**9**

votes

**1**answer

291 views

### Cohomology of vector bundles in families

Let $\pi \colon \mathfrak{X} \rightarrow B$ be a deformation of complex compact manifolds and $E$ be a holomorphic vector bundle on $\mathfrak{X}$ (or a coherent sheaf on $\mathfrak{X}$ that is flat ...

**3**

votes

**1**answer

91 views

### Can the $(n-1)$ power of the hermitian form on a compact complex $n$-fold be $\partial {\bar{\partial}}$-exact?

Let $\omega$ be the hermitian form for an hermitian metric on a compact complex manifold. Can $\omega^{n-1}$ be $\partial {\bar{\partial}}$-exact? (We know that our hermitian metric must necessarily ...

**3**

votes

**1**answer

268 views

### Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator?

A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition
\begin{equation}
...

**3**

votes

**0**answers

94 views

### Generalizing von Staudt's synthetic construction of the complex numbers

Starting from the real projective plane described synthetically/axiomatically, it is possible to construct the complex projective plane directly without passing through coordinates: one adjoins two ...

**1**

vote

**0**answers

122 views

### Homeomorphism of fibers of holomorphic maps

EDIT (after the comment by Jason Starr): Let $X$ be a complex algebraic (or, more generally, analytic) variety, possibly singular and non-compact. Let $f\colon X\to D^*$ be a proper algebraic morphism ...

**2**

votes

**0**answers

102 views

### Name for the variety of preimages of a finite morphism

If $f:X\to Y$ is a finite morphism of degree $d$ between two varieties, you get a closed subset of the symmetric product $X^{(d)}$ (or perhaps rather the Hilbert scheme $X^{[d]}$), defined as the ...

**4**

votes

**0**answers

111 views

### classification of homogenous complex manifolds

Suppose $X$ is a complex manifold (doesn't assume it's Kahler), and it's holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ?

**1**

vote

**0**answers

107 views

### Line bundles with vanishing cohomology on Calabi-Yau manifold

Suppose we have some line bundle $L(D)$ on Calabi-Yau threefold. Let's call this line bundle "rigid" if $H^0(X,L(D)) \simeq \mathbb{C}$ and $H^i(X,L(D))=0$ for $i=1,2,3$.
Is anything known about such ...

**0**

votes

**0**answers

101 views

### Stability of automorphism group of complex manifolds

Are there any stability theorems of analytic automorphism groups concerning the deformation of complex manifolds.
For example, in the case of K3 or Calabi-Yau.

**8**

votes

**0**answers

74 views

### Holomorphic natural bundles and operators

I am wondering up to what extent the classical theory of (smooth) natural bundles and natural operations extends to the holomorphic setting.
After a quick thought, I've gone through the standard ...

**2**

votes

**1**answer

152 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

215 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

180 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

299 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

110 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

160 views

### Is a G-invariant metric always Kähler-Einstein?

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

**1**

vote

**0**answers

157 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

205 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 \mathbb{Z}^{n+...

**58**

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

116 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

331 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 X_2\...

**8**

votes

**2**answers

629 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

204 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

177 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

177 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

149 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

84 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

68 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

124 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

**2**answers

173 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

151 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

182 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 ...

**16**

votes

**1**answer

391 views

### Is there an integrable complex structure on $\mathrm{SU}(3)$?

Is there a complex manifold diffeomorphic to $\mathrm{SU}(3)$?
This question arises in a StackExchange discussion by HK Lee, Ted Shifrin and Jason DeVito:
http://math.stackexchange.com/questions/...

**4**

votes

**1**answer

102 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

181 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

89 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

840 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

323 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

126 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

176 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

74 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

375 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

332 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

250 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 manifold?...

**2**

votes

**0**answers

41 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

88 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 almost-...

**2**

votes

**0**answers

119 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

121 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 ...

**7**

votes

**3**answers

306 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

108 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) $ ...