**2**

votes

**1**answer

146 views

### Control of a meromorphic function according to distance between its zeros

My question is rather philosophical : can a meromorphic function of normalized norm with simple zeros on the flat torus stay close to zero on a large set when its zeros are far from each other ?
The ...

**3**

votes

**1**answer

198 views

### Let $X$ be a projective variety and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable

Is there any approach for the following conjecture?
Let $X$ be a projective Fano manifold and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable.

**4**

votes

**0**answers

148 views

### Is there an example of a holomorphic vector bundle whose Atiyah class vanishes and does not admit a flat connection?

Let $E\to X$ be a holomorphic vector bundle over a Kähler manifold. The vanishing of the Atiyah class, $At(E)=0$, is equivalent to the existence of a holomorphic connection on $E$.
Moreover, it is ...

**2**

votes

**2**answers

281 views

### H. Cartan's “Variétés analytiques complexes et cohomologie”?

Does anyone know where I might find an online version (for free or purchase, translated or in french) of this paper by Henri Cartan from 1953? I know it was published in Colloque sur les fonctions de ...

**1**

vote

**1**answer

95 views

### Hermitian manifold with harmonic holomorphic volume form

Let M be a compact complex 3-manifold with trivial canonical line bundle and Ω be the non-vanishing holomorphic 3-form.
If the real and imaginary part of Ω are both harmonic with respect to the ...

**2**

votes

**0**answers

109 views

### If G-invariant metric is always Kahler-Einstein

Suppose there is an Hermitian symmetric space of compact type X. It is realized in the following way:
$X\hookrightarrow\mathbb{P}^N$ and equip it with induced Fubini-Study metric g. What's more, the ...

**1**

vote

**0**answers

95 views

### reference for weighted blow-up

Let $(0\in X)$ be a germ of a normal 3-fold with a singular point $0$
(over $\mathbb{C}$).
We think of $X$ as a small neighborhood of $0$ (for studying singularity).
If we can think $X$ as a ...

**4**

votes

**1**answer

177 views

### Neron-Severi group for product of curves

Let $C$ be a general genus $g$ curve, how can we describe the Neron-Severi group of its $n$-th self product $C^n=C\times \dots \times C$?
It is a lattice in $H^2(C^n,\mathbb{Z})\cong ...

**57**

votes

**1**answer

2k views

### Is there a complex surface into which every Riemann surface embeds?

This question was previously asked on Math SE.
Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma ...

**0**

votes

**0**answers

114 views

### A simple fundamental group of an hypersurface

Is there an example of analytic hypersurface in $C^n$ such that its fundamental group is simple i.e. does not have normal subgroups except the trivial group and the group itself ?
Thank you
EDIT : ...

**3**

votes

**1**answer

242 views

### Stratification of complex algebraic varieties

Let $V$ be a complex quasi-projective variety, we know from H. Whitney's and B Teissier works on stratifications of algebraic varieties that $V$ has an intrinsic stratification
$$X_0\subset ...

**8**

votes

**2**answers

465 views

### What is the obstruction to the existence of a global Kahler potential?

It is a well-know fact that if $(X,\omega)$ is Kahler then about every point $x \in X$ there exists a neighbourhood $U$ and a function $K \in C^{\infty}(U,\mathbb{R})$ such that $\omega|_U = i\partial ...

**11**

votes

**0**answers

185 views

### Do all simple factors of jacobians of curves come from correspondences?

For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface).
Let $C$ be a curve over ...

**0**

votes

**2**answers

148 views

### Classifying compact homogeneous Kähler manifolds

In this comprehensive answer to an old question, it is stated that
Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group.
...

**2**

votes

**0**answers

128 views

### Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function?
For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...

**3**

votes

**1**answer

131 views

### Are holomorphic quasi-positive line bundles on a Kähler manifold positive?

Holomorphic quasi-positive line bundles on a complex manifold $M$ are line bundles whose chern class can be represented by a closed $(1,1)$-form which is quasi-positive, that is, non-negative at all ...

**4**

votes

**0**answers

75 views

### Hodge-Weil Formula for Quaternionic-Kähler manifold

Let $M$ be a quaternionic-Kähler manifold, with fundamental form $\omega$, and let $L$ be the Lefschetz operator of $\omega$. In the Kähler and, more generally, symplectic cases, there is a mysterious ...

**1**

vote

**0**answers

44 views

### Construction of homogeneous Siegel domain from j-algebra

I am reading bounded homogeneous domain from Piatetski-Shapiro's
book ``Automorphic functions and the geometry of classical domains''
and have questions on how to construct homogeneous Siegel domain
...

**6**

votes

**1**answer

103 views

### Does the degeneracy of the Frölicher spectral sequence vary in families?

I would like to know if there are any known examples of families of complex manifolds for which the Frölicher spectral sequence of one fibre degenerates on the $E_m$ page and the spectral sequence of ...

**5**

votes

**1**answer

134 views

### Spin Structures for Quaternionic-Kaehler and Hyper-Kaehler Manifolds

As is well-known (see Friedrich's book for example) every Kähler manifold is spin (or at least spin$^c$) and the Dirac is given (up to a twist) by $\partial + \partial^*$. What happens in the ...

**4**

votes

**2**answers

143 views

### Are Wolf spaces flag manifolds?

It's all in the title: Are Wolf spaces flag manifolds? Both are group quotients of semi-simple Lie groups. In the Grassmannian case this is so, and I always tacitly assumed it extended to the general ...

**4**

votes

**0**answers

172 views

### What is behind the Hodge conjecture? [duplicate]

My question is quite naive, and my knowledge limited on the subject. I heard lot of talks about Hodge conjecture. I wanted to ask about an intuitive way to figure out why we should care about Hodge ...

**13**

votes

**1**answer

234 views

### Is there an integrable complex structure on SU(3)?

Is there a complex manifold diffeomorphic to SU(3)?
This question arises in a StackExchange discussion by HK Lee, Ted Shifrin and Jason DeVito:
...

**4**

votes

**1**answer

88 views

### Homogeneous Quaternionic-Kähler Structure of the Grassmannians?

Paraphrasing from Cortes' notes:
The quaternionic Kähler condition for a manifold $M$, means that $\operatorname{End}(T(M))$ admits a
parallel subbundle $Q$ which is locally spanned by $3$
...

**1**

vote

**0**answers

174 views

### Family $(X_y,D_y)$ with trivial canonical bundles

Let $i:D\hookrightarrow X$ and $f : X \to Y$ be holomorphic mappings of complex manifolds
such that $i$ is a closed embedding and $f$ as well as$ f \circ i$ are proper and smooth and $D$ is a divisor. ...

**2**

votes

**0**answers

80 views

### Rellich Embedding Theorem for the $2$-Sphere

I'm trying to understand the Rellich-Embedding Theorem in the non-flat case by looking at the $2$-sphere. To be precise, for $S$ the spinor bundle of $S^2$; $L^2(S^2)$ the space of square integrable ...

**6**

votes

**2**answers

795 views

### What is the role of projective spaces in GAGA?

The GAGA theorem is a celebrated elaboration of the idea that complex analytic and complex algebraic geometry are equivalent, at least for smooth projective varieties/manifolds.
I am aware why this ...

**8**

votes

**0**answers

315 views

### rings of modular functions on the upper half plane

Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index.
Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index.
Let $M_0(\Gamma_i)$ denote the ring of modular ...

**4**

votes

**1**answer

105 views

### what's the minimal embedding of orthogonal grassmannian

Suppose X is the orthogonal grassmanian. We know the plucker embedding does not span the whole background CP^N, just span Subspace CP^m. My question is that is there an expression of the isometric ...

**6**

votes

**1**answer

167 views

### Loci in the moduli space of K3 surfaces associated to lattices

The moduli space of K3 surfaces forms a 20-dimensional family with countably many 19-dimensional components $M_d$ corresponding to the polarized K3s $(X,L)$ with $L^2=d$. The moduli space $M_d$ has a ...

**3**

votes

**1**answer

69 views

### compact almost complex submanifolds of complex Lie groups

I find the following Corollary 1.21:
Question: does there exist any complex Lie groups $G$ such that there are some compact almost complex submanifolds (for example, $\mathbb{C}P^m$) of $G$? I want ...

**4**

votes

**1**answer

353 views

### vanishing theorem in algebraic geometry

This is a general question: As we know there are a lot of vanishing theorems like Fujita vanishing, kodaira Nakano vanishing, vanishing for big nef line bundle, Kollár vanishing, etc. Those ...

**8**

votes

**3**answers

294 views

### Trivialisation of vector bundles on Stein spaces

Does every vector bundle on a Stein space have a finite local trivialisation?
Definitions:
Stein space means either a complex analytic Stein space or a nonarchimedean Stein space in the sense of ...

**5**

votes

**1**answer

237 views

### When is the tangent bundle of a manifold naturally a complex manifold?

It is well-known that the cotangent bundle of a manifold is naturally a symplectic manifold. Inspired by mirror symmetry, when is the tangent bundle $TM$ of a manifold $M$ naturally a complex ...

**2**

votes

**0**answers

38 views

### cayley transformation of bounded symmetric domain

Can anyone write down the biholomorphic map between classical bounded symmetric domains(defiend by matrixs) with their siegel upperhalf plane models. I know if it's type 2, i.e $I-Z\bar{Z}^{t}>0$ ...

**4**

votes

**1**answer

80 views

### Almost complex manifold fibered by holomorphic sub-manifolds

Suppose $p:(M,J)\rightarrow (N,I)$ is a submersion between smooth manifolds M and N such that:
$(M,J)$ is an almost-complex manifold.
$(N,I)$ is a complex manifold where $I$ is the integrable ...

**2**

votes

**0**answers

81 views

### Examples of holomorphic Killing vector fields on compact Kahler manifolds

I'm looking for concrete examples of compact Kahler manifolds that admit global holomorphic Killing vector fields.
The only examples I can think of so far are quite trivial:
(i) CP^N with the Fubini ...

**2**

votes

**0**answers

116 views

### Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?

Let $X$ be a scheme of finite type over $\mathbb{C}$ and let $Z \hookrightarrow X$ be a closed subscheme. Consider the associated closed inclusion $Z_{an} \hookrightarrow X_{an}$ between their ...

**6**

votes

**3**answers

291 views

### Are compact, complex, affinely flat manifolds geodesically complete?

Let $M$ be a real, even dimensional, compact manifold endowed with a symplectic form $\omega$ and a flat, torsionless connection $\nabla$ compatible with $\omega$, that is $$\nabla \omega=0.$$
Under ...

**2**

votes

**0**answers

89 views

### Is there an algorithm to compute the intersection of tautological classes on the moduli space of genus one curves?

Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree
$d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let
$L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ ...

**2**

votes

**0**answers

57 views

### kahler einstein metric for exceptional compact type hermitian symmetric space

Can anyone write down the kahler einstein metric for exceptional compact type hermitian symmetric spaces($\frac{E_6}{SO(10)*SO(2)}$ and $\frac{E_7}{E_6*SO(2)}$). I can find the bergmann kernel for ...

**2**

votes

**0**answers

36 views

### Why generalized vectors can be written locally as sum of vectors and 1-forms?

I would like to understand better this point.
In generalized complex geometry the generalized bundle $E$ is defined as a non-trivial fibration of the cotangent bundle $T^*M$ over the tangent bundle ...

**1**

vote

**0**answers

74 views

### If there exists an immersion, then does a neighbourhood of a singular rational curve contain a genuine cuspidal point?

Let $X$ be a compact complex surface and $u_1, u_2: \mathbb{P}^1 \longrightarrow X$ be rational curves that are not multiply covered that represents a class $\beta \in H_2(X, \mathbb{Z})$. Suppose ...

**3**

votes

**0**answers

148 views

### Possible Betti numbers of smooth complex varieties

Given a smooth projective complex variety $X$ of dimension $n$, there are various restrictions on its sequence of Betti numbers $b_0, b_1, ..., b_{2n}$. Of course, $b_0=b_{2n}=1$ and $b_i=b_{2n-i}$ by ...

**0**

votes

**1**answer

95 views

### Unique Equivariant Symplectic Structure for the Full Flag Manifold of $SU(3)$?

I was looking at the following interesting question about the number of equivariant almost complex structures on the full flag manifold of $SU(3)$, and I began to wonder how many equivariant ...

**6**

votes

**1**answer

279 views

### Complex Geometry Consequesnces of Serre's Kahler-Zeta Function

Serre's famous paper Analogues K\"ahl\'eriens de Certaines Conjectures de Weil proves an analogue of the Weil conjectures for compact K\"ahler manifolds. It would go on to inspire the line of attack ...

**2**

votes

**0**answers

103 views

### Deformations of blow ups of $\mathbb{C}^{m}$

let $\mathbb{C}^{m}$ be the complex $m$-space with the standard complex structure and let
$$P:=\left\{p_{1},\ldots,p_{N} \right\}\subset \mathbb{C}^{m}$$
a finite set of points. Now we blow up ...

**2**

votes

**0**answers

90 views

### Can a class be represented by both a $(p,q)$ form and a $(p',q')$ form?

Suppose $X$ is a complex manifold.
If $X$ is Kahler, the cohomology groups decompose into subgroups represented by $(p,q)$ forms.
If $X$ is not Kahler, I think the decomposition may not hold?
Is ...

**4**

votes

**2**answers

251 views

### Riemannian metric of hyperbolic plane

I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector ...

**36**

votes

**0**answers

451 views

### Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here.
$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an ...