**6**

votes

**1**answer

962 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? ...

**21**

votes

**1**answer

445 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

**3**answers

290 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 ...

**7**

votes

**0**answers

137 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

910 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

138 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

95 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

67 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?

**5**

votes

**1**answer

53 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 ...

**2**

votes

**1**answer

91 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 ...

**0**

votes

**0**answers

64 views

### Use of algebraic topology in geometry(differential and complex analytic) [on hold]

I'm curious in seeing connections of algebraic topology with other areas of mathematics.
As far as I know, it's not really used in scheme-theoretic algebraic geometry and arithmetics(aside from ...

**1**

vote

**0**answers

70 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 ...

**4**

votes

**0**answers

155 views

### Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical

Let start with a definition
Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...

**-2**

votes

**0**answers

133 views

### Reference for algebraic manipulation of sheaves [closed]

I am currently playing with sheaves over families of algebraic varieties ($O_x$-modules) their torsion sub-sheaf, higher direct images and tensor products, I am looking for a good reference to learn ...

**0**

votes

**1**answer

434 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 ...

**2**

votes

**2**answers

208 views

### Cohomology of Homogeneous Complex Manifolds

Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all ...

**2**

votes

**0**answers

140 views

### When are Kähler potentials bounded from below?

The prototypical example of global Kähler potential is the one of the standard Kähler structure on $\Bbb C^n$ given by
$$f:\Bbb C^n\longrightarrow \Bbb R,\quad f(z_1,\ldots,z_n)=\sum_{k=1}^n|z_k|^2.$$
...

**2**

votes

**1**answer

86 views

### On some curves of real values of a rational function

For given parameters $a_{1},\dots,a_{k}\in\mathbb{R}$, define the rational function $\phi:\mathbb{C}\to\mathbb{C}$ as
$$\phi(z)=\frac{1}{z}-a_{1}z-a_{2}z^{2}-\dots-a_{k}z^{k}.$$
The domain of its real ...

**5**

votes

**0**answers

126 views

### Symplectic invariance of Hodge numbers?

Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$.
My ...

**15**

votes

**1**answer

815 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

228 views

### Classification of the Kähler Structures on the Sphere

Is there a classification of the Kähler structure on the sphere? More generally, is there a classification of the Kähler structures on the complex projective spaces? Even more generally, what about ...

**1**

vote

**0**answers

102 views

### Calabi-Yau with nodes

Suppose $X$ is a singular projective irreducible complex variety of dimension 3, and its singular loci are finite number of nodes, and its smooth locus $X_1$ is a Calabi-Yau quasi-projective variety, ...

**7**

votes

**1**answer

376 views

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

Hello,
Here's my question : 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 ...

**1**

vote

**0**answers

74 views

### Poincaré inequality for holomorphic line bundles

Let $M$ be a Riemann surface of genus >1, $g$ be an Hermitian metric on $M$. Let $E$ is a holomorphic negative line bundle over $M$, for example, the holomorhic tangent bundle of $M$. Let $h$ be an ...

**0**

votes

**1**answer

352 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 ...

**2**

votes

**0**answers

94 views

### Quantizable vs. integral Kahler form

Let $(M,\omega)$ be a (not necessarily compact) Kahler manifold. Then the form $\omega$ is integral if and only if $\omega \in c_1 (L) $ for some holomorphic line bundle $L$.
A Hermitian holomorphic ...

**2**

votes

**0**answers

100 views

### local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$

Let $X$ be a rational threefold (over the field of complex numbers) with terminal singularities. It is well-known that $X$ has only finitely many singular points $x_1,x_2, \ldots,x_n$.
To be more ...

**1**

vote

**2**answers

107 views

### Smooth, irreducible surface with real part containing two projective planes

Let $X$ be a smooth and irreducible projective variety over $\mathbb{R}$ of dimension two. I am looking for an instance of such a variety where two distinct connected components of $X(\mathbb{R})$ are ...

**23**

votes

**1**answer

591 views

### Biholomophic non-Algebraically Isomorphic Varieties

Recently, when writing a review for MathSciNet, the following question arose:
Is it true that two smooth complex varieties that are biholomorphic are algebraically isomorphic? The converse is true ...

**7**

votes

**0**answers

174 views

### A Hartogs-type criterion for flatness

Let $U$ be a smooth affine connected variety over $\mathbb C$ and let $V\subset U$ be an open whose complement is of codimension at least two.
Now, let $Y$ be a smooth quasi-affine connected variety ...

**6**

votes

**0**answers

399 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 ...

**2**

votes

**0**answers

37 views

### Estimates for hyperbolic metrics on nested surfaces

Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} ...

**4**

votes

**1**answer

48 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 ...

**3**

votes

**0**answers

251 views

### Tian's approach for solving the conjecture of invariance of plurigenera in Kahler setting

Let $f:X\to Y$ be a smooth holomorphic fibre space whose fibres $f^{-1}(y)$ have pseudoeffective canonical bundles. suppose that
$$\frac{\partial \omega(t)}{\partial ...

**5**

votes

**1**answer

235 views

### How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence
$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...

**3**

votes

**1**answer

204 views

### Confusion surrounding the Koszul-Malgrange theorem

I recently had the need to appeal to some complex geometry in my research and have been trying to unravel the various relationships surrounding the Koszul-Malgrange theorem.
According to nlab, the ...

**5**

votes

**1**answer

98 views

### Are square tiled surfaces dense in the moduli space of translation surfaces?

I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt.
At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is ...

**4**

votes

**1**answer

169 views

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

I want to understand compact complex manifolds $(M^{2n}, J)$ with the following property: there exists a collection $\{X_i\}_{i=1}^L$ of holomorphic vector fields (sections of $(T^{1,0}_{\mathbb C} ...

**4**

votes

**1**answer

91 views

### Chern-Einstein metrics on complex Hermitian manifolds

Metric on a Riemannian manifold $(M,g)$ is Einstein, if for some function $\lambda\colon M\to \mathbb R$
$$
Ric(g)=\lambda g.
$$
It is well know, that such $\lambda$ is, in fact, a constant.
The ...

**4**

votes

**0**answers

166 views

### Kähler quotients of affine varieties and GIT

Let $X\subseteq \Bbb C^n$ be a smooth affine variety and $G=K_{\Bbb C}$ a complex reductive group acting linearly on $\Bbb C^n$ preserving $X$ (where $K$ is a maximal compact subgroup of $G$). Suppose ...

**-1**

votes

**1**answer

63 views

### Zariski open set in orthogonal grassmanian [closed]

I am confused about the following question.
Consider $\mathbb C^4$ endowed with nondegenerate symmetric bilinear form ...

**0**

votes

**0**answers

134 views

### A question about invariance of plurigenera

Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So deformation invariance of plurigenera says that the ...

**2**

votes

**1**answer

144 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 $[\omega_0]=f^*(\omega_Y)+c_1(X)$ ...

**1**

vote

**0**answers

97 views

### Monodromy of Geometric Variation of Hodge Structures over punctured disc

We know that the monodromy action $T$ is a morphism of limiting MHS on cohomology of nearby fiber $H^n(X_{\infty})$ derived from the Geometric variation of hodge structure $\pi: \mathcal{X}\rightarrow ...

**1**

vote

**1**answer

105 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?

**6**

votes

**0**answers

79 views

### Fubini-Study form on weighted projective spaces

As it is known, $\mathbb CP^n$ with Fubini-Study symplectic form can be get by the symplectic reduction of $\mathbb C^{n+1}$ with a symplectic form $\sum_{i=0}^n dz_i\wedge d\bar z_i$ by a hamiltonian ...

**11**

votes

**2**answers

375 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 ...

**3**

votes

**0**answers

171 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

**0**answers

159 views

### Biholomorphic neighborhoods of the boundary of Stein domains

Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...