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# Questions tagged [chern-classes]

Characteristic classes associated to complex vector bundles.

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### Chern number of positive spinor bundle

What is the second chern number $c_2(V_+)$ of the positive spinor bundle on a 4-manifold, in particular $S^4$? Why is it that $V_+$ is the same as the quaternion line-bundle? Thanks,
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### Tautological ring for moduli of flat connections

Let $X$ be a smooth complex manifold and $G$ a connected complex algebraic group. Let $M$ denote the moduli stack of flat $G$-connections on $X$. Over $M\times X$, we have the tautological $G$-bundle, ...
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### Bound on norm of the curvature from Chern class

Let $M$ be an oriented closed $6$-manifold. $V$ be an hermitian complex vector bundle of dimension $4$ on $M.$ Hence $c^3(V)\in H^6(M,\mathbb{Z})\cong \mathbb{Z}$ can be thought of as an integer and ...
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### On Brylinski–McLaughlin's paper "Čech cocycles for characteristic classes"

In the paper "Čech cocycles for characteristic classes", the authors Brylinski and McLaughlin describe how to construct Čech cocycles with values in the Deligne smooth complex representing ...
1 vote
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### When the whole space $H^2(X,\mathbb Q)$ can be represented by $c(L_t)$ of $X_t$?

Let $X$ be a compact complex manifold, $\pi:\mathcal X\to B$ be a holomorphic family of $X$ with $X_t=\pi^{-1}(t),t\in B$, and $X_0=X$. Let $L_t$ be a holomorphic line bundle over $X_t$, then its ...
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Let $M^4$ be an orientable closed 4-manifold and $c_1$ be the first Chern class of a complex line bundle on $M^4$. Let $b$ be the mod 2 reduction of $c_1$, ie $b=c_1$ mod 2. We have a relation $w_2 b =... 6 votes 1 answer 418 views ### Action of Steenrod algebra on Chern classes This is question about result from Brown and Peterson$H^*(MO)$as an algebra over the Steenrod algebra. Unfortunately, the paper is not available on the Internet, so I can't find the proof. One of ... 18 votes 4 answers 1k views ### Analogy between Stiefel-Whitney and Chern classes There is a clear similarity between Stiefel-Whitney and Chern classes, if one replaces base field$\mathbb R$with$\mathbb C$, coefficient ring$\mathbb Z/2$with$\mathbb Z$and scales the grading ... 10 votes 1 answer 563 views ### Index of Dirac operator and Chern character of symmetric product twisting bundle I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text We are twisting the spinor bundle (on Einstein 4-manifold)$\Sigma$with an ... 3 votes 1 answer 650 views ### Chern classes of complex vector bundle I'm reading characteristic classes form the book Differential forms in Algebraic Topology by Bott and Tu. The Chern classes are defined as follows:$E\xrightarrow{\rho} M$is a vector bundle and$E_p$... 17 votes 1 answer 1k views ### Direct proof that Chern-Weil theory yields integral classes Suppose$E$is a complex vector bundle of rank$n$on a compact oriented manifold (both assumed smooth). Let$h$be a Hermitian metric on$E$, and let$A$be a Hermitian connection on$E$and$F_A$... 4 votes 1 answer 244 views ### Chern -Weil map for topological principal G bundles Let$G$be a Lie group. In the book Curvature and Characteristic classes, the author (Johan L. Dupont) mentiones in beginning of chapter 5 the following : The notion of a topological principal$G$... 11 votes 3 answers 640 views ### A binary operation on vector bundles that adds Chern classes? Let$E$and$F$be two complex vector bundles over a space$X$. There's a fairly well-known binary operation called the direct sum, written$E\oplus F$, which has the property that its first Chern ... 2 votes 1 answer 158 views ### Comparing the minimal Chern number and the cup-length of a symplectic manifold Let$(M,\omega)$be a symplectic manifold. One can define its minimal Chern number$N_M$as: $$N_M := \text{inf} \lbrace k > 0 \ |\ \exists A \in H_2(M; \mathbb{Z}), \langle c_1, A \rangle = k \... 4 votes 0 answers 442 views ### Chern classes of torsion-free sheaves Let X be a smooth projective variety and Z a closed subvariety of co-dimension k. The first k-1 chern classes of the ideal sheaf of Z vanishes and the k-th chern class is given by ... 6 votes 0 answers 220 views ### Equivariant Venice Lemma In the paper J. Simons and D. Sullivan. Structured vector bundles define differential K-theory, one of the key ideas is the so called Venice Lemma, which essentially can be stated as Theorem: For ... 6 votes 0 answers 155 views ### Does there exist a notion of Chern classes in intersection cohomology? First of all: I apologize for my mistakes, I'm a freshman in intersection cohomology. Let X be a (compact) complex analytic space, let L be a line bundle over X. Can one define a notion of ... 2 votes 0 answers 86 views ### Is chern classes of holomorphic vector bundles enough to generate Hodge cycles [duplicate] Let X ba a smooth projective variety of dimension n. Hodge Conjecture states that every Hodge cycle in Hdg^k(X,\mathbb{Q}) comes from a Chern class of codimension k in CH^k(X,\mathbb{Q}). ... 3 votes 1 answer 311 views ### Is the minimal Chern number of a toric manifold at least 2? I would like to show that the minimal Chern number N_M of a toric manifold M is at least 2, where$$ N_M := \underset{l>0}{\min} \lbrace \exists \ A \in H_2(M;\mathbb{Z}) \ : \ \langle c, A \... 6 votes 0 answers 266 views ### Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers? I'm interested in if it's possible to represent a spin structure and the Arf invariant associated to it in terms of some sort of local fields. For example, the first Chern class of a complex line ... 6 votes 0 answers 86 views ### Elliptic deformation of the second Chern class Second Chern class $$c_2 \in H^4(BGL,\mathbb{Q}(2))$$ admits a nice presentation using dilogarithm. The five term relation in this setting becomes a cocycle condition (details can be found here). ... 7 votes 1 answer 636 views ### What is the geometrical meaning of higher Chern forms and classes? Let$M$be a complex manifold,$R^{\nabla}$be the curvature operator for connections$\nabla$. Consider a polynomial function$f:\operatorname M_n(\mathbb{C})\to\mathbb{C}$. For the gauge group$\...
In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $G$ bundle. In this book, they use connection on a principal bundle to ...