**5**

votes

**0**answers

152 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

81 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 ...

**1**

vote

**0**answers

98 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 $\mathbb{...

**6**

votes

**1**answer

142 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 ...

**2**

votes

**2**answers

164 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 ...

**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. ...

**8**

votes

**0**answers

158 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 ...

**2**

votes

**1**answer

162 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

79 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 ...

**5**

votes

**3**answers

379 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 ...

**6**

votes

**2**answers

175 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\...

**1**

vote

**0**answers

73 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 ...

**2**

votes

**0**answers

160 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

95 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 ...

**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? ...

**5**

votes

**0**answers

138 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 ...

**1**

vote

**0**answers

105 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, ...

**4**

votes

**0**answers

172 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 ...

**1**

vote

**0**answers

79 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 ...

**2**

votes

**0**answers

100 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

138 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 ...

**7**

votes

**0**answers

180 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 ...

**1**

vote

**2**answers

122 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 ...

**2**

votes

**1**answer

89 views

### Quantifying the monotonicity property of the hyperbolic metric

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} &...

**5**

votes

**1**answer

100 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,...

**3**

votes

**1**answer

234 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

110 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 dense....

**2**

votes

**1**answer

105 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 \quad\iota_*\iota^*...

**4**

votes

**1**answer

187 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 $(T^{1,0}_{\mathbb C} M)$...

**4**

votes

**1**answer

115 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 ...

**-1**

votes

**1**answer

70 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 $J:=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&1&0&...

**0**

votes

**0**answers

148 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 ...

**4**

votes

**0**answers

206 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**

vote

**0**answers

101 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 ...

**2**

votes

**1**answer

166 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

106 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 ...

**1**

vote

**0**answers

229 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 ...

**4**

votes

**1**answer

175 views

### Embed a bordered Riemann surface into punctured Riemann surfaces?

Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ ...

**5**

votes

**1**answer

200 views

### Is the analytification functor part of a geometric morphism of topoi?

Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras.
A complex analytic space for our purpose is a locally ringed space locally ...

**1**

vote

**0**answers

90 views

### Orders of zeros of section of sheaf

We have a semistable family (fibers has normal crossings and they are reduced, multp 1) $f: X \rightarrow Y,$ of complex curves over a smooth curve $Y.$ The family is smooth over the set $U=Y\...

**7**

votes

**0**answers

163 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 ...

**1**

vote

**0**answers

130 views

### Is the category of mixed Hodge modules bi-filtered?

Let $X$ be a smooth complex algebraic variety and let $MHM(X)$ be the category of mixed Hodge modules on $X$, as defined in (Saito, "Mixed Hodge Modules", 1990), (Peters-Steenbrink, "Mixed Hodge ...

**8**

votes

**2**answers

282 views

### Bott Chern cohomology via currents

Let $X$ be a compact complex manifold. Is the space of $(p,p)$ $d$-closed currents modulo $\partial\bar{\partial}$-exact ones naturally isomorphic to the Bott-Chern cohomology (made in the same way ...

**2**

votes

**1**answer

262 views

### Existence of non-constant solutions for this equations

This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...

**4**

votes

**1**answer

274 views

### On the complexification of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $...

**1**

vote

**0**answers

112 views

### Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...

**1**

vote

**1**answer

279 views

### Local deformations of Lagrangian submanifolds in holomorphic symplectic manifold and their intersections

Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are ...

**3**

votes

**1**answer

141 views

### Do some kind of maximum principle exist on complex manifold?

Consider a holomorphic function $u:C\to C^n$, as $|u|^2 $is sub-harmonic, it satifies a maximum principle.
Do some general kind of complex manifold enjoy such property? Say, square of some distance ...

**2**

votes

**0**answers

110 views

### Generalized Isotropic almost complex structures

Let $(M,g)$ be a Riemannian manifold, $TM$ it's tangent bundle, $\mathcal{H}TM$ be the horizontal sub-space of $TTM$ with respect to $g$, $\mathcal{V}TM$ be the vertical sub-space of $TTM$ and $K$ be ...