Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

Faithfully flat descent of almost perfect complexes in rigid geometry

We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic information 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 with 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 of derived non-archimedean analytic geometry: derived Tate acyclicity, projection formula, and proper base change. These results also merit independent interest. Our main motivation comes from non-archimedean enumerative geometry. In our subsequent works, we will apply the derived mapping stacks to obtain non-archimedean analytic Gromov–Witten invariants.

In this work, I extend the theory of motives, as developed by Voevodsky and Morel-Voevodsky, to the context of rigid analytic geometry over a complete non archimedean field. The first chapter deals with the homotopical approach of Morel and Voevodsky. One finds there the construction of the motivic stable homotopy category of rigid analytic varieties and a complete description of this category in terms of algebraic motives when the base field has equal characteristic zero and its valuation is discrete. The second chapter deals with Voevodsky's approach based on transfers. One finds there the construction of the triangulated category of rigid analytic motives, and an extension to rigid analytic geometry of a large number of Voevodsky's fundamental results such as his theory of homotopy invariants presheaves with transfers. This is said, the present work is a lot more than just a mere copy of the classical theory and the reader will find a lot of results that are new and specific to rigid analytic geometry.

In this chapter we give a survey of rigid geometry over non-Archimedean fields. The foundation of the theory was laid by Tate in his private Harvard notes dating back to 1961, which were later published in Inventiones mathematicae (Tate in Invent. Math. 12:257–289, 1971). Here we explain the main results from the classical point of view as studied in the late sixties; for proofs we refer to Bosch (Lectures on Formal and Rigid Geometry, vol. 2105, 2014). At that time rigid geometry was mainly inspired by complex analysis. Fundamental results were achieved by Kiehl, who introduced the Grothendieck topology and proved the basic facts concerning coherent sheaves. Moreover, Kiehl makes Serre’s theory (Serre in Ann. Inst. Fourier 6:1–42, 1956) of Geometrie Algebrique et Geometrie Analytique available for rigid analytic geometry, often referred to as GAGA; cf. (Kopf in Uber eigentliche Familien algebraischer Varietaten uber affinoiden Raumen, vol. 7, 1974).

The workshop focused on recent developments in non-Archimede-an analytic geometry with various applications to other fields. The topics of the talks included foundational results on analytic spaces as well as applications to the local Langlands conjecture, p -adic cohomology theories, Shimura varieties, the non-Archimedean Simpson correspondence, and to questions on metrics in complex geometry, in the spirit of K -stability.

Blowing up is a useful technique in algebraic and analytic geometry. In particular, it is the main tool for proving resolution of singularities. Hironaka [2] proved in 1964 that every algebraic variety over a field of characteristic zero admits a resolution of singularities which is obtained by successive blowing ups of certain regular centers. Moreover, he proves the stronger version of embedded resolution of singularities, i.e., for every (singular) subvariety X of a smooth variety Z there exists a sequence of birational morphisms $${Z_N} \to {Z_{N - 1}} \to \cdots \to {Z_1} \to {Z_0} = Z,$$ (1.1) such that π i is the blowing up of Z i−1 at a regular center C i which is transversal to the exceptional divisor E i−1 of π i−1 ο⋯ο π1, and such that the strict transform X N of X at Z N is smooth and transversal (normal crossing) to the exceptional divisor E N .

This paper examines the problems and prospects of society’s transition to a green economy, considering methodological approaches, and substantiates the necessity of its integration into contemporary economic processes. The study explores current transformational processes and universal principles for implementing green economy practices at the regional level. Priorities for creating new green sectors and ecological modernization are considered. An analysis of scholars’ methodological approaches to assessing the impact of green economy on overall economic development is conducted. The main advantages and opportunities of green economy are outlined. The issue of methodology for assessing regional green economies is examined within the framework of interdisciplinary research that combines economic, environmental, spatial, and social analytics. Leading international organizations, including the Organization for Economic Co-operation and Development (OECD), the United Nations Environment Programme (UNEP), the European Commission (EC), the European Environment Agency (EEA), and the Global Green Growth Institute (GGGI), have proposed their own indicator systems for measuring the progress of green reforms. Based on the conducted analysis, it is demonstrated that all international frameworks are based on the principle of integrating economic, environmental, and social components of development, but differ in analytical objectives, structure, and level of data aggregation. The OECD methodology is the most balanced from the standpoint of macroeconomic analysis, oriented toward monitoring green growth and suitable for adaptation to national statistical systems. The UNEP approach emphasizes the social effects of green economy and its impact on population well-being. The European Commission, within the Circular Economy Monitoring Framework, focuses on assessing the effectiveness of green transition policy and provides the highest level of sectoral detail. The EEA methodology has a distinctly eco-informational character, ensuring ecosystem monitoring. The GGGI approach is the most contemporary tool for comprehensive assessment of green growth progress in terms of resource efficiency, social inclusivity, and innovation. The study proposes a comprehensive methodology for assessing the level of regional economy greening based on a multi-level system of analytical procedures. The implementation of systemic, index-integral, factorial, cluster, and scenario approaches enhances the objectivity of assessment, ensures inter-regional comparability, and scientific reliability of results. The research methodology encompasses eight consecutive stages: formation of an indicator system, data normalization, construction of an integral indicator using principal component analysis, regional classification, cluster analysis, examination of average values within clusters, scenario modeling, and determination of priority directions for enhancing ecological-economic potential of regions. The practical significance of this research lies in creating a scientific foundation for assessing green transformation of Ukrainian regions, identifying territorial disparities, and developing targeted strategies for environmentally oriented development in the context of European approaches to green economy.

Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L an , we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L an and that the non-archimedean Monge–Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over k. The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.

We show that non-Archimedean 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 define a topology on certain subcategories of the category of (relative) affine schemes. In the case that the monoidal category is the category of abelian groups, the topology reduces to the ordinary Zariski topology. By examining this topology in the case that the monoidal category is the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in rigid or Berkovich analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry. We demonstrate that the category of Berkovich analytic spaces (and also rigid analytic spaces) embeds fully faithfully into the category of (relative) schemes in our version of relative algebraic geometry. We define a notion of quasi-coherent sheaf on analytic spaces which we use to characterize surjectivity of covers. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

Monte Carlo simulations (MCS) applied in particle physics play a key role in medical imaging and particle therapy. In such simulations, particles are transported through voxelized phantoms derived from predominantly patient CT images. However, such voxelized object representation limits the incorporation of fine elements, such as artificial implants from CAD modeling or anatomical and functional details extracted from other imaging modalities. In this work we propose a new hYbrid Voxelized/ANalytical primitive (YVAN) that combines both voxelized and analytical object descriptions within the same MCS, without the need to simultaneously run two parallel simulations, which is the current gold standard methodology. Given that YVAN is simply a new primitive object, it does not require any modifications on the underlying MC navigation code. The new proposed primitive was assessed through a first simple MCS. Results from the YVAN primitive were compared against an MCS using a pure analytical geometry and the layer mass geometry concept. A perfect agreement was found between these simulations, leading to the conclusion that the new hybrid primitive is able to accurately and efficiently handle phantoms defined by a mixture of voxelized and analytical objects. In addition, two application-based evaluation studies in coronary angiography and intra-operative radiotherapy showed that the use of YVAN was 6.5% and 12.2% faster than the layered mass geometry method, respectively, without any associated loss of accuracy. However, the simplification advantages and differences in computational time improvements obtained with YVAN depend on the relative proportion of the analytical and voxelized structures used in the simulation as well as the size and number of triangles used in the description of the analytical object meshes.

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields. The topics of the talks included foundational results on analytic spaces as well as applications to the local Langlands conjecture, birational geometry, p-adic cohomology theories, Shimura varieties and the non-Archimedean Simpson correspondence.

In this work we develop a formalism of vanishing cycles for non-Archimedean analytic spaces which is an analog of that for complex analytic spaces from [SGA7], Exp. XIV. As an application we prove that in the equicharacteristic case the stalks of the vanishing cycles sheaves of a scheme X at a closed point x E X, depend only on the formal completion Spf(Ox,,) of X at x. In particular, any continuous homomorphism OX,x -+ y,y induces a homorphism from the stalks of the vanishing cycles sheaves of X at x to those of Y at y. Furthermore, we prove that, given Ox,x and O y, , there exists n > 1 such that, for any pair of continuous homomorphisms OX,x -Oy,y that coincide modulo the n-th power of the maximal ideal of Oy,y, the induced homomorphisms between the stalks of the vanishing cycles sheaves coincide. These facts generalize a result of G. Laumon from [Lau] (see Remark 7.6). Throughout the paper we fix a non-Archimedean field k (whose valuation is not assumed to be nontrivial). In ?1 we study etale Galois sheaves on k-analytic spaces. To define the vanishing cycles functor and to work with it, we use the language of pro-analytic spaces, i.e., pro-objects of the category of analytic spaces ([SGA4], Exp. I). Examples of such objects are the germs of analytic spaces as in [Ber2], ?3.4. Another example is considered in ?3. In ?4 we define the vanishing cycles functor and establish its basic properties. In ?5 we show that the vanishing cycles sheaves are trivial for smooth morphisms. In ?6 we prove a comparison theorem for vanishing cycles. This theorem is more general than its analog over C from [SGA7], Exp. XIV, and its proof does not use Hironaka's theorem on resolution of singularities. In ?7 we apply the comparison theorem to prove the properties of the vanishing cycles sheaves of schemes formulated above. It is worthwhile to note that this application is obtained by considering non-Archimedean analytic geometry over fields with trivial valuation. Like [Ber3], this work arose from a suggestion of P. Deligne to apply the etale cohomology theory from [Ber2] to the study of the vanishing cycles sheaves of schemes. I am very grateful to him for useful discussions on the subject. I also

Many problems in differential and analytic geometry seem to have something to do with the study of non-linear partial differential equations of elliptic type. For instance, the classical Weyl and Minkowski problems for a convex surface have been studied by H. Weyl, H. Lewy, and L. Nirenberg using the iteration method for the construction of solutions of certain non-linear equations of elliptic type (see [6]). Also, M. Kuranishi [3] constructed the effective complete family of deformations of complex analytic structures on a given compact complex manifold as the solution space of another non-linear equation of elliptic type; whereby the basic idea in his work is to apply an implicit function theorem to the non-linear operator of a Banach space, and to construct the bifurcation of solutions explicitly.

Trace formulas (Lagrange, Jacobi–Kronecker, Bergman–Weil) play a key role in division problems in analytic or algebraic geometry (including arithmetic aspects, see, for example, Berenstein and Yger in Am. J. Math. 121(4):723–796, 1999). Unfortunately, they usually hold within the restricted frame of complete intersections. Besides the fact that it allows one to carry local or semi-global analytic problems to a global geometric setting (think about Crofton’s formula), averaging the Cauchy kernel (from ℂn∖{z 1…z n =0}⊂ℙn(ℂ)), in order to get the Bochner–Martinelli kernel (in ℂn+1∖{0}⊂ℙn+1(ℂ)=ℂn+1∪ℙn(ℂ)), leads to the construction of explicit candidates for the realization of Grothendieck’s duality, namely BM residue currents (Passare et al. in Publ. Mat. 44:85–117, 2000; Andersson in Bull. Sci. Math. 128(6):481–512, 2004; Andersson and Wulcan in Ann. Sci. École Norm. Super. 40:985–1007, 2007), extending thus the cohomological incarnation of duality which appears in the complete intersection or Cohen–Macaulay cases. We will recall here such constructions and, in parallel, suggest how far one could take advantage of the multiplicative inductive construction introduced by Coleff and Herrera (Lecture Notes in Mathematics, vol. 633, Springer, Berlin, 1978), by relating it to the Stückrad–Vogel algorithm developed in Stückrad and Vogel (Queen Pap. Pure Appl. Math. 61:1–32, 1982), Tworzewski (Ann. Pol. Math. 62:177–191, 1995), Andersson et al. (arXiv:1009.2458, 2010) toward improper intersection theory. Results presented here were initiated all along my long-term collaboration with Carlos Berenstein. To both of us, the mathematical work of Leon Ehrenpreis certainly remained a constant and very stimulating source of inspiration. This presentation relies also deeply on my collaboration over the past years with M. Andersson, H. Samuelsson, and E. Wulcan in Göteborg.KeywordsComplete IntersectionGlobal SectionHolomorphic SectionCartier DivisorIntegration CurrentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In his 1944 paper Veranderliche Riemannsche Flachen , Teichmuller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space called today Teichmuller space. In the same paper, Teichmuller introduced the so-called universal Teichmuller curve -- a space over Teichmuller space where the fiber above each point is a Riemann surface representing that point. In fact, Teichmuller proved the existence of the Teichmuller curve as a space of Riemann surfaces parametrized by an analytic space, with an existence and uniqueness theorem establishing this analytic structure. This result was later reformulated and proved by Grothendieck in a series of ten lectures he gave at Cartan's seminar in 1960-1961. In his approach , Grothendieck replaced Teichmuller's explicit parameters by a general construction of fiber bundles whose base is an arbitrary analytic space. This work on Teichmuller space led him to recast the bases of analytic geometry using the language of categories and functors. In Grothendieck's words, the Teichmuller curve becomes a space representing a functor from the category of analytic spaces into the category of sets. In this survey, we comment on Grothendieck's series of lectures. The survey is primarily addressed to low-dimensional topologists and geometers. In presenting Grothendieck's results, we tried to explain or rephrase in more simple terms some notions that are usually expressed in the language of algebraic geometry. However, it is not possible to short-circuit the language of categories and functors. The survey is also addressed to those algebraic geometers who wish to know how the notion of moduli space evolved in connection with Teichmuller theory. Explaining the origins of mathematical ideas contributes in dispensing justice to their authors and it usually renders the theory that is surveyed more attractive. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichmuller theory. This volume is dedicated to the memory of Alexander Grothendieck.