scholarly journals A note on Thom classes in general cohomology

1980 ◽  
Vol 79 ◽  
pp. 69-78
Author(s):  
Ronald Kirk Haas ◽  
Hiroshi Uehara
Keyword(s):  
Direct Approach ◽  
Seventieth Birthday ◽  
Characteristic Classes ◽  
Classical Approach ◽  
Chern Classes ◽  
Kunneth Formula ◽  
Relative Version ◽  
Set Up ◽  
Entire Theory ◽  
General Cohomology

This note is dedicated to the second author’s teacher, Professor Atsuo Komatsu, in celebration of his seventieth birthday.It is well known [4], [7] that the theory of characteristic classes in general cohomology is essentially based upon one theorem, the Leray-Hirsch Theorem. We further claim [8] that the entire theory could be developed merely from the Künneth Formula if under suitable conditions a truly relative version of the Meyer-Vietoris sequence exists in the general cohomology theory. In his lectures delivered at Aarhus in 1968, Dold [6] set up the necessary machinery including the Leray-Hirsch Theorem to define Chern Classes with values in general cohomology. However, he then stated [6, p. 47] that he “found a difficulty here in choosing adequate orientations (Thorn Classes) for the bundles involved”, and proceeded differently, discarding the “classical” approach used in both ordinary cohomology [9] and K-theory [3]. Later, he [7] published a more categorical work, although the approach to Chern Classes was basically unchanged from his previous work. Consequently, the possibility of the classical direct approach to the theory has remained open.

scholarly journals HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES

2010 ◽  
Vol 02 (01) ◽  
pp. 1-55 ◽  
Author(s):  
JEAN-PAUL BRASSELET ◽  
JÖRG SCHÜRMANN ◽  
SHOJI YOKURA
Keyword(s):  
Chern Class ◽  
Blow Up ◽  
Homology Class ◽  
Grothendieck Group ◽  
Characteristic Classes ◽  
Chern Classes ◽  
Singular Spaces ◽  
Coherent Sheaves ◽  
Roch Theorem ◽  
Mixed Hodge Modules

In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformationmCy: K0( var /X) → G0(X) ⊗ ℤ[y], which generalizes the total λ-class λy(T*X) of the cotangent bundle to singular spaces. Here K0( var /X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G0(X) is the Grothendieck group of coherent sheaves of [Formula: see text]-modules. A first construction of mCy is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCy is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito. We define a natural transformation Ty* : K0( var /X) → H*(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty* is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = -1), the Todd class transformation in the singular Riemann-Roch theorem of Baum–Fulton–MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1). We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov–Libgober and the stringy Chern classes of Aluffi and De Fernex–Lupercio–Nevins–Uribe. All our results can be extended to varieties over a base field k of characteristic 0.

scholarly journals Forest ecotone survey by line intersect sampling

2004 ◽  
Vol 34 (8) ◽  
pp. 1776-1783 ◽  
Author(s):  
Piermaria Corona ◽  
Gherardo Chirici ◽  
Davide Travaglini
Keyword(s):  
Unit Area ◽  
Central Italy ◽  
Classical Approach ◽  
Visual Interpretation ◽  
Landscape Mosaic ◽  
Remotely Sensed Imagery ◽  
Ecological Relevance ◽  
Gis Environment ◽  
Set Up ◽  
Forest Ecotone

Given their ecological relevance, the survey of ecotones is of considerable interest in forest multiresource inventory. To this end, it is useful to set up survey procedures to provide efficient and reliable information about the length of such elements within the landscape mosaic. This note demonstrates a procedure based upon line intersect sampling on remotely sensed imagery. The estimate of ecotone length per unit area is obtained by visual interpretation of the changes from forest to other land use classes along each sampling line. The experimentation carried out in two test areas within forest landscapes of central Italy proves the operative soundness of the proposed procedure, which is more efficient than the classical approach by forest polygon delineation and perimeter mensuration in a GIS environment. Under the examined conditions, samples based on a moderately high number of lines characterized by relatively long length prove to be more efficient than those based on shorter survey units.

scholarly journals ON DIFFERENTIAL CHARACTERISTIC CLASSES

2014 ◽  
Vol 99 (1) ◽  
pp. 30-47 ◽  
Author(s):  
MAN-HO HO
Keyword(s):  
Explicit Formula ◽  
Chern Class ◽  
Natural Transformation ◽  
Euler Class ◽  
Characteristic Classes ◽  
Characteristic Class ◽  
Chern Classes ◽  
Explicit Formulas ◽  
Differential Characteristic ◽  
Differential Characters

In this paper we give explicit formulas for differential characteristic classes of principal $G$-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and the differential Euler class. Furthermore, we show that the differential Chern class is the unique natural transformation from (Simons–Sullivan) differential $K$-theory to (Cheeger–Simons) differential characters that is compatible with curvature and characteristic class. We also give the explicit formula for the differential Chern class on Freed–Lott differential $K$-theory. Finally, we discuss the odd differential Chern classes.

scholarly journals Bounded sets of sheaves on Kähler manifolds

2016 ◽  
Vol 2016 (710) ◽  
Author(s):  
Matei Toma
Keyword(s):  
Kähler Manifold ◽  
Coherent Sheaf ◽  
Connected Components ◽  
Hilbert Polynomial ◽  
Chern Classes ◽  
Boundedness Criterion ◽  
Kahler Manifold ◽  
Set Up ◽  
Compact Kähler Manifold ◽  
Bounded Sets

AbstractWe show that any set of quotients with fixed Chern classes of a given coherent sheaf on a compact Kähler manifold is bounded in a sense which we define. The result is proved by adapting Grothendieck's boundedness criterion expressed via the Hilbert polynomial to the Kähler set-up. As a consequence we obtain the compactness of the connected components of the Douady space of a compact Kähler manifold.

scholarly journals Generating Series in the Cohomology of Hilbert Schemes of Points on Surfaces

2007 ◽  
Vol 10 ◽  
pp. 254-270 ◽  
Author(s):  
Samuel Boissière ◽  
Marc A. Nieper-Wisskirchen
Keyword(s):  
Smooth Surface ◽  
Vertex Algebra ◽  
Characteristic Classes ◽  
Vertex Operators ◽  
Chern Classes ◽  
Hilbert Schemes ◽  
Rewriting Logic ◽  
Rational Cohomology ◽  
Generating Series ◽  
Logic System

In the study of the rational cohomology of Hilbert schemes of points on a smooth surface, it is particularly interesting to understand the characteristic classes of the tautological bundles and the tangent bundle. In this note we pursue this study. We first collect all results appearing separately in the literature and prove some new formulas using Ohmoto's results on orbifold Chern classes on Hilbert schemes. We also explain the algorithmic counterpart of the topic: the cohomology space is governed by a vertex algebra that can be used to compute characteristic classes. We present an implementation of the vertex operators in the rewriting logic system MAUDE, and address observations and conjectures obtained after symbolic computations.

scholarly journals An Integral Formula for the Chern from of a Hermitian Bundle

1971 ◽  
Vol 42 ◽  
pp. 135-172 ◽  
Author(s):  
Hideo Omoto
Keyword(s):  
Vector Bundle ◽  
Chern Class ◽  
Basic Characteristic ◽  
Grassmann Manifold ◽  
Integral Formula ◽  
Characteristic Classes ◽  
Characteristic Class ◽  
Chern Classes ◽  
Simplicial Decomposition ◽  
Hermitian Bundle

We shall consider a Hermitian n-vector bundle E over a complex manifold X. When X is compact (without boundary), S.S. Chern defined in his paper [3] the Chern classes (the basic characteristic classes of E) Ĉi(E), i = 1, · · ·, n, in terms of the basic forms Φi on the Grassmann manifold H(n, N) and the classifying map f of X into H(n, N). Moreover he proved ([3], [4]) that if Ek denotes the k-general Stiefel bundle associated with E, the (n — k + 1)-th Chern class Ĉn-k+1(E) coincides with the characteristic class C(Ek) of Ek defined as follows: Let K be a simplicial decomposition of X and K2(n-k)+1 the 2(n — k) + 1 — shelton of K.

ON A THEORY OF CHERN CLASSES FOR SUPER VECTOR BUNDLES

2008 ◽  
Vol 05 (03) ◽  
pp. 287-295
Author(s):  
S. VARSAIE
Keyword(s):  
Vector Bundles ◽  
Characteristic Classes ◽  
Chern Classes

In this paper, a theory of characteristic classes for super vector bundles over [Formula: see text] is studied. Some properties of even and odd Chern classes, constructed in this theory, are established. At last, the relevance of the theory to supergeometry is discussed.

The Effects of Drying Techniques on Clay-Rich Soil Texture

1974 ◽  
Vol 32 ◽  
pp. 466-467
Author(s):  
T. G. Naymik
Keyword(s):  
Critical Point ◽  
Liquid Nitrogen ◽  
Liquid Nitrogen Temperature ◽  
Drying Methods ◽  
Air Drying ◽  
Freeze Dried ◽  
Critical Point Drying ◽  
Set Up ◽  
The Relationship ◽  
Quartz Grains

Three techniques were incorporated for drying clay-rich specimens: air-drying, freeze-drying and critical point drying. In air-drying, the specimens were set out for several days to dry or were placed in an oven (80°F) for several hours. The freeze-dried specimens were frozen by immersion in liquid nitrogen or in isopentane at near liquid nitrogen temperature and then were immediately placed in the freeze-dry vacuum chamber. The critical point specimens were molded in agar immediately after sampling. When the agar had set up the dehydration series, water-alcohol-amyl acetate-CO2 was carried out. The objectives were to compare the fabric plasmas (clays and precipitates), fabricskeletons (quartz grains) and the relationship between them for each drying technique. The three drying methods are not only applicable to the study of treated soils, but can be incorporated into all SEM clay soil studies.

Evaluation of Freezing Methods by Low Temperature X-Ray Diffraction

1977 ◽  
Vol 35 ◽  
pp. 330-333
Author(s):  
T. Gulik-Krzywicki ◽  
M.J. Costello
Keyword(s):  
Biological Materials ◽  
Molecular Organization ◽  
X Ray Diffraction ◽  
Glass Capillary ◽  
X Ray ◽  
Rate Dependent ◽  
Before And After ◽  
Freeze Etching ◽  
Diffraction Patterns ◽  
Set Up

Freeze-etching electron microscopy is currently one of the best methods for studying molecular organization of biological materials. Its application, however, is still limited by our imprecise knowledge about the perturbations of the original organization which may occur during quenching and fracturing of the samples and during the replication of fractured surfaces. Although it is well known that the preservation of the molecular organization of biological materials is critically dependent on the rate of freezing of the samples, little information is presently available concerning the nature and the extent of freezing-rate dependent perturbations of the original organizations. In order to obtain this information, we have developed a method based on the comparison of x-ray diffraction patterns of samples before and after freezing, prior to fracturing and replication.Our experimental set-up is shown in Fig. 1. The sample to be quenched is placed on its holder which is then mounted on a small metal holder (O) fixed on a glass capillary (p), whose position is controlled by a micromanipulator.

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