# Tagged Questions

**4**

votes

**0**answers

232 views

### Hirzebruch's ICM talk

Is there an English translation of Hirzebruch's 1958 ICM lecture note:
KOMPLEXE MANNIGFALTIGKEITEN
Thank you very much!

**1**

vote

**0**answers

70 views

### Boundary of fibers of submersions

Let $M$ be a smooth manifold with boundary (say of dimension $m$) and let $N$ be a smooth manifold with no boundary (say of dimension $n$ with $m\geq n$). We have the following classical result:
...

**3**

votes

**3**answers

303 views

### Finiteness of De Rham cohomology of smooth quasi-projective varieties

Let $U$ be a smooth quasi-projective variety over $\mathbf{C}$. Let $U^{\infty}$
be $U$ but thought of as a smooth manifold.
Q1: Is there a simple proof (so it should avoid Hironaka's ...

**-1**

votes

**4**answers

293 views

### An isomorphism on space of smooth sections

Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to
$$A=\{f:L^{\times}\to ...

**0**

votes

**1**answer

106 views

### Is the space of degree $d$ curves with marked smooth points dense inside the space of curves with marked points?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d} $ be the space of nonzero
homogeneous degree $d$ polynomials in three variables upto scaling, where
$\delta_d = \frac{d(d+3)}{2} $
(basically degree ...

**3**

votes

**1**answer

184 views

### General position argument for reasonable spaces

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of non zero homogeneous degree $d$ polynomials in three variables up to scaling, where
$\delta_d:= \frac{d(d+3)}{2} $. Given a point $p ...

**4**

votes

**3**answers

510 views

### What is the easiest way to show that three lines in two dimensional space do not intersect?

I have two similar questions:
1) Let $X$ and $Y$ be two measure spaces. Suppose for every point
$x \in X $ there exists a set $ \mathcal{U}_x \subset Y $ of full
measure in $Y$. Suppose $V \subset ...

**3**

votes

**2**answers

270 views

### Is it impossible for the dimension of a topological space to increase under a smooth map?

First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots ...

**4**

votes

**2**answers

360 views

### Is there an analogous concept for the degree of a map, when the spaces are singular?

Let $M$ and $N$ be two smooth compact, oriented manifolds and
$X\subset M$ an oriented submanifold of $M$ of dimension $k$
(not necessarily closed). Suppose in addition that $\bar{X}-X$ is contained ...

**1**

vote

**0**answers

208 views

### Does the closure of a ``nice'' smooth submanifold define a homology class?

Let $M$ be a smooth compact, oriented manifold. Let
$X$ be a submanifold which is of the following type
$$X := \{ p \in M: \psi(p) =0, ~~\varphi(p) \neq 0 \} $$
where
$$ \psi: M \rightarrow V, ...

**0**

votes

**1**answer

444 views

### Does the closure of a smooth algebraic always define a homology class?

Let $X\subset \mathbb{C} \mathbb{P}^{N}$ be a smooth,
algebraic (locally closed) complex
submanifold of $\mathbb{C} \mathbb{P}^N$
of complex dimension $k$. More concretely, $X$ is of the
...

**8**

votes

**2**answers

321 views

### Are there analogous statements for the number of zeros of a section in terms of the Euler class, even when the relevant spaces are not manifolds?

Let $V \rightarrow M$ be an oriented rank $k$ vector bundle over a compact orientd manifold $M$. Let $X \subset M$ be a
compact topological subspace of $M$ that is a smooth oriented submanifold of
...

**1**

vote

**1**answer

118 views

### If you perturb a polynomial by a smooth function, then is the signed number of small zeros of the perturbed equation the same as the lowest non zero derivative?

Let $f: \mathbb{C} \rightarrow \mathbb{C} $ be a function of the form
$$ f(z) = z^n + z^{n+ 1} g(z) $$
where $g$ is a $\textbf{smooth}$ function (not necessarily holomorphic).
Is it true that the ...

**5**

votes

**2**answers

313 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 ...

**2**

votes

**1**answer

480 views

### General position argument

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be ...

**2**

votes

**2**answers

268 views

### Does a generic curve inside the space of curves with a node at a specific point have only finitely many nodes?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be ...

**1**

vote

**1**answer

199 views

### Does the diffeomorphism group preserving a particular section act transitively?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$, be the space of homogeneous
degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where
$\delta_d = \frac{d(d+3)}{2}$. ...

**0**

votes

**1**answer

244 views

### What is the simplest way to show that a section of a vector bundle is transverse to the zero set

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$, be the space of homogeneous
degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where
$\delta_d = \frac{d(d+3)}{2}$. ...

**3**

votes

**1**answer

356 views

### Does passing through a point in general position cut down the dimension by one?

Let $\mathcal{D} \approx P^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$. A point
$p\in \mathbb{P}^2$ gives ...

**0**

votes

**3**answers

297 views

### If you take the closure of two smooth varieties and then take their intersections, is the singular locus still small?

Let
$$ X, Y \subset \mathbb{P}^N$$
be two non singular algebraic varieties of dimensions $k$ and $l$ that
intersect transversally. Is it true that the ``dimension'' of the variety
$\overline{X} \cap ...

**3**

votes

**2**answers

558 views

### Can you ``perturb'' a submanifold to intersect transversally with any other smooth submanifold of projective space?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous
degree $d$ polynomials in three vriables, where
$\delta_d = \frac{d(d+3)}{2}$. Let
$$ X \subset \mathcal{D} \times ...

**2**

votes

**2**answers

427 views

### Does the Bertini Theorem imply that there exists $k$ points such that passing through them imposes linearly independent conditions?

Consider the space of all homogeneous degree $d$ polynomials in three variables
$[X,Y,Z]$, i.e.
$$ f[X,Y,Z] = f_{d00} X^d + f_{d10} X^{d-1}Y + \ldots .$$
This can be thought of as a section of the ...

**4**

votes

**1**answer

285 views

### Unique Almost Complex Structure with a Two-Form

I am studying complex structures $I$ on [real] 4-manifolds $M$, and in particular uniqueness properties. Through one example, it seems that this property (written below) guarantees the existence (and ...

**22**

votes

**8**answers

3k views

### What's the difference between a real manifold and a smooth variety?

I am teaching a course on Riemann Surfaces next term, and would like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular ...

**12**

votes

**3**answers

1k views

### A topological consequence of Riemann-Roch in the almost complex case

This question originated from a conversation with Dmitry that took place here
Is there a complex structure on the 6-sphere?
The Hirzebruch-Riemann-Roch formula expresses the Euler characteristic of ...