1
vote
0answers
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
2answers
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
2answers
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
1answer
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
3answers
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
3answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
3answers
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
3answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
5answers
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
3answers
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
2answers
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
2answers
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
2answers
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
2answers
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
2answers
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
3answers
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
1answer
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
0answers
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
2answers
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 ...