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**16**

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692 views

### Poincaré-Hopf and Mathai-Quillen for Chern classes?

One. The Poincaré-Hopf theorem is usually stated as a formula for the Euler characteristic of the tangent bundle TM. Is there a version for Euler classes, of oriented real vector bundles?
It seems ...

**10**

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522 views

### What is Quillen's contribution to index theorem?

In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...

**6**

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164 views

### Bundles over Grassmanian with given top Chern class

So, I have been working on Chern classes for my master thesis and apparently (My proofs could be wrong and a few things are still vague) I was able to give a construction method and exhibit, via ...

**5**

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96 views

### Flatness of Chern classes for flat family of sheaves

Let $Q$ be a quasi-projective $k$-scheme (not necessarily smooth), $X$ a smooth projective $k$-variety and $\mathcal E$ a family of (torsion free) sheaves on $X$ parametrized by $Q$. Suppose that ...

**5**

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1k views

### How to compute the Chern class of a projective bundle?

For example, what is the first Chern class of $X:=\mathbb{P}(T\mathbb{P}^3)$ and $Y:=\mathbb{P}(T^*\mathbb{P}^3)$?
I am asking this question because I saw an essay today by F.Hirzebruch, saying that ...

**4**

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168 views

### Why write GRR with the relative tangent sheaf?

The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form
$$
\operatorname{ch}(f_!\alpha).\operatorname{Td}(Y)
=
...

**4**

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120 views

### Strategy to prove formula for top chern class from knowlege of chern character

I am trying to prove a conjecture that involves an enumerative problem. In the course of doing so, the following situation came up.
I have a sequence of (smooth, complex, rationally connected) ...

**4**

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219 views

### What is the structure of the stack of complexes supported in dimension less than r?

Let $X$ be something. (smooth and projective variety over C are my assumptions)
The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed substacks $M_\alpha$, ...

**4**

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157 views

### Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic

It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...

**4**

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346 views

### Atiyah--Singer for the Complex Projective Line

I'm trying to understand Atiyah--Singer by looking at the usual starting point of $CP^1$ and the Dirac--Dolbeault operator. If I've reduced everything down correctly, then in this case the theorem ...

**3**

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192 views

### What is known about analogous results of Kazdan and Warner in higher dimensions?

First let me state a Theorem due to Kazdan and Warner:
``Let M be a compact two dimensional orientable manifold. Let
$f: M \rightarrow \mathbb{R}$ be a function that has the same
sign as ...

**2**

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95 views

### Chern character of finite $CW$-complexes and rational Pontrjagin class of vector bundles

Let $K$ be a finite $CW$-complex. Could you give any references or explanations for the following two items? I do not understand. Thanks!
(1). The Chern character from $\tilde{KO}^0(K)$ to the ...

**2**

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79 views

### degree of Chern class of logarithmic differentials

Let $X$ be a smooth complex projective variety of dimension $n$ and $D$ a normal crossings divisor. I know that the following holds:
$$
\mathrm{deg}\ c_n(\Omega^1_X(\log D))=(-1)^n \chi(U),
$$ where ...

**1**

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83 views

### Chern classes of a resolution of singularities

Let $j:X\subset \mathbb P_{\mathbb C}^n$ ($n\geq 3$) be a hypersurface, defined by a section of a very ample line bundle $\mathcal L$, with a ordinary double point $P$ as the only singularity and ...

**0**

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95 views

### Chern classes of a family and Chern classes of a member

Let $X$ be a smooth projective variety over an algebraically closed field $k$ and $\mathcal E$ a family on torsion free coherent sheaves on $X$ parametrized by a smooth curve (over $k$) i.e. a ...

**0**

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110 views

### Chern classes of ideal sheaf of locally complete intersection

Let $C\varsubsetneq X$ be a reducible locally complete intersection closed pure $1$-dimensional subscheme in a smooth projective variety of dimension $3$ over a algebraically closed field $k$ of ...

**0**

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107 views

### Chern classes, vanishing of smooth sections or vanishing of holomorphic?

I have seen both definitions and this is getting me more and more confused.
Are Chern classes dual to the degeneracy cycles of smooth sections or holomorphic?
They can't be the same thing, can they?
...

**0**

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254 views

### About first Chern class and Poincaré duality in case of an ample divisor

Led $D$ be a very ample divisor in $X$ projective variety.
I can't understand why the first Chern class $c_1(\mathscr{O}_X(D))$ equals the Poincaré dual of $D$, $\mathscr{P}(D)$