**0**

votes

**0**answers

86 views

### Constructing special holomorphic functions

I would appreciate any help with this question as I am not sure how I should approach it.
Suppose $ D$ is the unit disk and that $A(x)$ is a real valued smooth function on $D$.
Does there exist a ...

**0**

votes

**1**answer

262 views

### The injection of direct image sheaf

Let $f:X \longrightarrow Y$ be a smooth holomorphic fibration between K\"ahler manifolds and $L$ be a holomorphic line bundle on $X$. Let $m$ be a positive integer. We denote by ...

**5**

votes

**2**answers

710 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 ...

**11**

votes

**1**answer

1k views

### Theorem of Bryant in higher dimensions

I have the following question. I read about Bryant's theorem which says that: any real-analytic 3-dimensional Riemannian manifold $(Y,g)$ with real-analytic metric $g$ can be isometrically embedded as ...

**7**

votes

**0**answers

286 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

**0**answers

146 views

### Almost complex structures on compact surfaces

Let $M$ be a real manifold of dimension $n$ and $E\rightarrow M$ be a rank two vector bundle above it. Let $J_0$ and $J_1$ be two almost complex structures on $E.$ If there is a continuous ...

**5**

votes

**0**answers

345 views

### When a Spherical variety is $K$-stable

Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical.
My question is When a Spherical variety is $K$-stable? Is ...

**0**

votes

**1**answer

234 views

### Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...

**4**

votes

**1**answer

79 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 ...

**4**

votes

**0**answers

85 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

60 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 ...

**2**

votes

**0**answers

245 views

### Orbital integral by using symplectic quotient

Let $(M,\omega)$ be a compact symplectic Hamiltonian $G$-manifold and $\Gamma_{\text hol}(M,L)$ be the space of holomorphic sections of the line bundle $L\to M$
I am looking for a proof for ...

**4**

votes

**1**answer

301 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 ...

**4**

votes

**1**answer

344 views

### Affine space structure on the space of Hermitian connections

I'm reading Gauduchon's paper Hermitian connections and Dirac operators.
For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = ...

**8**

votes

**3**answers

235 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

218 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 ...

**0**

votes

**1**answer

444 views

### Artin approximation of a diagram

Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods ...

**1**

vote

**0**answers

28 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 $I-Z\bar{Z}^{t}>0$ ...

**6**

votes

**3**answers

267 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 ...

**4**

votes

**1**answer

74 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 ...

**1**

vote

**0**answers

52 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

109 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 ...

**2**

votes

**0**answers

74 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) $ ...

**0**

votes

**0**answers

54 views

### Complex structure of a torus [migrated]

Given the definition of complex structure for a complex manifold:
the real $(1,1)$ type tensor $J_p : T_p M \rightarrow T_p M $ defined by
$$ J_p (\frac{\partial}{\partial x^\mu}) = ...

**1**

vote

**0**answers

47 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 ...

**1**

vote

**0**answers

30 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 ...

**4**

votes

**2**answers

224 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 ...

**6**

votes

**1**answer

240 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 ...

**1**

vote

**0**answers

69 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 ...

**22**

votes

**4**answers

2k views

### de Rham vs Dolbeault Cohomology

For a complex manifold $M$, one can consider (A) its de Rham cohomology, or (B) its Dolbeault cohomology. I'm looking for some motivation as to why one would bother introducing Dolbeault cohomology. ...

**0**

votes

**1**answer

77 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 ...

**3**

votes

**0**answers

134 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 ...

**2**

votes

**0**answers

102 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

82 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 ...

**7**

votes

**1**answer

505 views

### Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{n-1}$

Let $f,g:\mathbb{C}^n\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^n\to \mathbb{C}^n$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...

**5**

votes

**0**answers

509 views

### Almost-Kahler Einstein four manifolds

Are the odd-degree Betti numbers of a compact Almost-Kahler Einstein four manifold necessarily even ?

**1**

vote

**1**answer

783 views

### Hamiltonian potentials of holomorphic vector fields on modifications of Kahler manifolds

let $(M,\omega)$ be a compact Kähler manifold. Let $\mathfrak{g}=H^{0}(M,T_{M})$ be the Lie algebra of holomorphic vector fields on $M$.We can decompose $\mathfrak{g}$ as
...

**-2**

votes

**0**answers

80 views

### almost complex embedding of $S^2$ and $S^6$ into $\mathbb{C}^N$ [migrated]

In Which Spheres are Complex Manifolds? , I find that $S^2=\mathbb{C}P^1$ is a complex manifold and $S^6$ is an almost complex manifold.
Are there references about:
What is the smallest integer $N$ ...

**28**

votes

**0**answers

322 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 ...

**2**

votes

**0**answers

102 views

### Lie algebra of holomorphic vector fields

It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly.
For example, take $X=\mathbb{P}^n$, ...

**4**

votes

**0**answers

121 views

### Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

Does every compact complex manifold of complex dimension greater than or equal two
possess a nontrivial rank 2 holomorphic vector bundle?

**7**

votes

**2**answers

377 views

### Generalising the Penrose Twistor Fibration

As is well known, there exists a fibration $\mathbb{CP}^3 \to S^4$, of the four sphere by complex projective $3$-space, called the Penrose twistor fibration. Does this fibration admit a "canonical" ...

**3**

votes

**2**answers

90 views

### Has anyone developed a technique to generate a polytope given (possibly redundant) inequality constraints? [closed]

I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm just trying to find the vertices for a feasible region, given a set of inequality ...

**2**

votes

**1**answer

76 views

### Higher dimensional analogue of Ahlfors covering surface theory

It is well known that Ahlfors covering surface theory in one dimensional is very powerful in dealing with many problems. I wonder whether there exists some generalization of this theory into higher ...

**4**

votes

**1**answer

138 views

### Equivariant Almost Complex Structures on the Full Flag Manifolds

On complex projective space ${\bf CP}^m$, there exists a unique $SU(m+1)$-equivariant almost-complex structure. What happens for the case of the full flag manifold of $SU(m+1)$, which is to say the ...

**0**

votes

**1**answer

198 views

### Torsion in the (co-)homology of a smooth projective variety - what is known in general?

There are lots of ways in which the complex singular (co-)homology of a smooth projective variety over $\mathbb{C}$ is "special" among complex manifolds - the hard Lefschetz theorem, the Hodge ...

**2**

votes

**0**answers

53 views

### Group of real analytic isometries of $g$-fold product of the Poincare upper half plane

Let $\mathfrak{h}^g$ be the cartesian product of $g$ copies of the Poincare upper half plane. We endow $\mathfrak{h}^g$ with the usual Poincare metric given in local coordinates by $ds^2=\sum_{i=1}^g ...

**0**

votes

**0**answers

161 views

### Bigness of a symplectic form on pair $(X,D)$

Let $(M,\omega_M)$ be a compact Kähler manifold. We say that a semi-positive $(1,1)$ form $\omega$ is big iff $$\int_M\omega^n>0$$.
Now let we have the pair $(X,D)$ where $D$ is a divisor on ...

**3**

votes

**1**answer

115 views

### Stokes-like Theorem for Dolbeault Operator

I have a simple question regarding complex geometry: is there an analog for the Stokes Theorem for the Dolbeault Operator $\bar{\partial}$? For instance, suppose that $M$ is a closed complex manifold ...

**0**

votes

**0**answers

42 views

### What is an induced Abel Jacobi map?

Let $X$ be a compact Kahler threefold, $C\subset X$ be a smooth curve, $\tau\colon\tilde{X}\to X$ be the blow up of $X$ along $C$, $j\colon E\to \tilde{X}$ be rthe exceptional divisor, $\tau_E$ be the ...