# Tagged Questions

**1**

vote

**0**answers

86 views

### A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof?
Statement. ...

**10**

votes

**2**answers

1k views

### The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?
Please see : Most interesting mathematics mistake?
Added : According to their method, ...

**0**

votes

**0**answers

279 views

### A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...

**2**

votes

**1**answer

387 views

### Topological degree and polynomial degree

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ...

**1**

vote

**2**answers

239 views

### The Schottky group and the fundamental group of a compact Riemann surface

I am quoting the following description from a paper,
"...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...

**0**

votes

**1**answer

187 views

### Birkhoff decomposition vanishing of the Chern numbers

Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...

**3**

votes

**1**answer

117 views

### Existence of a map between automorphism group of universal covers

Let $f:X\to Y$ be a holomorphic map of holomorphic manifolds. You can assume that $dimY=1$. Let $\tilde X$ and $\tilde Y$ be universal covers of $X$ and $Y$ with group of holomorphic automorphisms ...

**3**

votes

**2**answers

453 views

### Is there a “by hand” proof on the symmetry of the Atiyah class of $TX$?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...

**5**

votes

**1**answer

319 views

### Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane)

It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all ...

**6**

votes

**3**answers

304 views

### Are non-isomorphic covers of riemann surfaces also generally nonisomorphic as riemann surfaces?

Suppose you've got a Riemann surface $E$, and two topological covers $X,Y\rightarrow E$. Suppose $X,Y$ are nonisomorphic topological covers of $E$, then would you expect $X,Y$ as Riemann surfaces ...

**6**

votes

**3**answers

418 views

### Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$

Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What ...

**2**

votes

**1**answer

359 views

### Question about a Lefschetz hyperplane type theorem

Let $X$ be a simply connected projective manifold of dimension $n$ over $\mathbb{C}$ and $D = \cup D_i$ be a divisor with normal crossings such that its all components $D_i$ are smooth and ample.
I ...

**5**

votes

**1**answer

113 views

### Fundamental groups of normal complex quasi-projective varieties

I would like to know if there is an explicit example of a finitely presented group that can not be realised as the (topological) fundamental group of a normal complex quasi-projective variety?

**4**

votes

**2**answers

359 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 ...

**9**

votes

**1**answer

364 views

### Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to ...

**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

442 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

319 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

243 views

### Milnor number in terms of minimal resolution of an isolated singularity.

Suppose $F$ is a holomorphic (or polynomial if you prefer) function on $\mathbb C^3$ and $0$ is an isolated singularity of the surface $F=0$. Then on the one hand we can define Milnor number of this ...

**1**

vote

**0**answers

102 views

### Existence of open dense subset in a Lie group

Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group
and $\Gamma$ a discrete subgroup of $G$ such that the subgroups
$\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all ...

**4**

votes

**1**answer

471 views

### Is a simply connected Ricci-flat Kaehler manifold a Calabi-Yau manifold?

Hi,
I have the following question: Let $(M,\omega, J)$ be a simply connected Kaehler manifold with Ricci-flat Kaehler metric. How can one show that $M$ is a Calabi-Yau manifold. By Calabi-Yau ...

**7**

votes

**2**answers

393 views

### Injective maps on cohomology and Kahler manifolds

Compact Kahler manifolds have the property that surjective maps induce injections on cohomology with coefficents in $\mathbb{Q}$ (That is, if $X,Y$ compact Kahler, then a surjective map $\phi: X ...

**6**

votes

**1**answer

399 views

### Fundamental group of an analytic hypersurface

Let $M$ denote a complex manifold of dimension $n$ and let $X\subset M$ denote an analytic hypersurface. Then it is a standard fact from several complex variables that around a given point $p\in X$ ...

**2**

votes

**3**answers

309 views

### connectedness of the complement of the zero set of a polynomial $P: SL(N,\mathbb{C})^n \rightarrow \mathbb{C}$

I know that the complement of the zero set of a polynomial $P: \mathbb{C}^n \rightarrow \mathbb{C}$ is connected in $\mathbb{C}^n$ (by the way, can anybody suggest a reference?).
Is it possible to ...

**13**

votes

**3**answers

769 views

### Geometric meaning of L-genus

Is there any reasonable geometric meaning of the L-genus for smooth manifolds? or perhaps easier, for complex algebraic surfaces?
The question came up after a friend and I realized that we don't ...

**11**

votes

**1**answer

590 views

### $\mathbb{Z}/2$ is to $\mathbb{Z}/3$ as K3 is to what?

I'd like to know "what" (say, in the classification of complex surfaces) the following complex manifold $X$ is:
Construction: Let $\Lambda$ be the hexagonal lattice in $\mathbb{C}$; that is, the ...

**2**

votes

**1**answer

157 views

### $b_2$ of the blow up of a complex $3$-fold in a curve

Suppose that $V$ is a complex analytic manifold of dimension 3 with mild singularities, say it is an orbifold (i.e. has only quotient singularities). Let $C$ be a complex irreducible curve in $V$. ...

**3**

votes

**0**answers

505 views

### Euler Characteristic in a neighborhood of a Singularity of Complex Curve, and Deformations

Hello all.
Let's say we have a one parameter (complex) family of complex curves on a family of corresponding surfaces. i.e, think of the whole thing living inside some threefold, fibered along ...

**1**

vote

**2**answers

391 views

### On delta complex structures of complex quasi-projective varieties

Q1.Given a quasi-projective variety $X$ over $\mathbf{C}$, is it always possible to find a $\Delta$-complex structure on $X$?
Q2. What is a good reference which gives a survey about what we know of ...

**6**

votes

**1**answer

228 views

### Homotopy type of the complement to a subvariety of $\mathbb C^n$

Let $V^k\subset \mathbb C^n$ be a sub variety, such that all its irreducible components have dimension $\ge k$. Is it true that $\mathbb C^n\setminus V^k$ has homotopy type of a CW complex of ...

**6**

votes

**5**answers

2k views

### An example of a complex manifold without a finite open cover

Are there non-compact complex manifolds that
a) Don't embed in C^n (holomorphically)
and
b) Cannot be covered by a finite number of coordinate open sets?
If b) can be satisfied, then I think so can a) ...

**6**

votes

**3**answers

1k views

### Lefschetz Hyper-plane theorem for singular projective varieties?

Lefschetz hyper-plane theorem for smooth projective varieties, $X\subset \mathbb{P}^{n+1}$ says:
For smooth hyperplane section $Y= X\cap H$, the restriction map
$H^i(X) \rightarrow H^i(Y)$ is an ...

**2**

votes

**2**answers

613 views

### Fundamental group of a product of two curves

Let $S$ be a complex surface whose fundamental group is isomorphic to the fundamental group of a product of two curves of genera $>1$. Does $S$ have to be a product of two curves?

**5**

votes

**2**answers

433 views

### Pushing Complex Structure Forward

Let $p: E\to B$ be a covering map of $C^\infty$ manifolds, where $E$ has a complex structure. There are many cases when we want to know whether $B$ has a complex structure (which is obviously unique) ...

**3**

votes

**2**answers

517 views

### Torsion in the Betti cohomology of complex surfaces

Q1. So what is the "simplest example" of a compact complex manifold of dimenension $2$, say $X$, for which $H_B^2(X,\mathbb{Z})$ has non-trivial torsion.
Q2. How do we think about these torsion ...

**5**

votes

**2**answers

494 views

### On the fundamental group of hypersurfaces

Let $H$ be a smooth projective hypersurface in $\mathbb{P}^n(\mathbb{C})$ where $n\geq 3$. Then by the Lefschetz hyperplane theorem we have that $H^1(H,\mathbb{C})=
...

**5**

votes

**2**answers

511 views

### Topology of the preimage of a point for degree one holomorphic maps

Let $M^n$ and $N^n$ be two compact complex (or complex projective) manifolds. Let $f: M\to N$
be a holomorphic map of degree one. How to prove that for each $x\in N$ the set
$f^{-1}(x)$ is ...

**5**

votes

**3**answers

2k views

### Griffiths and Harris reference

Trying to read the section on Poincare duality from Griffiths and Harris is a nightmare. I want to know if there is a place where Poincare duality and intersection theory are done cleanly and ...

**3**

votes

**1**answer

808 views

### Chern classes generating cohomology

The fact that Chern classes are Hodge classes (and are rational combinations of algebraic cycles) is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my ...

**8**

votes

**0**answers

742 views

### Weight filtration over the integers

This is a follow up question to Weight filtration for smooth analytic manifolds
As mentioned in that question, the integral cohomology of some smooth complex analytic manifolds is equipped with a ...

**9**

votes

**2**answers

576 views

### Are spaces of holomorphic maps manifolds?

Hello,
Let $X$ and $Y$ be two smooth (probably projective) algebraic varieties defined over $\mathbf{C}$.
What is known in general about the (topological) space of holomorphic maps ...