**13**

votes

**1**answer

295 views

### List of Applications of the $\partial\overline{\partial}$-lemma

Quoting from Huybrecht's book Complex Geometry on the $\partial\overline{\partial}$-lemma for Kaehler manifolds:
Although it looks like a rather innocent technical statement, it is
crucial for ...

**6**

votes

**2**answers

156 views

### $G$-invariant holomorphic vs. polynomial functions

Let $X\subseteq\Bbb C^n$ be a smooth affine variety and $G$ a complex reductive group acting algebraically on $X$.
Let $x_0\in X$. If there is a non-constant $G$-invariant holomorphic function ...

**6**

votes

**1**answer

273 views

### Is polynomial convexity a topological invariant?

Is the property of being polynomially convex a topological invariant?
In other words, let $M$ and $N$ be two homeomorphic, compact subsets of $n$-dimensional complex Euclidean space, and assume in ...

**5**

votes

**1**answer

95 views

### Density of non-algebraic leaves in the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of $D$, the restriction of $\omega$ to $D$ has ...

**1**

vote

**0**answers

63 views

### Decompositon of the Euler class in the ideal generated by Weyl-invariant polynomials

Let $G$ be a complex reductive Lie group, $B$ be a Borel subgroup, $T\subset B$ be a maximal torus, $W$ be the Weyl group. Then the space $X:=G/B$ is a complex manifold of dimension $n$, denote by ...

**0**

votes

**1**answer

470 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 ...

**16**

votes

**1**answer

376 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:
...

**6**

votes

**1**answer

321 views

### The Hypercomplex Structure of $SU(3)$

(A) In this really stylish answer it is shown that one can define a family of complex structures $J_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash ...

**4**

votes

**2**answers

525 views

### Alternative Almost Complex Structures

Originally posted on Maths Stack Exchange.
Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure ...

**5**

votes

**1**answer

364 views

### Holomorphic vector bundles over $\mathbb{C}^{n}\setminus 0$

Is it true that every holomorphic vector bundle over $\mathbb{C}^{n}\setminus 0$ is trivial? If not true, how can one construct a counterexample?
And just a small note here (wrong):
For $n\leq 2$, ...

**15**

votes

**1**answer

1k views

### What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

**3**

votes

**1**answer

230 views

### How to tell if it's a Moishezon morphism

Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...

**-4**

votes

**0**answers

38 views

### How does x+jy represent a 2D coordinate? [closed]

Let $z = x + jy = |\overrightarrow{z}|e^{j\theta}$ where $j = \sqrt{-1}$ and $\angle{\overrightarrow{z}} = \theta$
How does $z$ represent a 2D coordinate?
Where does the exponential $e$ come from?
...

**0**

votes

**1**answer

396 views

### Positivity of forms

I'm getting stuck with a proposition. Please can someone help me?
Let $E$ be a holomorphic vector bundle with hermitian metric $h$ over
a connected compact Kähler manifold $X$ with Kähler form ...

**3**

votes

**0**answers

67 views

### Open Period Integrals of Elliptically Fibered K3 surfaces

Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, ...

**4**

votes

**1**answer

160 views

### Hyper-Kaehler Strucutre for Compact Lie Groups?

We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia page.
I am asking if it ...

**2**

votes

**1**answer

170 views

### Deformation of $\mathbb{P}^1 \times \mathbb{P}^1$

I learnt that $\mathbb{P}^1 \times \mathbb{P}^1$ is rigid, but can be deformed to a non-rigid Hirzebruch surface $S$. Suppose $\pi: M \to B$ is such deformation such that $\mathbb{P}^1 \times ...

**2**

votes

**0**answers

153 views

### Normal bundle to fibers of a rational morphism

Let $f:X\dashrightarrow C$ be a rational fibration from a 3-dimensional variety $X$ to a curve $C$ such that generic fiber is smooth and different fibers intrsect in smooth curves. Take $S$ to be a ...

**1**

vote

**1**answer

69 views

### Find the Picard Fuchs operator of a four parameter fundamental period

In my research, we have constructed a Calabi-Yau as a hypersurface of a toric variety and we could compute the fundamental period of the complex moduli, which is a series in four parameters. Say
...

**2**

votes

**1**answer

215 views

### Relative tangent bundle and trivilization, tautological foliation

Let $T_{X}\rightarrow X$ be the tangent bundle over a complex manifold $X.$ Let $\pi:PT_{X}\rightarrow X$ be a projectivization of that bundle. Let $L$ be the tautological line bundle of $PT_{X}.$
...

**4**

votes

**1**answer

235 views

### Algebraicity and non-algebraicity of leaves of the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of
$D$, the restriction of $\omega$ to $D$ has ...

**30**

votes

**2**answers

919 views

### Complex manifold with subvarieties but no submanifolds

I previously asked this question on MSE and offered a bounty but received no responses.
There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. ...

**5**

votes

**1**answer

72 views

### Complex manifolds with spanning sets of holomorphic tensor fields

This question is an extension of this one. Consider a complex manifold $(M^{2n}, J)$. Fix $1 \leq p \leq n-1$, and suppose that the space of holomorphic sections of $\Lambda^{p,0}$ spans ...

**1**

vote

**0**answers

43 views

### Hypercomplex, hyperKahler, or quaternion-Kähler from Joins/Connected Sums

I am looking for examples of (compact) hypercomplex, hyperKahler, or quaternion-Kähler manifolds which can be constructed as joins/connected sums of manifolds which do are not hypercomplex, ...

**5**

votes

**3**answers

640 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 ...

**3**

votes

**2**answers

212 views

### Chern classes and singular hermitian metrics on vector bundles

Let $L$ be a holomorphic line bundle on a complex manifold $X$, and assume it is equipped with a singular hermitian metric $h$ with local weight $\varphi$. Then, one can show that the de Rham class of ...

**2**

votes

**1**answer

155 views

### Kahler Ricci flow in Fano fibration

Let $f:X\to Y$ be a Fano fibration of Kahler manifolds $X, Y$. Then why the Kahler Ricci flow
$$\frac{\partial \omega}{\partial t}=-Ric(\omega(t))$$
starting of $[\omega_0]=f^*(\omega_Y)+c_1(X)$ ...

**2**

votes

**2**answers

156 views

### Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...

**7**

votes

**2**answers

1k views

### Riemannian metrics as sections of a vector bundle

Let $\pi \colon E \to M$ be a smooth vector bundle. A Riemannian metric on $E$ can be regarded as a global section of the vector bundle $(E\otimes E)^{\ast}$, or more specifically, the subbundle ...

**1**

vote

**1**answer

129 views

### A necessary condition for existence of Ricci flat metric on pair (X,D)

Let $X$ be a complex compact manifold with simple normal crossing divisor $D$. Is the condition $K_X +D = 0$ necessary for the existence of Ricci-flat metric?

**13**

votes

**2**answers

455 views

### regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that
$$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$
is constant on $L.$ Could somebody ...

**1**

vote

**0**answers

96 views

### Applications of Iss'sa's theorem on homomorphisms between algebras of meromorphc functions

In Remmert's book Funktionentheorie II, the following theorem, apparently due to Hironaka under the pseudonym Iss'sa, is proved:
Let $U,V \subseteq \mathbb{C}$ be open subsets. Every ...

**2**

votes

**0**answers

116 views

### Kodaira fibration and moduli space of Riemann surfaces

Here we mean Kodaira fibration $f: X \rightarrow C$ where $f$ is a
holomorphic submersion with maximal rank everywhere, but not a
complex fiber bundle map. Such a surface has been constructed by
...

**1**

vote

**0**answers

47 views

### Hyper-Complex Connected Sums of Grassmannians?

As we all know, the Grassmannians are Kaehler manifolds. Is there anyway to take connected sums of Grassmannians and produce a examples of hypercomplex manifolds, hyper-Kaehler or Calabi--Yau exmples ...

**5**

votes

**0**answers

63 views

### On K-theory of blow-ups of compact complex manifolds

Is there a long exact sequence for the K-theory of (coherent sheaves on) blow-ups of compact complex manifolds? Does it split? What can one say on (possibly, singular) complex analytic spaces here?
...

**5**

votes

**0**answers

147 views

### When a proper smooth fibration is isotrivial?

Let $\pi : X \to Y$ be a proper smooth morphism between complex quasi-projective varieties. Assume there exists a connected and Zariski dense analytic subvariety $Z$ of $Y$ such that for any two ...

**0**

votes

**1**answer

61 views

### Sequence of translation surfaces and length of saddle connections

A sequence of translation surfaces $(X_n,\omega_n)$ is said to "go to infinity" if it leaves every compact set in the space of translation surfaces as $n$ goes to infinity. I know that this is ...

**8**

votes

**1**answer

1k views

### How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry?
What do I mean by complex geometry? ...

**6**

votes

**1**answer

138 views

### First Chern class vanishes on a Lagrangian submanifold

Suppose $(M,\omega)$ is a symplectic manifold and $L \subset M$ is a compact Lagrangian submanifold. Is it the case that the first Chern class $c_1(TM) \in H^2(M)$ vanishes when restricted to the ...

**8**

votes

**3**answers

2k views

### Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?

Let $X$ be a complex analytic space. It is a 'well known fact' that the categories of local systems on $X$ (i.e. locally constant sheaves with stalk $C^n$), and of (holomorphic) vector bundles on $X$ ...

**25**

votes

**2**answers

5k 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 this related post and also the following post.
Added : According to their ...

**5**

votes

**3**answers

340 views

### Families of abelian varieties on the line (or more generally simply connected varieties)

I'm curious whether the following is true:
Question 1: Let $V/\mathbb{C}$ be a smooth connected variety such that $V^\text{an}$ is simply connected. Then, is every abelian scheme $f:\mathscr{A}\to ...

**8**

votes

**0**answers

157 views

### Holomorphic contractibility of GL(H)?

Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later ...

**3**

votes

**1**answer

912 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

**1**answer

157 views

### The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold

Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...

**1**

vote

**0**answers

119 views

### Atiyah-Guillemin-Sternberg Theorem for current

The Atiyah-Guillemin-Sternberg theorem says that the image of a moment map of a toric action is independent of the choice of symplectic form in the cohomology class. So all is well for a smooth ...

**0**

votes

**0**answers

77 views

### How to investigate the harmonocity of holomorphic vector fields?

Let $(M,g,J)$ be a Kahler manifold and $\nabla$ be its Levi-Civita connection. We know that $\Delta _gX=||\nabla X||^2X$ is the characterizing equation for harmonic unit vector fields.
I dont know ...

**22**

votes

**3**answers

3k views

### Which Kahler Manifolds are also Einstein Manifolds?

Is it known which Kahler manifolds are also Einstein manifolds? For example complex projective spaces are Einstein. Are the Grassmannians Einsein? Are all flag manifolds Einstein?

**2**

votes

**1**answer

102 views

### Kernel of projection formula

For a closed embedding of compact complex manifolds
$$
\iota : Y \hookrightarrow X
$$
and any $\alpha \in H^*(X,\mathbb Q)$, we have trivially:
$$
\iota^*(\alpha)=0\quad \Rightarrow ...

**1**

vote

**0**answers

72 views

### How many real-analytic forms exist on a real-analytic manifold?

I hope my question isn't too vague:
Let $M$ be a real-analytic compact manifold (say surface for simplicity). How big is $\Omega^k(M)$ (space of global real-analytic $k$-forms)?
Let me explain ...