# Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

**1**

vote

**1**answer

43 views

### On extensions of holomorphic mappings with image in a projective algebraic variety

I am reading the paper "Two Theorems on Extensions of Holomorphic Mappings" by PHILLIP A. GRIFFITHS. In Example 2 of the paper, there is a proposition saying that:
Let $N$ be a complex manifold, $S\...

**1**

vote

**1**answer

61 views

### Integrability of an almost complex structure vs holomorphicity of the section $M\rightarrow \mathcal{J}(M)$

Let's say we have an almost complex manifold $(M, J)$. Consider the complex vector bundle $V\rightarrow M$ whose fiber over $x$ is the space of almost complex structures on $T_x M$.
Is there any ...

**0**

votes

**3**answers

96 views

### Almost complex structure corresponds to unique complex structure up to biholomorphism [on hold]

Assume I have a smooth manifold which has two different holomorphic atlases that induce the same almost complex structure on it. How to show that two complex manifolds (corresponding to the two ...

**2**

votes

**1**answer

92 views

### Pushing forward a complex structure by submersion

I have a surjective smooth map with surjective differential between two balls $\phi:B^{2n}\rightarrow B^{2k}$. Fix an integrable almost complex structure $J$ on $B^{2n}$. Assume that $\mathrm{Ker}\:d\...

**4**

votes

**0**answers

46 views

### Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains

Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and ...

**1**

vote

**0**answers

70 views

### Interpretation of deformation of complex structure

Let $X$ be a smooth complex analytic space and let $D$ be the unit disk in $\mathbb{C}$. Let $\omega:Y \to D$ be a deformation of complex structures of $X$ in the sense that (1) $\omega^{-1}(0) \simeq ...

**1**

vote

**0**answers

112 views

### Example of open manifold with no free integer homology non-homeomorphic to a ball

I would like to state that if an open oriented even-dimensional (complex) manifold $M$ is such that $dim(H_k(M,\mathbb{Z}))=0$ for $k>0$, and 1 for $k=0$, then $M$ is homeomorphic to an open ball.
...

**6**

votes

**1**answer

253 views

### Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$

Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.)
Is it true that for any ...

**3**

votes

**0**answers

119 views

### Cohomology of a tubular neiborhood of submanifold vs cohomology of the formal neighborhood

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$.
Is ...

**8**

votes

**0**answers

230 views

### Cohomology of complex manifold vs cohomology of its complex submanifold

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that
$$H^i(Z, A|_Z)=0 \mbox{ for any } ...

**1**

vote

**1**answer

52 views

### Minimal complex surfaces with pseudo-effective canonical bundles

A complex line bundle $L$ over a complex surface $X$ is said to be pseudo-effective if it admits a (possibly singular) Hermitian metric $h$ whose curvature is positive semi-definite in the sense of ...

**2**

votes

**0**answers

237 views

### Path lifting property of holomorphic unbranched map

Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on ...

**2**

votes

**0**answers

122 views

### Finding the torsion of the Neron Severi group in the first homology group

Let X be a variety over $\mathbb C$. I will implicitly identify this with us complex analytification. Consider the exponential sequence:
$0 \to \mathbb Z \to \mathcal O_X \to \mathcal O_X^* \to 0$
...

**0**

votes

**0**answers

46 views

### Differential operator of globally unbounded order on connected complex manifold?

Let $X$ be a connected complex manifold. Consider the module $\mathcal{D}_X$. I recall hearing somewhere that one has to be careful with regards to differential operators that are not globally of ...

**7**

votes

**1**answer

271 views

### How does one complexify a real $n$-dimensional Riemannian manifold $(M,g)$?

If $V$ is a real vector space, then the complexification of $V$ is formally defined as $V^{\mathbb{C}}=V\otimes_{\mathbb{R}}\mathbb{C}$. Is there an analogous complexification operation for a real $n$-...

**7**

votes

**0**answers

77 views

### When is Radon-Nikodym derivative induced by a proper map of manifolds bounded?

Let $X,Y$, be compact complex manifolds, and let $f:X\to Y$ be a smooth, proper (i.e. for each $y\in Y$, $f^{-1}(y)$ is a compact set) and surjective map. Choose metrics on $X,Y$ and let $\mu_X, \mu_Y$...

**5**

votes

**0**answers

112 views

### Criterium for algebraicity of an analytic map

Let $X$ and $Y$ be algebraic varieties over $\mathbb{C}$. Let $f:X^{an}\to Y^{an}$ be a holomorphic map.
Is the following statement correct?
If there is an algebraic variety $V$ over $\mathbb{...

**4**

votes

**0**answers

104 views

### Lagrangian foliation for a holomorphic symplectic manifold

I am interested in gathering as many examples as possible for Lagrangian foliations of holomorphically symplectic manifolds $(X, \omega)$, where $X$ is a $2n$-dimensional complex manifold equipped ...

**3**

votes

**1**answer

112 views

### Definition of quotient manifolds, and $\Gamma \backslash \mathscr H$ as a quotient manifold

I have just encountered some subtlety with quotient manifolds and now I don't think I understand some things as well as I thought I did.
Let $X$ be a real or complex analytic manifold, and $\sim$ ...

**3**

votes

**1**answer

115 views

### Clarification on Beltrami Differentials

I have troubles with the theory of existence of quasi-conformal homeomorphisms realizing Beltrami coefficients. Let $X$ be a (compact) Riemann surface and $f \colon X \rightarrow \mathbb{C}$ be smooth....

**1**

vote

**1**answer

70 views

### Upper bound of the dimension of automorphism group of compact Kähler manifolds

It is well-known that the dimension of the isometry group of an $n$-dimensional compact Riemannian manifold is no larger than $\frac{1}{2}n(n+1)$, which is attained precisely by $S^n$ and $\mathbb{R}P^...

**0**

votes

**1**answer

46 views

### Saddle-point method for complex functional of real field

If I have a path integral of the form
$$\int D\phi e^{i S}$$
where the action S is a complex functional of a real-valued field $\phi$, say
$$S=\int dt ( \phi^2 + i \phi )$$
what is the correct way ...

**5**

votes

**0**answers

72 views

### Holomorphic vector fields and derivations

Let $M$ be a complex manifold and $U\subset M$ a domain.
Question: Is every derivation of the complex algebra of holomorphic functions $\mathcal{O}(U)$ induced by a holomorphic vector field defined ...

**2**

votes

**0**answers

97 views

### Duality of Mixed Hodge Structures without compactness

Let $X$ be a smooth separated algebraic variety over $\mathbb{C}$ and $Z \subset X$ a subvariety of codimension $p$. There are no compactness assumptions. I am looking for an isomorphism of mixed ...

**10**

votes

**1**answer

267 views

### Is every endomorphism of the sheaf of holomorphic functions on a disk a differential operator?

Let $D= \{z\in \mathbb{C}:|z| < 1\}$ be the unit disk. And consider the sheaf of holomorphic functions $\mathcal{O}_{D}$.
Question (?) : Is there a sheaf endomorphisms $\phi : \mathcal{O}_D \to \...

**1**

vote

**0**answers

21 views

### First return map in complex 2DOF Hamiltonian systems

The standard way to construct the first return map around a periodic orbit in real 2DOF Hamiltonian systems is the following:
We choose a periodic orbit and a point on it.
We restrict the system on ...

**7**

votes

**1**answer

314 views

### Elliptic operator on compact Hermitian manifold

Let $X^n$ be a compact complex manifold, and $\omega$ be a Hermitian metric on $X$.
Define an operator $P:=i\Lambda_\omega \bar{\partial} \partial$ on the space of the smooth function $C^\infty(X, \...

**3**

votes

**0**answers

201 views

### About the exponential sequence

For a complex analytic space $X$, we have the exponential sequence
$$0\to\mathbf{Z}(1)_X\to\mathcal{O}_X\to\mathcal{O}_X^{\times}\to 1$$
the last map being the exponential $\text{exp}$.
For $d>0$ ...

**1**

vote

**1**answer

137 views

### What is a sufficient condition for summability of formel power series? [closed]

There are several kind of summability , i accrossed differents conditions for applying for example Borel summation or laplace transform which let me mixed and confused , really i don't know if i have ...

**16**

votes

**0**answers

295 views

### Relationship between the signs of different notions of curvature in complex geometry

Let $(X,\omega)$ be a complex hermitian manifold, and call $\Theta$ its Chern curvature tensor. Out of this we can consider different notions of curvature, namely the holomorphic bisectional curvature ...

**2**

votes

**0**answers

85 views

### Stein subspaces of polydiscs and balls

Let $D$ be a either an open polydisc or an open ball in $\mathbf{C}^n$.
(1) Let $\mathcal{O}$ be the $\mathbf{C}$-algebra of holomorphic functions on $\mathbf{C}^n$, resp. $D$, and let $f_1,\ldots, ...

**0**

votes

**1**answer

254 views

### de Rham closed harmonic form on a Kähler manifold

For a compact Kähler manifold, we say that a form is primitive if it is contaned in the kernel of the dual Lefschetz operator, or the co-Lefschetz operator. For all examples I know, a primitive form $\...

**4**

votes

**0**answers

119 views

### Hermitian and Kähler Metrics for Projective Stiefel Manifolds

Let $V(n,r)$ denote the complex Stiefel manifold of orthonormal complex
$r$-frames in complex $\mathbb{C}^n$-space. Each $V(n,r)$ admits a canonical $U(1)$-action, and the quotient with respect to ...

**2**

votes

**0**answers

57 views

### Tangent space of a linear section

Let $X\subset\mathbb{C}^n$ be an embedded smooth manifold, $Y = X\cap H$ a smooth hyperplane section of $X$, $p\in Y\subset X$ a point, and $T_pX, T_pY$ the embedded tangent spaces at $p$ of $X$ and $...

**4**

votes

**1**answer

196 views

### Shrinking the boundary of a Riemann surface

Let $X$ be a compact Riemann surface with boundary. Let us shrink each connected component of the boundary into a point. We get a closed topological surface $Z$ with several marked points (which came ...

**1**

vote

**0**answers

88 views

### Is every monotone Lagrangian Hamiltonian isotopic to minimal Lagrangian?

Assume we have a closed Lagrangian submanifold $L$ in Kaehler-Einstein manifold of positive scalar curvature (for instance, complex projective space). Dazord has proved that 1-form $\alpha=\omega(\...

**2**

votes

**1**answer

125 views

### Real diffeomeorphism preserving the space of Holomorphic vector fields

Assume that $M$ is a complex manifold.
Let $G$ be the group of all (real) smooth diffeomorphisms $\phi$ of $M$ such that $\phi^* (X)$ is a holomorphic vector field for all holomorphic ...

**3**

votes

**0**answers

88 views

### Restriction of a singular metric with minimal singularities

Let $X $ be a smooth complex algebraic variety and $L $ a pseudo-effective line bundle on $X $, consider $h $ to be a singular Hermitian metric with minimal singularities on $L$ and $|A|$ be the ...

**3**

votes

**1**answer

119 views

### Almost Hermitian manifolds of constant curvature

Let $(M,g)$ be a $2n$-dimensional almost Hermitian manifold ($n\geq 2$) with a almost complex structure $\cal J$ (not necessary integrable). i.e.,
$${\cal J}^2=-I,\quad\qquad g({\cal J} X,{\cal J} Y)=...

**2**

votes

**0**answers

76 views

### Is the affine hypersurface defined by ((xz+1)^2/z)-((yz+1)^3/z)+xyz=1 a “known” hypersurface?

The hypersurface $\lbrace (x,y,z) \in \mathbb{C}^3:\frac{(xz+1)^2}{z}-\frac{(yz+1)^3}{z}=1 \rbrace$ is a well-known example of a contractible hypersurface in $\mathbb{C}^3$. See for instance, Example ...

**1**

vote

**0**answers

128 views

### Does $\mathfrak{m}_z/\mathfrak{m}_z^2\cong\overline{\mathfrak{m}_z}/\overline{\mathfrak{m}_z}^2 $ on all complex manifolds?

Let $M$ be a complex manifold with its sheaf $\mathcal{O}_M$ of holomorphic functions.
Fix a point $z\in M$ and denote by $\mathcal{O}_z$ the stalk of $\mathcal{O}_M$ at $z.$
Cosider ideals $\...

**11**

votes

**2**answers

1k views

### Can the group of holomorphic automorphisms of an open subset of the complex plane be isomorphic to the additive group of real numbers?

We can construct open sets in the complex plane $\mathbb{C}$ whose automorphism group is isomorphic to $\mathbb{Z}$, but is there an open set whose automorphism group is isomorphic to $\mathbb{R}$?

**1**

vote

**0**answers

110 views

### On $G$-gerbes over the punctured disk

Let $G$ be a finite (not necessarily abelian) group and let $\mathcal{X}\to D^*$ be a $G$-gerbe over the punctured disk $D^*$.
Is there a finite etale cover $D^*\to \mathcal{X}$?
I think of $G$-...

**1**

vote

**0**answers

58 views

### Largest elementary neighbourhood

The real projective space $\mathbb{R}P^n$ can be defined as the quotient space of $\mathbb{S}^n$by the equivalence relation that identifies antipodal points. The largest open set of $\mathbb{S}^n$ ...

**3**

votes

**1**answer

393 views

### Proof of the holomorphic frobenius theorem in Voisin's book on Hodge theory (Theorem 2.26)

I'm trying to understand the proof of the holomorphic version of the frobenius integrability theorem given in p. 51-52 of Voisin's text "Hodge Theory and Complex Algebraic Geometry I".
Statement: ...

**5**

votes

**2**answers

248 views

### fibers of birational contraction for complex manifolds - are they Moishezon?

Let $X$ be a smooth complex manifold and
$\phi:\; X \mapsto Y$ a proper holomorphic
map which is birational ("birational contraction"),
and $Z= \phi^{-1}(y)$ its fiber in a point $y$.
The variety $Y$ ...

**8**

votes

**1**answer

245 views

### Families of curves on compact complex surfaces and algebraicity

Let $S$ be a compact complex manifold of dimension $2$ and assume that there exists a two-dimensional family of curves on $S$. Is it true then that the algebraic dimension of $S$ is $2$, i.e. that $S$ ...

**6**

votes

**0**answers

185 views

### A topological property of flat morphisms

Let $f\colon X\to Y$ be a faithfully flat morphism of smooth projective varieties over $\mathbb{C}$. Assume that the generic fiber of $f$ is smooth. Then there exists a non-empty Zariski open subset $...

**3**

votes

**0**answers

87 views

### Isotropy symmetric holomorphic functions

Let $G$ be a bounded homogeneous domain in $\mathbb{C}^{n}$ and let $z\in G$.
Assume that $f$ is a holomorphic function on $G$, which is isotropy symmetric, i.e. $f\circ \varphi=f$ for any ...

**6**

votes

**0**answers

154 views

### A relation of convergence in Hilbert scheme to convergence in sense of currents

Let $\{X_i\}$ be a sequence of closed irreducible $k$-dimensional subvarieties of $\mathbb{C}\mathbb{P}^n$ of degree $d$ (they may be assumed to be smooth if necessary). Assume that this sequence ...