Several approaches to nonarchimedean geometry brian conrad1 introduction let kbe a nonarchimedean eld. There is a classical theory of k analytic manifolds often used in the theory of algebraic groups with k a local. This contrasts with synthetic geometry analytic geometry is widely used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. The theory of nonarchimedean analytic spaces was developed by berkovich in two works in 1990 and 1993.
The result in 1 is formulated in terms of berkovich spaces but may be useful in the study of k ahler metrics. The nonarchimedean syz fibration compositio mathematica. In fact, flatness has already been considered in nonarchimedean analytic ge ometry, but in the rigidanalytic context 7 and 8, where its definition is as simple. Gromov compactness in non archimedean analytic geometry 5 ofthegeometryofstablecurvesinspiredbytheworkofdejongonalterations 23,24,2. Gromov compactness in nonarchimedean analytic geometry core.
An introduction to berkovich analytic spaces and nonarchimedean potential theory on curves matthew baker1 introduction and notation this is an expository set of lecture notes meant to accompany the authors lectures at the 2007 arizona winter school on padic geometry. In this article, we apply the approach of relative algebraic geometry towards analytic geometry to the category of bornological and indbanach spaces nonarchimedean or not. Finally, we reduce the nonarchimedean problem to the known compactness results in algebraic geometry. Numerous and frequentlyupdated resource results are available from this search. Spectral theory and analytic geometry over nonarchimedean fieldsvladimir g. Oct 08, 2014 in the upcoming uli14 the author intends to study the non archimedean analytic geometry of artin fans, certain locally to ric artin stacks that hav e been introduced in acmw14 and a w.
Organized by matthew baker georgia tech and sam payne yale university, the symposium will focus on setting a clear agenda for future developments in the related fields of tropical and non archimedean analytic geometry. M of codimension j gives a nonnegative closed j, jcurrent. This paper provides an elementary introduction to vladimir berkovichs theory of analytic spaces over nonarchimedean elds, focussing on topological aspects. In a narrower sense, nonarchimedean geometry describes the geometrical properties of a straight line on which archimedes axiom is not true the nonarchimedean line. It is the foundation of most modern fields of geometry, including algebraic. We demonstrate that the category of berkovich analytic spaces embeds fully faithfully into the category of varieties in our version of relative algebraic geometry. The correct analytic geometry over nonarchimedean field, however, is not a straightforward analogue of the complex. In spite of the fact that such fields are totally disconnected, non archimedean analytic spaces, which possess many of the topological properties of complex. More generally one may replace the complex numbers by non archimedean fields in which case one speaks of rigid analytic geometry.
Berkovich space, analytic stack, higher stack, pregeometry, structured. Geometrical considerations are used to obtain some applications, and the analytic spaces are used to construct the foundations of a non archimedean spectral theory of bounded linear operators. Organized by matthew baker georgia tech and sam payne yale university, the symposium will focus on setting a clear agenda for future developments in the related fields of tropical and nonarchimedean analytic geometry. This is a consequence of a theorem by gelfand and mazur. Tropical varieties for nonarchimedean analytic spaces. Nonarchimedean analytic geometry department of mathematics. Finally, we reduce the non archimedean problem to the known compactness results in algebraic geometry. Berkovich department of mathematics, the weizmann institute of science p. Local monodromy in nonarchimedean analytic geometry 169 of kummer type. Nonarchimedean analytic geometry is an analog of complex analytic geometry over nonarchimedean fields, i. Brian conrad, several approaches to nonarchimedean geometry, lectures at arizona winter school 2007, pdf. To investigate geometrical relationships in nonarchimedean geometry, one introduces a calculus of segments a nonarchimedean number system, regarded as a special number system. Berkovich american mathematical society providence, rhode island. Several approaches to nonarchimedean geometry brian conrad1 introduction let k be a nonarchimedean.
We show that berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of toenvaquievezzosi over the category of non archimedean banach spaces. Nonarchimedean geometry is geometry over nonarchimedean fields. Pdf a geometric theory of nonarchimedean analytic stacks. The purpose of these notes is to introduce the basic notions of rigid analytic geometry, with the aim of discussing the non archimedean uniformizations of certain abelian varieties. In fact, it is a nontrivial theorem that they are the only complete archimedean valued. Namely, the spectrum of such an operator is a nonempty compact. Classical rigid geometry may be viewed as a theory of analytic functions over local. This symposium focused on setting a clear agenda for future developments in the related fields of tropical and nonarchimedean analytic geometry. Then we construct the moduli stack of nonarchimedean analytic stable maps using formal models, artins representability criterion and the geometry of stable curves. Nonarchimedean analytic geometry as relative algebraic. This book requires a background at the level of basic graduate courses in algebra and topology, as well as some familiarity with algebraic geometry. Several approaches to non archimedean geometry brian conrad1 introduction let kbe a non archimedean eld. This gives a functor of points approach to nonarchimedean analytic geometry and in this way we also get definitions of higher nonarchimedean analytic stacks.
The purpose of these notes is to introduce the basic notions of rigid analytic geometry, with the aim of discussing the nonarchimedean uniformizations of certain abelian varieties. Ber1 and ber2 for berkovichs theory of kanalytic spaces. Similarly to an algebraic variety, an analytic variety is locally given as a zero locus of a finite set of analytic functions, i. However, formatting rules can vary widely between applications and fields of interest or study. Nonarchimedean geometry encyclopedia of mathematics. Several approaches to nonarchimedean geometry stanford. Now describe two types of results in non archimedean geometry and dynamics. While the concrete results are quite different, the basic formalism of algebraic schemes and formal schemes over a nonarchimedean field k k is the special case of the standard formalism over any field. Oversimplifying, its contribution to rigid nonarchimedean geometry was in saturating tates rigid spaces with many generic points that made analytic spaces to honest topological spaces. In arithmetic, one seeks to construct adelic metrizations, which involve simultaneously archimedean and nonarchimedean valuations. We also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry. There is a classical theory of kanalytic manifolds often used in the theory of algebraic groups with k a local.
A systematic approach to rigid analytic geometry, 1984 pdf. For any closed symmetric monoidal quasiabelian category we can define a topology on certain subcategories of the of the category of affine schemes with respect to this category. For any closed symmetric monoidal quasiabelian category we can define a topology on certain subcategories of the of the category of affine schemes with respect. Several approaches to nonarchimedean geometry brian conrad1 introduction let k be a non archimedean.
An introduction to berkovich analytic spaces and non archimedean potential theory on curves matthew baker1 introduction and notation this is an expository set of lecture notes meant to accompany the authors lectures at the 2007 arizona winter school on padic geometry. Siegfried bosch lectures on formal and rigid geometry. Non archimedean analytic geometry is an analog of complex analytic geometry over non archimedean fields, i. We show that berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of to\envaqui\evezzosi over the category of non archimedean banach spaces. There are two senses in which the term may be used, referring to geometries over fields which violate one of the two senses of the archimedean property i. Nonarchimedean analysis i assume k eld equipped with nonarchimedean norm. Now describe two types of results in nonarchimedean geometry and dynamics. We show that berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of toenvaquievezzosi over the category of nonarchimedean banach spaces.
One of the goals of the meeting was to produce highquality expository material presenting the methods, results and ambitions of these active research areas. The correct analytic geometry over nonarchimedean field, however, is not a. Nonarchimedean analytic geometry as relative algebraic geometry. Non archimedean geometries may, as the example indicates, have properties significantly different from euclidean geometry. Guntzer, reinhold remmert, nonarchimedean analysis a systematic approach to rigid analytic geometry, 1984. Oren benbassat, kobi kremnizer, nonarchimedean analytic geometry as relative algebraic geometry arxiv. While the concrete results are quite different, the basic formalism of algebraic schemes and formal schemes over a non archimedean field k k is the special case of the standard formalism over any field.
Mathematical surveys and monographs, issn 00765376. Nonarchimedean analytic spaces the hebrew university. Koblitz 1980 mathematics subject classification 1985 revision. In spite of the fact that such fields are totally disconnected, nonarchimedean analytic spaces, which possess many of the topological properties of complex. Theory and practice 2428 august 2015 papeete, french polynesia scienti. Rigid analytic geometry and abelian varieties mihran papikian abstract. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. In 2 the formulation of the problem does not involve berkovich. There is a classical theory of k analytic manifolds often used in the theory of algebraic groups with ka local.
A nonarchimedean montels theorem compositio mathematica. Introduction complex analytic geometry is a powerful tool in the study of algebraic geometry. Second, we introduce formal stacks and non archimedean analytic stacks. Spectral theory and analytic geometry over nonarchimedean fields. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. We also discuss realizations of bruhattits buildings in nonarchimedean groups and ag varieties. The correct analytic geometry over non archimedean field, however, is not a straightforward analogue of the complex. Then we construct the moduli stack of non archimedean analytic stable maps using formal models, artins representability criterion and the geometry of stable curves. An introduction to berkovich analytic spaces and non. Nonarchimedean analytic spaces annette werner abstract.
Singularities berkovich spaces tropical geometry di. Analytic geometry is widely used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. There is a classical theory of kanalytic manifolds often used in. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. American mathematical society, 1990 ocolc802299757. First steps 3 to view the space a1 as the ane line was the fact that it provided an answer to the question on the spectrum of a bounded linear operator i posed to myself at the very beginning. Second, we introduce formal stacks and nonarchimedean analytic stacks. Non archimedean geometry is geometry over non archimedean fields. In classical mathematics, analytic geometry, also known as coordinate geometry or cartesian geometry, is the study of geometry using a coordinate system.
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