Mixed-characteristic shtukas

2020 ◽  
pp. 90-97
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Vector Bundle ◽  
Hodge Theory ◽  
Mixed Characteristic ◽  
Object A ◽  
Fiber Product ◽  
Perfectoid Spaces ◽  
Test Objects

This chapter looks at mixed-characteristic shtukas. Much of the theory of mixed-characteristic shtukas is motivated by the structures appearing in (integral) p-adic Hodge theory. The chapter assesses Drinfeld's shtukas and local shtukas. In the mixed characteristic setting, X will be replaced with Spa Zp. The test objects S will be drawn from Perf, the category of perfectoid spaces in characteristic p. For an object, a shtuka over S should be a vector bundle over an adic space, together with a Frobenius structure. The product is not meant to be taken literally (if so, one would just recover S), but rather it is to be interpreted as a fiber product over a deeper base. Motivated by this, the chapter then defines an analytic adic space and shows that its associated diamond is the appropriate product of sheaves on Perf.

scholarly journals -DIVISIBILITY FOR COHERENT COHOMOLOGY

2015 ◽  
Vol 3 ◽  
Author(s):  
BHARGAV BHATT
Keyword(s):  
Hodge Theory ◽  
Mixed Characteristic ◽  
Rational Singularities ◽  
Proper Morphism

We prove that the coherent cohomology of a proper morphism of noetherian schemes can be made arbitrarily$p$-divisible by passage to proper covers (for a fixed prime$p$). Under some extra conditions, we also show that$p$-torsion can be killed by passage to proper covers. These results are motivated by the desire to understand rational singularities in mixed characteristic, and have applications in$p$-adic Hodge theory.

scholarly journals -ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIES

2013 ◽  
Vol 1 ◽  
Author(s):  
PETER SCHOLZE
Keyword(s):  
Hodge Theory ◽  
Analytic Varieties ◽  
De Rham ◽  
Perfectoid Spaces

AbstractWe give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-étale site, which makes all constructions completely functorial.

scholarly journals THE RELATION BETWEEN VISUAL ACUITY AND ILLUMINATION

1937 ◽  
Vol 21 (2) ◽  
pp. 165-188 ◽  
Author(s):  
Simon Shlaer
Keyword(s):  
Visual Acuity ◽  
Test Object ◽  
State Equation ◽  
Limiting Factor ◽  
Object A ◽  
Fixed Distance ◽  
Different Types ◽  
The Difference ◽  
Trained Observers ◽  
Test Objects

1. An apparatus for measuring the visual acuity of the eye at different illuminations is described. The test object is continuously variable in size and is presented at a fixed distance from the eye in the center of a 30° field. Observation of the field is through an artificial pupil. The maximum intensity obtainable is more than enough to cover the complete physiological range for the eye with white light though only 110 watts are consumed by the source. Means for varying the intensity over a range of 1:1010 in small steps are provided. 2. The relation of visual acuity and illumination for two trained observers was measured, using two different types of test object, a broken circle and a grating. The measurements with both test objects show a break at a visual acuity of 0.16, all values below that being mediated by the rods and those above by the cones. The grating gives higher visual acuities at intensities less than about 30 photons and lower visual acuities above that. The maximum visual acuity attainable with the grating under the same conditions is about 30 per cent lower than that with the C. It is shown that the limiting factor in the resolution of the eye for the grating is the diameter of the pupil when it is less than 2.3 mm. and the size of the central cones when the pupil is larger than that. The value of the diameter of the cone derived on that basis from the visual acuity data agrees with that derived from direct cone count in a unit of area. 3. The data for the cones made with both test objects are adequately described by one and the same form of the stationary state equation derived by Hecht for the photoreceptor system. This fact, together with certain considerations about the difference in the nature of the two test objects with regard to the resolvable area, leads to the conclusion that detail perception is a function of a distance rather than an area. All the data for the rods can likewise be described by another variety of the same equation, although the data are too fragmentary to make the choice of the form as certain as might be desired.

scholarly journals Toric vector bundles: GAGA and Hodge theory

2020 ◽  
Author(s):  
Jonas Stelzig
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Hodge Theory ◽  
Algebraic Construction ◽  
Smooth Base

Abstract We prove a GAGA-style result for toric vector bundles with smooth base and give an algebraic construction of the Frölicher approximating vector bundle that has recently been introduced by Dan Popovici using analytic techniques. Both results make use of the Rees-bundle construction.

Berkeley Lectures on p-adic Geometry

2020 ◽  
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Moduli Spaces ◽  
Reductive Group ◽  
Arithmetic Geometry ◽  
Mixed Characteristic ◽  
Algebraic Spaces ◽  
New Ideas ◽  
The Subject ◽  
Perfectoid Spaces ◽  
Divisible Groups ◽  
The University

This book presents an important breakthrough in arithmetic geometry. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, the author introduced the concept of “diamonds,” which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. This book shows that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. The book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained.

scholarly journals Nonabelian Hodge theory for klt spaces and descent theorems for vector bundles

2019 ◽  
Vol 155 (2) ◽  
pp. 289-323 ◽  
Author(s):  
Daniel Greb ◽  
Stefan Kebekus ◽  
Thomas Peternell ◽  
Behrouz Taji
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Hodge Theory ◽  
Resolution Of Singularities ◽  
Independent Interest ◽  
Restriction Theorem ◽  
Projective Varieties ◽  
To Come ◽  
Smooth Locus

We generalise Simpson’s nonabelian Hodge correspondence to the context of projective varieties with Kawamata log terminal (klt) singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety$X$and any resolution of singularities, any vector bundle on the resolution that appears to come from$X$numerically, does indeed come from $X$. Furthermore, and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established.

Reconstruction of Objects from their Projections Using Orthogonal Functions

1973 ◽  
Vol 31 ◽  
pp. 268-269
Author(s):  
Elrnar Zeitler
Keyword(s):  
Unit Area ◽  
Three Dimensional ◽  
One Dimension ◽  
Spatial Density ◽  
Dimensional Representation ◽  
Orthogonal Functions ◽  
Object A ◽  
Plane Normal ◽  
Dimensional Object ◽  
Spatial Density Distribution

Considering any finite three-dimensional object, a “projection” is here defined as a two-dimensional representation of the object's mass per unit area on a plane normal to a given projection axis, here taken as they-axis. Since the object can be seen as being built from parallel, thin slices, the relation between object structure and its projection can be reduced by one dimension. It is assumed that an electron microscope equipped with a tilting stage records the projectionWhere the object has a spatial density distribution p(r,ϕ) within a limiting radius taken to be unity, and the stage is tilted by an angle 9 with respect to the x-axis of the recording plane.

Bright-Field Images (Ctem) of Phase Objects at High Resolution

1976 ◽  
Vol 34 ◽  
pp. 490-491 ◽  
Author(s):  
Sumio Iijima
Keyword(s):  
High Resolution ◽  
Electron Beam Irradiation ◽  
Fine Particles ◽  
High Resolution Electron Microscopy ◽  
Good Test ◽  
Resolution Electron ◽  
Single Crystal Films ◽  
Crystal Films ◽  
By Products ◽  
Test Objects

We have developed a technique to prepare thin single crystal films of graphite for use as supporting films for high resolution electron microscopy. As we showed elsewhere (1), these films are completely noiseless and therefore can be used in the observation of phase objects by CTEM, such as single atoms or molecules as a means for overcoming the difficulties because of the background noise which appears with amorphous carbon supporting films, even though they are prepared so as to be less than 20Å thick. Since the graphite films are thinned by reaction with WO3 crystals under electron beam irradiation in the microscope, some small crystallites of WC or WC2 are inevitably left on the films as by-products. These particles are usually found to be over 10-20Å diameter but very fine particles are also formed on the film and these can serve as good test objects for studying the image formation of phase objects.

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