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.

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 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 Goresky–Pardon lifts of Chern classes and associated Tate extensions

2017 ◽  
Vol 153 (7) ◽  
pp. 1349-1371 ◽  
Author(s):  
Eduard Looijenga
Keyword(s):  
Vector Bundle ◽  
Hodge Structure ◽  
Holomorphic Vector Bundle ◽  
Mixed Hodge Structure ◽  
Chern Classes ◽  
Analytic Variety ◽  
Algebraic Setting ◽  
Geometric Conditions ◽  
Stable Cohomology ◽  
Complex Analytic Variety

Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

scholarly journals OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

2019 ◽  
Vol 7 ◽  
Author(s):  
A. ASOK ◽  
J. FASEL ◽  
M. J. HOPKINS
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Algebraic Variety ◽  
Vector Bundles ◽  
Necessary Condition ◽  
Chern Classes ◽  
Smooth Complex ◽  
Steenrod Operations ◽  
Complex Algebraic Variety ◽  
Cycle Class

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

scholarly journals Chern Classes of Projective Modules

1963 ◽  
Vol 23 ◽  
pp. 121-152 ◽  
Author(s):  
Hideki Ozeki
Keyword(s):  
Vector Bundle ◽  
Cross Sections ◽  
Chern Class ◽  
Vector Bundles ◽  
Stein Manifold ◽  
Projective Modules ◽  
Singular Variety ◽  
Differentiable Functions ◽  
Holomorphic Vector Bundles ◽  
Differentiable Vector

In topology, one can define in several ways the Chern class of a vector bundle over a certain topological space (Chern [2], Hirzebruch [7], Milnor [9], Steenrod [15]). In algebraic geometry, Grothendieck has defined the Chern class of a vector bundle over a non-singular variety. Furthermore, in the case of differentiable vector bundles, one knows that the set of differentiable cross-sections to a bundle forms a finitely generated projective module over the ring of differentiable functions on the base manifold. This gives a one to one correspondence between the set of vector bundles and the set of f.g.-projective modules (Milnor [10]). Applying Grauert’s theorems (Grauert [5]), one can prove that the same statement holds for holomorphic vector bundles over a Stein manifold.

scholarly journals On vector bundles on algebraic surfaces and 0-cycles

1993 ◽  
Vol 130 ◽  
pp. 19-23 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Algebraic Structure ◽  
Vector Bundles ◽  
Algebraic Surfaces ◽  
Projective Surface ◽  
Classical Theorem ◽  
Chern Classes ◽  
Complex Projective Surface ◽  
Holomorphic Structure ◽  
Rank 2

Let X be an algebraic complex projective surface equipped with the euclidean topology and E a rank 2 topological vector bundle on X. It is a classical theorem of Wu ([Wu]) that E is uniquely determined by its topological Chern classes . Viceversa, again a classical theorem of Wu ([Wu]) states that every pair (a, b) ∈ (H (X, Z), Z) arises as topological Chern classes of a rank 2 topological vector bundle. For these results the existence of an algebraic structure on X was not important; for instance it would have been sufficient to have on X a holomorphic structure. In [Sch] it was proved that for algebraic X any such topological vector bundle on X has a holomorphic structure (or, equivalently by GAGA an algebraic structure) if its determinant line bundle has a holomorphic structure. It came as a surprise when Elencwajg and Forster ([EF]) showed that sometimes this was not true if we do not assume that X has an algebraic structure but only a holomorphic one (even for some two dimensional tori (see also [BL], [BF], or [T])).

Analytic Evaluation of Certain Characteristic Classes(1)

1973 ◽  
Vol 16 (2) ◽  
pp. 219-223
Author(s):  
Clark D. Jeffries
Keyword(s):  
Vector Bundle ◽  
Orthonormal Frame ◽  
Alternative Proof ◽  
Characteristic Classes ◽  
Real Vector ◽  
Analytic Evaluation ◽  
Riemann Structure

We have an alternative proof of the following result of Kervaire [2]:Let V→M be a real vector bundle with fibre dimension n≥4k+l over a compact 4k-manifold. Suppose V restricted to M — {x} is trivial. Choose a Riemann structure for V and an orthonormal frame for V restricted to M —{x}. Thus the obstruction to extending the frame smoothly over M is an element λ in π4k+1SO(n))≅Z. Then up to sign the evaluation of the kth Pontrayagin class Pk on M is ak(2k— 1)!. λ, where ak is 1 or 2 depending upon whether k is even or odd.

Basic Characteristic Classes

2007 ◽  
pp. 111-125
Author(s):  
D. Husemöller ◽  
M. Joachim ◽  
B. Jurčo ◽  
M. Schottenloher
Keyword(s):  
Basic Characteristic ◽  
Characteristic Classes
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