**2**

votes

**0**answers

164 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

226 views

### Canonical metric on moduli space of singular Calabi-Yau varieties

Let $\pi:X\to Y$ be a surjective holomorphic map with connected fibers
and let fibers are singular Calabi-Yau varieties (i.e. numerical dimension is zero) then is it possible to construct canonical ...

**7**

votes

**3**answers

775 views

### Are there examples of Fano manifolds such that Tian's alpha invariant satisfies $\alpha_G(X)=\frac{n}{n+1}$ but without a Kähler-Einstein metric?

The alpha invariant $\alpha(X)$ of a Fano manifold $X$ of dimension $n$ is defined as the infimum of log canonical thresholds of (effective) $\mathbb{Q}$-divisors $D\sim_{\mathbb{Q}}-K_X$. Similarly, ...

**1**

vote

**1**answer

71 views

### Hyperplane generic to a given arrangement

At the moment, I am reading the paper "on the connectivity of the realization spaces of line arrangements" of Nazir and Yoshinaga.
I would like to extend their Lemma 3.2 to higher dimension. However, ...

**0**

votes

**1**answer

507 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 ...

**1**

vote

**0**answers

82 views

### Ellipticity of Bott-Chern Laplacian

I want to prove that Bott-Chern Laplacian
$$\tilde{\Delta}_{BC}^{p,q}=(\partial\bar\partial)(\partial\bar\partial)^*+(\partial\bar\partial)^*(\partial\bar\partial)+(\bar\partial^*\partial)(\bar\...

**9**

votes

**1**answer

216 views

### Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure?

Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure? Every paper or article I looked at, talk only about singular laminations on $\mathbb{CP}^2$. I was wondering why. If you know ...

**5**

votes

**1**answer

186 views

### Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on ...

**5**

votes

**2**answers

217 views

### Alternative construction of the first Chern class map

Let $X$ be a compact Kähler manifold. Consider the exponential sequence $0 \to \mathbf Z \to \mathscr O_X \to \mathscr O_X^* \to 0$. The boundary map gives a map $H^1(X, \mathscr O_X^*) \to H^2(X, \...

**4**

votes

**1**answer

212 views

### Hodge decomposition for Bott-Chern cohomology

$\DeclareMathOperator{im}{im}$
I want to prove that Bott-Chern cohomology group $H^{p,q}_{BC}=\frac{\ker\partial\cap \ker\bar{\partial}}{\im\partial\bar{\partial}}$ has finite dimension via Hodge ...

**3**

votes

**1**answer

158 views

### A very general complex torus is simple

Let us parametrize the set of lattices inside $\mathbb{C}^g$ with the open dense subset $U = \text{GL}_{2g}(\mathbb{R})$ of $\mathbb{R}^{4g^2}$. Does there exist a countable family $(Z_n)_{n \in \...

**4**

votes

**1**answer

279 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 $X_s$, ...

**0**

votes

**1**answer

66 views

### Reference request: Existence of plurisubharmonic potential for positive (1,1)-current on Stein (affine) manifold

I am looking for a reference for the following fact, which I believe should be true.
Let $X$ be a Stein manifold (or smooth affine variety over $\mathbb{C}$).
If $\omega$ is a positive closed $(1,...

**-4**

votes

**0**answers

81 views

### Everywhere holomorphic functions over C [closed]

Let $f, g$ be two everywhere holomorphic functions over ${\Bbb C}$. We consider the local representation of $f, g$ at the origin of ${\Bbb C}$, i.e. $z = 0$. That is, we can consider $f, g \in {\Bbb C}...

**27**

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 ...

**4**

votes

**0**answers

117 views

### Question on Weil-Petersson metric on Teichmuller space

I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space".
The tangent space at ...

**1**

vote

**1**answer

45 views

### On the limit set of eigenvalues of banded Toeplitz Hessenberg matrices

Let $T_{n}(b)$ be the $n\times n$ Toeplitz matrix determined by the symbol
$$
b(z)=\frac{1}{z}+\sum_{j=0}^{k}a_{j}z^{j}
$$
where $k\in\mathbb{N}$ and $a_{0},\dots,a_{k}\in\mathbb{R}$, $a_{k}\neq0$. ...

**5**

votes

**1**answer

89 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 $\Lambda^{p,...

**2**

votes

**2**answers

153 views

### Kähler forms arising as the curvature form of a singular metric on a line bundle

The Fubini-Study metric on complex projective space $\mathbb{P}^n$ is a smooth metric $h = e^{-\phi}$ on the line bundle $\mathcal{O}(1)$ and it is a standard calculation to check that its curvature ...

**1**

vote

**0**answers

94 views

### On dimension of the moduli space of abelian differentials on Riemann surfaces

I fear I'm missing something important here, so forgive me if my question is stupid.
Consider $\mathcal{M}_g$ the moduli space of Riemann surfaces of genus $g>2$ and $\mathcal{H}_g$ the moduli ...

**32**

votes

**2**answers

1k 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. ...

**2**

votes

**1**answer

162 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?

**8**

votes

**1**answer

442 views

### Chern classes of ideal sheaf of an analytic subset

Let $X$ be a Kähler manifold of dimension $n$, and $Z \subset X$ an analytic subset of codimension $k$. I have read in a paper the following result, a proof of which I cannot find:
$$c_k(\mathscr{I}...

**14**

votes

**2**answers

501 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 ...

**0**

votes

**0**answers

77 views

### Action of a lattice on abelian varieties

Let $\pi\colon Y\to\mathbb{P}_\mathbb{C}^1$ be a ramified cover of degree two of $\mathbb{P}_{\mathbb{C}}^1$ such that $Y$ is smooth. I fix a point $x$ on $\mathbb{P}^1$, over which the cover is étale,...

**0**

votes

**1**answer

70 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 ...

**2**

votes

**1**answer

213 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 ...

**3**

votes

**0**answers

92 views

### How many compact complex 3-folds with $b^1 = b^2=h^{1,2}=0$?

Are there any compact complex 3-folds with Betti numbers, $b^1 = b^2 = 0 $ and Hodge number, $h^{1,2}=0$? If yes, then how plentiful are they?

**7**

votes

**2**answers

448 views

### Spin^c structures on manifolds with almost complex structure

Let $M$ be a smooth even-dimensional manifold.
Is it true that for each almost-complex structure $J$ on $M$ there exists a canonical spin$^c$ structure $S_J$ associated to $J$ ?
(I've read this ...

**10**

votes

**1**answer

257 views

### Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?

Is there a closed subscheme $D$ in $\mathbb P^2_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map
$$ \phi: X(\mathbb C) \to \mathbb P^...

**2**

votes

**0**answers

246 views

### $G$-equivariant coherent sheaves on Bott$-$Samelson resolutions

Let $G$ be a Lie group and $B$ a Borel subgroup. $G/B$ is the corresponding flag variety.
Let $w$ be an element of the Weyl group $W$ with a reduced expression
$w = s_1 \cdots s_n$. Let $X_w$ be ...

**3**

votes

**1**answer

930 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
$$\mathfrak{g}=\mathfrak{h}\...

**0**

votes

**1**answer

150 views

### Normal bundle of a fiber of the family of curves

If we have the family of complex curves $f:X\rightarrow Y$, over a complex smooth curve $Y$ , we consider a fiber $C=f^{-1}(y)$ and its tangent bundle $T_{C}$. We know that $df: f^{*}{T_{Y}}_{|C}\...

**6**

votes

**2**answers

174 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 $f:X\...

**6**

votes

**1**answer

290 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

123 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 one-...

**1**

vote

**0**answers

94 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 $\...

**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/...

**6**

votes

**1**answer

347 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

543 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 ...

**6**

votes

**1**answer

398 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?

**0**

votes

**1**answer

398 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

80 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

169 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 ...

**3**

votes

**1**answer

179 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 \mathbb{...

**1**

vote

**1**answer

74 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

238 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

247 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 one-...

**1**

vote

**0**answers

47 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, ...