# Tagged Questions

**3**

votes

**1**answer

298 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 $s^{-1}(0)\neq \emptyset$, ...

**8**

votes

**1**answer

312 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, ...

**5**

votes

**2**answers

264 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 ...

**3**

votes

**1**answer

290 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, ...

**4**

votes

**0**answers

335 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 ...

**4**

votes

**1**answer

472 views

### Tensor product of a line bundle with a large multiple of another positive line bundle also positive?

Let $X$ be a complex manifold and $\mathcal{L}$ be a positive line bundle on $X$. If $E$ is any other line bundle on $X$, then is it true that for all sufficiently large $m$, $\mathcal{L}^m \otimes E$ ...

**5**

votes

**1**answer

358 views

### Can one bound the todd class of a 3-dimensional variety polynomially in c_3

This question is on bounding the degree of the Todd class. Let me explain where this comes from.
Suppose that $X$ is a smooth compact connected complex surface. Let $c_i=c_i(TX)$ be its $i$-th Chern ...

**6**

votes

**1**answer

585 views

### How does f_* O_X measure ramification and Grothendieck-Riemann-Roch

Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective varieties over a field $k$ of characteristic zero, where $\dim X=\dim Y$. Then $f$ is flat. Hence $f_\ast \mathcal{O}_X$ is a ...

**15**

votes

**1**answer

1k views

### GAGA and Chern classes

My question is as follows.
Do the Chern classes as defined by Grothendieck for smooth projective varieties coincide with the Chern classes as defined with the aid of invariant polynomials and ...