**2**

votes

**0**answers

167 views

### When is the analytification of a variety homeomorphic to $\mathbb C$

Let $X$ be the analytification of a variety $V$ such that $X$ is homeomorphic to $\mathbb R^2$.
What can we say about $X$? Can $V$ be singular?

**1**

vote

**1**answer

89 views

### How far do conjugated Mobius transforms move points?

I start with an automorphism $f$ of the complex unit disc $S^1 = \{ z \in \mathbb{C} : |z| \leq 1\}$. I assume that such a map is given by a Mobius transform, namely
$$
f(z) = \frac{z - ...

**2**

votes

**3**answers

207 views

### Lattice polarized K3 surfaces

I've recently encountered the definition of a lattice polarized K3 surface. What is the idea behind the definition? Surely, there's something deeper to it than merely being a natural generalization of ...

**3**

votes

**1**answer

163 views

### Deformation of Hitchin-Simpson correspondence

Let $X$ be a compact Riemann surface, Hitchin-Simpson correspondence over $X$ says that irreducible representations of $\pi_1(X)$ one-to-one correspondend to stable Higgs bundles with vanishing Chern ...

**13**

votes

**1**answer

761 views

### Are all holomorphic vector bundles on a contractible complex manifold trivial?

It is true that over a contractible manifold all differentiable vector bundles are trivial. However the method of proof does not apply in the holomorphic category.
It is also true that a contractible ...

**2**

votes

**0**answers

192 views

### Generalizing a result of Paul Andi Nagy

I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact
$$
...

**3**

votes

**1**answer

136 views

### Harmonic function osculating a given subharmonic function

Let $\Omega\subseteq\mathbb C$ be an open set and let $\phi:\Omega\to\mathbb R_{\geq 0}$ be an exhausting (i.e. proper) smooth subharmonic function. Fix $p\in\Omega$. Does there exist a harmonic ...

**4**

votes

**2**answers

777 views

### A question about Marsden-Weinstein reduction theory

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J:M\to \frak g^*$ be its moment map then the reduced ...

**2**

votes

**1**answer

210 views

### The behaviour of holomorphic mapping of curves

Given a polynomial, (or rational function, transcendental entire (meromorphic) function) $f$, and a smooth closed Jordan curve $\gamma$, can we give a complete description of the image of $f(\gamma)$ ...

**3**

votes

**3**answers

181 views

### Is it possible to find $h$ hermitian metric such that $Isom_{h}(X) \cong Aut(X)$?

I suspect this is true for some class of analytic manifolds (Riemann surfaces maybe), but my knowledge in differential geometry is very poor, so I could not conclude it. For complex manifolds, is it ...

**1**

vote

**2**answers

670 views

### Spicing up Riemann surfaces course (revised)

I am a master's student planning to write a master's thesis on Riemann surfaces. I plan to study Forster's Lectures on Riemann surfaces. What side topics could one study to spice up the thesis? I am ...

**4**

votes

**1**answer

396 views

### What is the “complex third derivative”?

Background
I am including this information about real higher order derivatives because it does not seem to be common knowledge. I also include a review of the complex Hessian.
If $f:\mathbb{R}^n ...

**4**

votes

**2**answers

374 views

### Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...

**3**

votes

**0**answers

116 views

### Moment map in the singular case

The moment map is defined on the symplectic manifold $(M,\omega)$, or particularly, $(M,\omega)$ is Kahler. While $\omega$ is smooth or differential enough, the definition is obvious to understand, in ...

**0**

votes

**0**answers

123 views

### Moment map of CP^1 as rational normal curve

I am little confused about some basic symplectic geometry about Hamiltonian actions on sphere. I appreciate your comments.
Consider sphere $S^2 = \mathbb{C}P^1$ with its standard symplectic (Kaehler) ...

**3**

votes

**1**answer

131 views

### 3d-analog of “every 2d oriented manifold is complex”

Is there an analog of the statement of "every 2d oriented surface is a complex manifold"?
I saw a theorem in Blair's book, that "every 3d contact metric manifold is a strongly pseudo convex CR ...

**4**

votes

**1**answer

201 views

### Smooth mixed hodge modules - representations of fundamental group?

I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as ...

**2**

votes

**1**answer

280 views

### Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits.
A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:
$X$ is holomorphically convex, ...

**2**

votes

**1**answer

421 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 $L$ ...

**0**

votes

**1**answer

319 views

### Kahler structure on holomorphic principal bundles

Let $G$ be a compact complex Lie group and $M$ be a compact Kähler manifold.
Does there exist any example of a holomorphic principal $G$-bundle over $M$ admitting Kähler structures?

**1**

vote

**1**answer

197 views

### Arithmetic property of a surface of general type

In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...

**2**

votes

**2**answers

274 views

### Etale covers of a hyperelliptic curve

Let $X$ be a hyperelliptic curve of genus at least two.
Let $Y\to X$ be a finite etale morphism with $Y$ connected.Then $Y$ is a smooth projective connected curve.
Is $Y$ hyperelliptic?
More ...

**1**

vote

**1**answer

148 views

### A family of examples of (Brody) hyperbolic surfaces

Let $P(x)$ and $Q(x)$ be two general polynomial of the same degree $d$ (d can be arbitrary large). Consider the surface $S : z^2 = P(x) Q(y)$ in the projective space $\mathbb{P}^3$. I want to prove ...

**2**

votes

**1**answer

114 views

### Uniform estimate for the Cauchy-Riemann equations on a hyperbolic Riemann surface

I have been trying to find the answer to this question in the literature, but have not succeeded. The question is as follows.
Suppose $X$ is a Riemann surface that admits a green's function (i.e. ...

**3**

votes

**1**answer

271 views

### Atiyah classes of holomorphic vector bundles with trivial Chern classes

Let $X$ be compact Kahler and $E \to X$ a holomorphic vector bundle. Then $E$ has an Atiyah class, $At(E)$, valued in the sheaf cohomology $H^1(\Omega_X \otimes \operatorname{End} E)$. Suppose the ...

**1**

vote

**0**answers

81 views

### Translation of “Über kompakte homogene Kählersche Mannigfaltigkeiten”

Has anyone translated Borel and Remmert's 1962 paper titled:
Über kompakte homogene Kählersche Mannigfaltigkeiten?

**2**

votes

**1**answer

138 views

### What are the known examples of compact complex $n$-folds that do not have a strongly Gauduchon metric for $n >2$?

Gauduchon showed that every conformal hermitian structure on a compact complex $n$-fold contains an hermitian metric such that the associated 1,1-form $\omega$ satisfies $\partial ...

**4**

votes

**0**answers

240 views

### Hirzebruch's ICM talk

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

**3**

votes

**0**answers

139 views

### Is there any advantage to knowing that Gauss-Manin is Hermitian flat?

Let $S$ be a complex manifold and let $p : E \to S$ be a holomorphic vector bundle. Is there any advantage to knowing that $E$ carries a flat Hermitian metric $h$, ie a smooth Hermitian metric with ...

**1**

vote

**0**answers

97 views

### algebraic varieties whose fundamental group is subgroup separable wrt subvariety subgroups

Call an algebraic variety $\pi_1$-subgroup separable iff,
for every $Y\subseteq X$ a closed subvariety and $\hat Y\xrightarrow{i} Y$ a normalisation of $Y$,
and subgroup $\Gamma=Im(\pi_1(\hat ...

**0**

votes

**0**answers

109 views

### filling by holomorphic disks method

Can you give me a reference for the proof of the filling by holomorphic disks method, besides Bishop's original paper?

**2**

votes

**1**answer

163 views

### mixed Hodge structure on Cohomology with compact support

Let $X$ be a smooth quasi-projective variety of dimension $n$ over the complex numbers (let us assume it is the complement of a normal crossings divisor in a smooth projective variety).
Then how does ...

**11**

votes

**0**answers

468 views

### What is the holomorphic sectional curvature?

Let $X$ be an $n$-dimensional complex manifold, let $\omega$ be a Kahler metric on $X$ and let $R$ be the $(4,0)$ curvature tensor of $\omega$. We can simplify the tensor $R$ in different ways, two of ...

**4**

votes

**1**answer

120 views

### Weyl group action on complexified Iwasawa decomposition

Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. ...

**1**

vote

**2**answers

268 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 ...

**1**

vote

**1**answer

239 views

### A question about the first eigenvalue for two Kahler metrics

While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies ...

**8**

votes

**4**answers

318 views

### Automorphism group of flag manifolds?

If $F=F(n_1, \ldots, n_k)$ is the (complex) flag variety associated to the partition $(n_1, \ldots, n_k)$, what is the automorphism group of $F$? Here I mean holomorphic and/or variety automorphisms.
...

**0**

votes

**0**answers

77 views

### moduli space of equivariant holomorphic embeddings into the quintic

I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc.
Fix a non-negative integer $g$ and consider
the space
...

**1**

vote

**1**answer

275 views

### A question about complex polarization

Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar P ...

**2**

votes

**1**answer

189 views

### almost holomorphic line bundles

Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...

**1**

vote

**1**answer

85 views

### Do Hermitian metrics also split on the Riemann sphere?

Maybe this is well known, but i could not find a pointer to some literature:
Let us assume $E$ is a rank n vector bundle on the Riemann sphere $\mathbb{C}\mathbb{P}^1$. We know that ...

**9**

votes

**2**answers

362 views

### Newlander-Nirenberg in dimension 2

What is the easiest (and what is the most elementary) way of proving
Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce
it to existence of non-trivial harmonic functions ...

**5**

votes

**2**answers

291 views

### Making Hironaka's theorem explicit for hypersurfaces

Given a smooth hypersurface $H$ in $\mathbb{C}^n$, a theorem of Hironaka promises that one can find a strict normal crossings compactification $\bar{H}$ inside of a projective variety $X$. For me, ...

**3**

votes

**2**answers

236 views

### $\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational

We know that if $X$ is a smooth complex projective variety and we assume that there is a dominant morphism $f : X \to Y$ with $Y$ and the general fibers of $f$ rationally connected. Then $X$ itself is ...

**1**

vote

**0**answers

94 views

### Algebraic set given by sequence of polynomials

When working on some problem, I have end up with a following situation. Suppose
$P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...

**5**

votes

**1**answer

165 views

### The proof of Belyi theorem by Lando and Zvonkin

I'm sorry for asking such a specific question, but i have trouble understanding one detail in the proof of Belyi's theorem in the book "Graphs on surfaces and their applications" by Lando and Zvonkin" ...

**7**

votes

**1**answer

280 views

### Log forms and Tate classes

Let $X$ be a smooth finite type variety over $\mathbb{C}$. Suppose that $\theta$ is a closed algebraic $1$-form whose cohomology class is weight $2$.
Can we always express $\theta$ as
$$\theta = ...

**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:
...

**11**

votes

**2**answers

668 views

### Vector bundles on vector bundles

Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base?
In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X ...

**-1**

votes

**1**answer

143 views

### Order of the zero of a meromorphic function under the action of $Gal(\mathbb{C},\mathbb{Q})$

Let's take $X$ a Riemann surface as an algebraic curve in $\mathbb{P}^n$.
The group $Gal(\mathbb{C},\mathbb{Q})$ (automorphisms of $\mathbb{C}$ which act as the identity on $\mathbb{Q}$) acts on $X$ ...