The chern-classes tag has no usage guidance.

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**1**answer

148 views

### Universal family of grassmannian as projective bundle over $\mathbb P^n$

Let $p:X=\mathbb P(\mathcal E_2)\rightarrow Gr(2,n+1)$ be the universal family of the lines in $\mathbb P^n$. If we denote $e:X\rightarrow \mathbb P^n$ the natural projection, we have $\mathcal O_{\...

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104 views

### Finding the Chern Class of a the pushfoward of a invertible sheaf

I am trying to understand what happens to the Chern Classes of an invertible sheaf $F$ over a complete intersection reduced curve of genus $g$ and degree $d$, when viewed as a invertible sheaf of $\...

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**2**answers

238 views

### Chern classes and singular hermitian metrics on vector bundles

Let $L$ be a holomorphic line bundle on a complex manifold $X$, and assume it is equipped with a singular hermitian metric $h$ with local weight $\varphi$. Then, one can show that the de Rham class of ...

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**1**answer

141 views

### Chern classes of PU(n)-bundles

Let $PU(n) = U(n)/U(1)$ be the projective unitary group and denote by $BPU(n)$ its classifying space. Consider the algebra $M_n(\mathbb{C})$ as an $n^2$-dimensional Hilbert space equipped with the ...

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88 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 $\...

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99 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 $\...

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101 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 ...

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96 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 ...

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**2**answers

272 views

### 1st Chern class is invariant under choice of section?

How do I see that the 1st Chern class is invariant under choice of section? I know metric invariance follows from how two metrics on line bundle have to be conformally equivalent, but how do we show ...

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**1**answer

216 views

### characteristic classes of tangent bundle of 2-nd unordered configuration space

Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space
$$
B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2
$$
where
$$
\Delta=\{(m,m)\mid m\...

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120 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 ...

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166 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 ...

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111 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?

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**0**answers

171 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)
=
f_*\left(\operatorname{ch}(\alpha).\...

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**1**answer

949 views

### Why is the first chern class of a line bundle $c_1(L) = 1-L$ in complex K-theory?

I'm trying to understand why on earth the first chern class of a line bundle in K-theory $c_1(L) = 1-L$.
I understand that the first Chern class of the trivial bundle is zero, and that $H-1$ ...

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**1**answer

137 views

### Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?

Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n_1:= \...

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**1**answer

235 views

### How does one compute the first Chern class of a Line bundle defined as the Kernel of a linear map?

Let $M$ and $N$ be compact complex manifolds of the same dimension ($m$) and
$\mu: M \rightarrow N$ a holomorphic map. Let $D \subset M$ be the subset of
points of $M$, where $d\mu|_p$ fails to be ...

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**2**answers

406 views

### Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?

Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V ...

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81 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 ...

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**0**answers

122 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) ...

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**1**answer

356 views

### Computing the Chern class of $S^6$ [closed]

I am trying to calculate the Chern class of the tangent bundle of the sphere $S^6$. I am told that this is an interesting case, since $S^6$ is not a complex manifold, but it has an almost complex ...

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**1**answer

289 views

### Chern-Weil Theory for $p_1$

I'm studying Riemannian manifolds that admits a almost-complex manifold, thus
$$3\tau+2\chi=c_1^2,$$
where $\tau$ is the signature, $\chi$ is the euler characteristic and $c_1$ is the first Chern ...

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**1**answer

184 views

### Chern and Segre classes

I've recently started to learn about Chern and Segre classes, and it seems to me that they are very similar, sharing the same important properties and having closely related definitions.
Fulton's ...

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**2**answers

134 views

### Chern classes of reduced space for Hamiltonian circle action

I have a question about Chern class of symplectic reduction.
Let $(M,\omega)$ be a smooth compact symplectic manifold with a Hamiltonian circle action.
Let $H : M \rightarrow \mathbb{R}$ be the ...

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**1**answer

158 views

### Chern classes of the sheaf of LOG differentials

Let $\Omega_X^1(\log D)$ be the (locally free) of logarithmic differentials on a smooth projective variety $X$ with respect to a simple normal crossing divisor $D$.
What are the Chern classes of $\...

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**1**answer

351 views

### top chern class

Let $E\to M$ be a holomorphic vector bundle ($M$ is compact) and assume $s\colon M\to E$ is a non-trivial section transverse to the zero section.
Is it be possible that $s^{-1}(0)\neq \emptyset$, ...

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352 views

### chern classes of push-pulled vector bundles

Let $f:X\to Y$ be a finite cover of smooth algebraic varieties, branched along a divisor $R\subset Y$. Let $E$ be a vector bundle on $Y$. What is the relation between the chern classes of $E$ and the ...

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183 views

### Preimage of $1 \in H^n(M^n)$ under Chern character

Let $M$ be a closed, oriented manifold of dimension $n$. We know that the Chern character induces an isomorphism $K^\ast(M) \otimes \mathbb{Q} \cong H^\ast(M; \mathbb{Q})$ and now I was wondering how ...

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**1**answer

479 views

### Does there always exist a line bundle whose Chern class represents an integer symplectic form?

Let $(M, \omega, J)$ be a compact symplectic manifold with a
compatible almost complex structure $J$, such that the symplectic
form determines an integer cohomology class, ie
$$ [\omega] \in H^2(M, ...

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**0**answers

526 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 ...

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**1**answer

226 views

### Pull-back of algebraic cycles under holomorphic maps

Let $f:X \to Y$ be a holomorphic map between two smooth complex projective manifolds. Is there a good notion of pull-back of algebraic cycles by $f$ which preserves degree in the following sense: ...

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221 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 $M_\alpha$, ...

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**2**answers

463 views

### Top chern class under finite, unramified, dominant morphism

Situation: Let $\Bbbk$ be an algebraically closed field. Assume that $\pi:Y\to X$ is an finite, dominant, unramified morphism between nonsingular varieties of dimensions $n$. Let $d=\deg(\pi)$.
What ...

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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 $\chi(M)$,...

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**2**answers

531 views

### Is there a natural form representing the Thom class of a vector bundle, which when pulled back via the zero section represents the Euler class on the level of forms?

Let $V \rightarrow M$ be an oriented vector bundle over a compact
oriented manifold $M$ equipped with a metric $h$ (the metric $h$
is a metric on the Vector bundle $V$, not on the manifold $M$).
Is ...

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**0**answers

259 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 $D$, $\mathscr{P}(D)$

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**3**answers

633 views

### How does one go from Chern--Weil to cohomology classes on BGL(n,C)?

Let's assume we start with Chern--Weil theory in the following form:
Given a manifold $M$ and a complex vector bundle $V$ over $M$, we can equip $V$ with a $\mathfrak g\mathfrak l_n(\mathbb C)$ ...

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**1**answer

236 views

### Schur polynomials in the Chern classes as direct images

Let $E\to X$ be a rank $r$ holomorphic vector bundle on a $n$-dimensional compact complex manifold. Then, it is well known that one can recover the Segre classes of $E$ as follows.
Let $\pi\colon P(E)\...

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**1**answer

558 views

### Short time existence on Hyperbolic Ricci flow in non-compact case

We know
Laplace equation (elliptic equations)
$ Δ u = 0$
Heat equation (parabolic equations)
$u_t − Δu = 0$
Wave equation (hyperbolic equations)
$u_{tt} − Δu = 0$
we have
- Hyperbolic geometric ...

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**1**answer

485 views

### sign of the First chern class fundamental group of Kahler Manifolds

We know by some facts from Kobayashi, if the Kahler manifold $M$ has positive first Chern class, i.e., $c_1 (M)>0$ then $M$ is simply connected. So if $c_1 (M)<0$ under which assumption on $M$ ,...

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**1**answer

282 views

### recurrence formula for *i*-th Chern class of $CP^n$

one can show that the relation between first Chern class and second Chern class of $CP^n$ is
$\frac{2(n+1)}{n} c_2 (M)=c_1 (M)^2$
here $c_1 (M)^2=c_1 (M)∧c_1 (M)$.
So is there any recurrence ...

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**1**answer

1k views

### First chern class

I know some examples that first Chern class has not sign(negative, positive or zero). But I am looking for a necessary and sufficient condition that first Chern class has sign.

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1k views

### Chern classes of a blow-up at a point

Let $X$ be a nonsingular projective variety over $\mathbb{C}$, and let $\widetilde{X}$ be the blow-up of X at a point $p\in X$.
What relationships exist between the degrees of the Chern classes of $X$...

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**1**answer

369 views

### Does bundle with torsion Chern classes admit flat connection?

I want to know something about torsion in topological k-theory. So, consider complex bundle with chern classes lying in torsion part of integer homologies and my question is : does it admit a flat ...

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**1**answer

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### Calculating chern numbers yields a contradiction, why?

I am really stuck on this one. Let $Y=\mathbb{P}^n$ be the complex projective space and let $\tilde Y$ be the blow-up of $Y$ along a linear subvariety $X$ of codimension $d$. We get the following blow-...

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838 views

### (Second) Chern class of projective space, blown up in a linear subvariety

I already asked the same question at stack exchange but got no response for quite a while, so I thought I'd ask here. I also know that this has a certain resemblance to this question, but I cannot ...

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**1**answer

335 views

### A simple question about the degree of some vector bundle over rational curve.

Let $E$ be a holomorphic vector bundle (infact complex vector bundle is enough) over $\mathbb{P}^1$. Let $c: \mathbb{P^1} \rightarrow \mathbb{P^1}$ be the anti-holomorphic involution, $c(z)=\frac{-1}{...

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**1**answer

522 views

### Is there a simplicial volume definition of Chern Simons invariants?

Suppose we have some compact hyperbolic 3-manifold $M=\Gamma\backslash\mathbb H^3$. Now we know that the hyperbolic volume of $M$ can be defined as (a constant times) the simplicial volume of the ...

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**1**answer

605 views

### Top chern class in positive characteristic

Given a nonsingular, projective variety $X$ of dimension $n$ over an algebraically closed field $k$.
Over $k=\mathbb{C}$, the top chern class $c_n(T_X)$ of the tangent sheaf is the Euler ...

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**0**answers

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 ...