably closed maps. It enables application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses i.a. the local behavior of definable functions of one variable and fiber shrinki…
We are concerned with rigid analytic geometry in the general setting of Henselian fields $K$ with separated analytic structure, whose theory was developed by Cluckers--Lipshitz--Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore the usual global resolution of singularities from rigid analytic ge…
This paper is concerned with algebraic geometry over complete discretely valued fields $K$ of equicharacteristic zero. Several results are given including: the canonical projection $K^{n} \times K\mathbb{P}^{m} \longrightarrow K^{n}$ and blow-ups of the $K$-points of smooth $K$-varieties are definably closed maps, a descent principle, a version of the Lojasiewicz inequality for continuous rational functions, curve selection for semialgebraic sets and the theorem on extending continuous hereditarily rational functions, established for the real field in our joint paper with J. Kollar. Our approa…
The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over $\FF_q((t))$ of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields $K$ with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over $K$. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper…
Let R be a Henselian discrete valuation ring with field of fractions K. If X is a smooth variety over K and G a torus over K, then we consider X-torsors under G. If XX/R is a model of X then, using a result of Brahm, we show that X-torsors under G extend to XX-torsors under a Néron model of G if G is split by a tamely ramified extension of K. It follows that the evaluation map associated to such a torsor factors through reduction to the special fibre. In this way we can use the geometry of the special fibre to study the arithmetic of X.
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 …
This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry.
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 give a formulation for derived analytic geometry built from commutative differential graded algebras equipped with entire functional calculus on their degree 0 part, a theory well-suited to developing shifted Poisson structures and quantisations. In the complex setting, we show that this formulation recovers equivalent derived analytic spaces and stacks to those coming from Lurie's structured topoi. In non-Archimedean settings, there is a similar comparison, but for derived dagger analytic spaces and stacks, based on overconvergent functions.
Geometry over non--existent "field with one element" $F_1$ conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in at least two related but different guises. In this paper I analyze the crucial role of roots of unity in this geometry and propose a version of the notion of "analytic functions" over $F_1$. The paper combines a focused survey with some new constructions. In new version, several local additions and changes are made, references added.
We give conclusive answers to some questions about definability in analytic languages that arose shortly after the work by Denef and van den Dries, [DD], on $p$-adic subanalytic sets, and we continue the study of non-archimedean fields with analytic structure of [LR3], [CLR1] and [CL1]. We show that the language $L_K$ consisting of the language of valued fields together with all strictly convergent power series over a complete, rank one valued field $K$ can be expanded, in a definitial way, to a larger language corresponding to an analytic structure (with separated power series) from [CL1], he…
The tempered fundamental group of a p-adic analytic space classifies coverings that are dominated by a topological covering (for the Berkovich topology) of a finite etale covering of the space. Here we construct cospecialization homomorphisms between (p') versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log reduction. In particular, we will have to study invariance of the geometric log fundamental group of…
We define a notion of global analytic space with overconvergent structure sheaf. This gives an analog on a general base Banach ring of Grosse-Kloenne's overconvergent p-adic spaces and of Bambozzi's generalized affinoid varieties over R. This also gives an affinoid version of Berkovich's and Poineau's global analytic spaces. This affinoid approach allows the introduction of a notion of strict global analytic space, that has some relations with the ideas of Arakelov geometry, since the base extension along the identity morphism on Z (from the archimedean norm to the trivial norm) sends a strict…
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.
In the first half of twentieth century the theory of complex analytic functions and of their zerosets was fully developed. The definition of holomorphic function has a local nature. Germs of holomorphic functions form a distinguished subring of the ring of germs of continuous functions. This way came out the notion of analytic space. The definition of a complex analytic set is by local models as in the case of complex manifolds. But while local models for manifolds are open sets of ${\mathbb C}^n$, a local model of an analytic space is the zeroset of finitely many analytic functions on an open…
This paper constructs cospecialization homomorphisms between the (p') versions of the tempered fundamental group of the fibers of a smooth morphism with polystable reduction (the tempered fundamental group is a sort of analog of the topological fundamental group of complex algebraic varieties in the p-adic world). We studied the question for families of curves in another paper. To construct them, we will start by describing the pro-(p') tempered fundamental group of a smooth and proper variety with polystable reduction in terms of the reduction endowed with its log structure, thus defining tem…
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…
We describe the Lipschitz geometry of complex curves. For the most part this is well known material, but we give a stronger version even of known results. In particular, we give a quick proof, without any analytic restrictions, that the outer Lipschitz geometry of a germ of a complex plane curve determines and is determined by its embedded topology. This was first proved by Pham and Teissier, but in an analytic category.
We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps app…
The goal of the present text is to state and prove a generalization of Raynaud localization theorem in the setting of derived geometry. More explicitly, we show that the $\infty$-category of quasi-paracompact and quasi-separated derived $k$-analytic spaces can be realized as a localization of the $\infty$-category of admissible derived formal schemes. We construct a derived rigidification functor generalizing Raynaud rigidification functor. In order to construct the latter we will need to formalize derived formal $\mathcal{O}_k$-adic formal geometry via a structured spaces approach. We prove t…
I extend the framework of rigid analytic geometry to the setting of algebraic geometry relative to monoids, and study the associated notions of separated, proper, and overconvergent morphisms. The category of affine manifolds embeds as a subcategory defined by simple algebraic (normal) and topological (overconvergent) criteria. The affine manifold of a rigid space can be recovered either as a set of `Novikov field' points or as a universal Hausdorff quotient. After base change to any topological field, one obtains a `toric' analytic space that fibres over the affine manifold.
This paper describes an algorithm for determining the branching geometry of algebraic functions. The graphs of these complex-valued functions have a complicated interweaving structure that can be described by analytic branches separated by singular points. Power expansions for the branches in discs centered at a point can be computed using the Newton Polygon method, and expansions around annular regions centered at the origin computed using a version of Laurent's Theorem applied to algebraic functions. However, neither of the methods enable a determination of the region of convergence of the p…
We present a new notion of decomposition of semialgebraic sets by introducing a mode of irreducibility based on arc-analytic functions. The result is a refinement of the decomposition of such sets with respect to the Zariski topology as well as a refinement of the decomposition in each of the recent approaches based on Nash and continuous rational functions. In addition, by pairing the ring of arc-analytic functions with semialgebraic sets, we obtain a theory of algebraic geometry equipped with strong tools such as the Identity Principle and the Nullstellensatz.
We develop the foundations of higher geometric stacks in complex analytic geometry and in non-archimedean analytic geometry. We study coherent sheaves and prove the analog of Grauert's theorem for derived direct images under proper morphisms. We define analytification functors and prove the analog of Serre's GAGA theorems for higher stacks. We use the language of infinity category to simplify the theory. In particular, it enables us to circumvent the functoriality problem of the lisse-étale sites for sheaves on stacks. Our constructions and theorems cover the classical 1-stacks as a special ca…
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…
We use Kiehl-Verdier's and Houzel's finiteness theorems in the setting of local analytic geometry, and the Whitney-Thom theory of stratified spaces, to prove that fibrewise constructible complex of sheaves have coherent direct images. We review the notions of stratification theory which are needed.
We continue the explorations of derived \canal geometry started in [DAG-IX] and in http://arxiv.org/abs/1506.09042. We describe the category of $\mathcal O_X$-modules over a derived complex analytic space $X$ as the stabilization of a suitable category of analytic algebras over $\mathcal O_X$. Finally, we apply this description to introduce the notion of analytic square-zero extension and prove a fundamental structure theorem for them.
We prove that the outer Lipschitz geometry of a germ $(X,0)$ of a normal complex surface singularity determines a large amount of its analytic structure. In particular, it follows that any analytic family of normal surface singularities with constant Lipschitz geometry is Zariski equisingular. We also prove a strong converse for families of normal complex hypersurface singularities in $\mathbb C^3$: Zariski equisingularity implies Lipschitz triviality. So for such a family Lipschitz triviality, constant Lipschitz geometry and Zariski equisingularity are equivalent to each other.
If $X$ is a smooth scheme of characteristic zero or a complex analytic manifold, and $S$ is a locally split infinitesimal thickening of $X$, we compute explicitly the derived self-intersection of $X$ in $S$.
We show that every compact complex analytic space endowed with a fine logarithmic structure and every morphism between such spaces admit a semi-universal deformation. These results generalize the analogous results in complex analytic geometry first independently proved by A. Douady and H. Grauert in the '70. We follow Douady's two steps process approach consisting of an infinite-dimensional construction of the semi-universal deformation space followed by a finite-dimensional reduction.
This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry canbe used to compactify buildings. We discuss in detail the example of the special linear group. Moreover, we give an intrinsic description of Bruhat-Tits buildings in the framework of non-Archimedean analytic geometry.
This text is the write-up of a talk at the Bellairs Workshop in Number Theory on Tropical and Non-Archimedean Geometry that took place at the Bellairs Research Institute, Barbados, in May 2011. The goal of this text is to present recent work by in Diophantine Geometry over function fields due to Gubler and Yamaki, where analytic geometry in the sense of Berkovich plays a significant place. I also give an introduction to basic concepts and notions on Diophantine Geometry, such as heights, the Manin-Mumford conjecture, the Bogomolov conjecture, and its proof by Ullmo and Zhang.
This is our sequel to our previous work on the corresponding generalized Frobenius modules over some big multivariate Robba rings. We will go beyond our previous discussion where we focused on the corresponding analytic functions on polydiscs and polyannuli in the strictly affinoid situation, and general Hodge-Frobenius structures which are admissible in the corresponding context in our previous work.
We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.
We prove a rigid analytic analogue of the Artin vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric etale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$. Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove a general comparison theorem relating the algebraic and analytic etale cohomologies of any affinoid rigid space.
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.
We prove the representability theorem in derived analytic geometry. The theorem asserts that an analytic moduli functor is a derived analytic stack if and only if it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack. Our result applies to both derived complex analytic geometry and derived non-archimedean analytic geometry (rigid analytic geometry). The representability theorem is of both philosophical and practical importance in derived geometry. The conditions of representability are natural expectations for a moduli functor.…
In this paper, we expand the foundations of derived complex analytic geometry introduced in [DAG-IX] by J. Lurie. We start by studying the analytification functor and its properties. In particular, we prove that for a derived complex scheme locally almost of finite presentation $X$, the canonical map $X^{\mathrm{an}} \to X$ is flat in the derived sense. Next, we provide a comparison result relating derived complex analytic spaces to geometric stacks. Using these results and building on the previous work arXiv:1412.5166, we prove a derived version of the GAGA theorems. As an application, we pro…
In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f_1,...,f_n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f_1,...,f_n. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the…
This memoir is devoted to the study of formal-analytic arithmetic surfaces. These are arithmetic counterparts, in the context of Arakelov geometry, of germs of smooth complex-analytic surfaces along a projective complex curve. Formal-analytic surfaces provide a natural framework for arithmetic algebraization theorems, old and new. Formal-analytic arithmetic surfaces admit a rich geometry which parallels the geometry of complex analytic surfaces. Notably the dichotomy between pseudoconvexity and pseudoconcavity plays a central role in their geometry. Our study of formal-analytic arithmetic surf…
We introduce and study a notion of analytic loop group with a Riemann-Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity with non trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf-Poisson structure is isomorphic to the semi-classical limit of the center of the quantum affine algebra (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters, are parameterized by certain G-bundles on an el…
