# Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

334
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### Is there a Cauchy integral formula for complex manifolds?

Is there a Cauchy integral formula for holomorphic functions on complex manifolds?

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### Contractible real analytic varieties

If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point?
Here a real analytic variety is the set of zeros of a real analytic ...

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### Teichmuller interpretation of unbounded holomorphic quadratic differentials

For a closed Riemann surface $\Sigma$ of genus $g \geq 2$, the space of holomorphic quadratic differentials on $\Sigma$ can be identified with the cotangent space $T_\Sigma^* \mathcal{T}_g$: in other ...

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### Complex structure on the product of two real torus

Let $T^{2n}_{\mathbb{R}}$ be a real torus of dimension $2n$, and let $Z_n$ be the space consisting of all possible complex structures on $T^{2n}_{\mathbb{R}}$. It is known that:
$$Z_n = \mathrm{GL}(2n,...

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### Is there a maximum principle for CR functions over domains inside CR manifolds?

I am new to this area and I am a bit confused by the literature. Is there a strong maximum principle for CR functions over domains in a CR manifold, please? If so, could someone please state it (...

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### Given a smooth hyperplane section Y of a variety X there exists a Lefschetz pencil of hyperplane sections of X containing Y

Let $X$ be a variety contained in $\mathbb{P}^N$ and let $Y$ be a smooth hyperplane section of $X$. I have read in page 54 of Voisin's book "Hodge theory and complex algebraic geometry II" ...

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### Classification of surface bundles over surfaces

Can anyone recommend one place or a few places that describe what is known about the classification of (real) surface bundles over (real) surfaces?
Now, if the fibre F and the base B are both ...

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### Top integer homology of compact analytic variety

Let $V$ be a compact connected complex analytic subvariety (possibly singular) of a complex smooth manifold. Let $n$ denote its complex dimension.
Is it true that $H_{2n}(V,\mathbb{Z})\simeq \mathbb{Z}...

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### Intersection of Stein opens admits a Stein neighborhood basis?

Let $X$ be a Stein manifold, $K$ be a compact subset of $X$. Consider the following conditions:
1.$K$ admit an open neighborhood basis in $X$ whose members are Stein;
2.$K=\cap_{j\ge 1}V_j$, where $...

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### Kawamata BPF applied to a semi-positive line bundle using Demailly's holomorphic Morse inequalities

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic ...

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### Vertical Fourier decomposition for skew-Hermitian 1-forms

In an arXiv preprint [2108.05125v1], the authors use the following vertical Fourier decomposition (page 7 therein).
Let $(M,g)$ be a Riemannian surface and $SM$ be its unit tangent bundle. Denote by $...

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### Is there a projective bundle formula for Deligne cohomology?

Given a projective bundle $\mathbb{P}(E) \to X$ on a complex manifold $X$, is there a projective bundle formula for Deligne cohomology? That is, can Deligne cohomology $H_D^n(\mathbb{P}(E),\mathbb{Z}(...

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150
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### Betti numbers of threefolds with trivial canonical class

I am interested in a simply-connected compact complex manifold $M$ of dimension three with trivial canonical class.
Note that if it is K"ahler, then it is a Calabi-Yau threefold.
Its independent ...

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110
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### Are two notions of generalized solution of Monge-Ampere equation equivalent?

Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to \mathbb{R}$ be a continuous plurisubharmonic (psh) function. The theorem of Chern-Levine-Nirenberg defines a non-negative ...

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### Threefolds with the same Betti numbers and the same Chern numbers

By a threefold, I mean a compact complex manifold of dimension three.
My question is a simple one:
Are there known INFINITELY many non-homeomorphic threefolds that have the same Betti numbers and the ...

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### Complex manifold with conjugate complex structure

Let $(M,J)$ be a complex manifold with complex structure $J$. It is clear that $(M,-J)$ is also a complex manifold.
Under what condition is $(M,J)$ biholomorphic to $(M,-J)$?

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### Do non-projective K3 surfaces have rational curves?

Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K_X \simeq \mathcal{O}_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov ...

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### How do we define the type of a singularity on a cubic surface?

Nine different types of singularities are possible on a cubic surface, according to Wikipedia. How exactly is the "type" of singularity defined? I know that the number corresponding to the ...

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### Is there a non-singular cubic surface that has a point where four lines intersect?

Every non-singular complex projective cubic surface has $27$ lines. Many such surfaces contain points where three lines intersect (called Eckardt points). There are even surfaces with many Eckardt ...

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### Modern reference for a theorem by Bott on the Dolbeault cohomology of compact homogeneous manifolds

I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology ...

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### Can a punctured ball $(B\setminus\{0\})\subset\mathbb{C}^n$ be foliated by complete leaves?

Recently Antonio Alarcón proved that in the case of the unit ball $B\subset\mathbb{C}^n$ for $n\geq 2$ every smooth closed complex submanifold of dimension $q\leq n$, $V\subset\mathbb{C}^n$ defines a ...

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### Is there an extension of Ogg's results to surfaces of Genus 1

The first hints of moonshine appeared around 1974 when Andrew Ogg noticed that quotienting the hyperbolic plane by normalizers of the Hecke Congruence subgroups $\Gamma_{0}(p)$ has genus zero iff p is ...

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### Real analytic subvariety in complex manifold which is complex outside of its singular set

Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic?...

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### Is every smooth projective variety contained in a chain of smooth projective varieties of increasing dimension?

Let $X ⊆ \mathbb{P}^n$ be a smooth projective variety (over $\mathbb{C}$). I think we can find a chain of irreducible varieties $X = X_0 ⊆ X_1 ⊆ X_2 ⊆ \cdots ⊆ X_k = \mathbb{P}^n$ whose dimension ...

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### Fixed-point free holomorphic involutions

Here is the new version of the question which is more explicit. The older version is below.
I am looking for complex projective varieties (in dimensions $2$ and higher) admitting a fixed-point free ...

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### “Logarithmic” form of Kodaira Embedding

Suppose you have a non-compact complex manifold $X$ with a Hodge metric, whose associated Kahler form has integral cohomology class. In the compact case, one would be able to conclude that $X$ is ...

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### Approximating an infinite family of holomorphic functions by polynomials in relative error

I think I just proved a theorem I haven't found in the literature, and I think it must generalize. I therefore have two questions. First, if this is in the literature, what is it called? Second, what ...

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### Constructions of complex surfaces covered by the ball of $\mathbb{C}^2$

Let $S$ be a compact complex surface. It is well-known that the following two facts are equivalent
$c_1^2(S) = 3 c_2(S)$ and $S \neq \mathbb{CP}^2$
The universal cover of $S$ is biholomorphic to the ...

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### Extension of a Szegő Kernel to the boundary

Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$.
Recall: the Szegő kernel is a kernel of the Szegő projection $P: L^{2}(\partial\...

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### Manifolds whose tangent spaces have a special behavior

Consider an $n$-dimensional complex manifold $M\subset\mathbb{C}^N$ and let
$$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$
be a local parametrization of $M$.
...

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139
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### Maximal analytic continuation

I'm currently reading about the concept of a “maximal analytic continuation” from Forster's book Lectures on Riemann Surfaces (see Section 7). There are a bunch of definitions to unpack before I can ...

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### Isometries of the complex projective space for the Fubini Study metric

$\DeclareMathOperator\SU{SU}$I am trying to understand a geometric proof in our mathematical quantum mechanics lecture regarding Wigner's theorem in finite dimensions. We have already shown that it ...

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### Complete intersections in complex manifolds

Let $X$ be a complex manifold of dimension $n$ and $Y\subset X$ a closed submanifold of codimension $k$.
a) Say that $Y$ is a complete intersection if the ideal $I(Y)\subset \mathcal O(X)$ of global ...

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### Phase (argument) of a complex top form

Let $\xi\in\Omega^{n,n}(X,\mathbb{C})$ be a volume form on a complex manifold $X$ of dimension $n$. For any other non vanishing top form $\eta\in\Omega^{n,n}(X,\mathbb{C}) $, there exists a function $...

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### Viewing an algebraic subset through hyperplane sections

Suppose $n \ge 3$ and $X$ is a path-connected subset of $\Bbb{CP}^n$ under the manifold topology. If for every complex hyperplane $H \subset \Bbb{CP}^n$, $H \cap X$ is a degree $k$ algebraic ...

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### Complement of complex submanifolds of codimension $\ge1$ is connected

Let $X,Y$ be complex manifolds of $\dim X=n$, $\dim Y=m>1$, $U\subset X$ open and $g\colon U\to Y$ holomorphic embedding. Then $g(U)$ is a submanifold of codimension $m-n\ge1$. It seems clear that $...

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### Riemann-Roch theorem for higher-dimensional complex manifolds

Does an analogue of the Riemann-Roch theorem hold for higher-dimensional complex manifolds? (Hirzebruch-Riemann-Roch theorem is for algebraic manifolds, but not for general complex manifolds, right?)

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### Do we know any examples of complex surfaces where we have explicit knowledge of the Chern–Weil functions?

Let $X$ be a compact complex surface (smooth). Let $\gamma_1, \gamma_2$ denote the Chern–Weil functions. That is, if $\omega$ is a Kähler form on $X$ with volume form $\omega^2$, then $\gamma_1, \...

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### Automorphism group of compact almost complex manifold

Does the automorphism group of a compact almost complex manifold carry a (canonical) Lie group structure? Part 3 of Theorem 4.1 in
*"The automorphism group of a homogeneous almost complex ...

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### Kahler property and finite covering

Let $(M,\omega)$ be a compact symplectic manifold and $\pi:\tilde M\to M$ a finite covering. Clearly $(\tilde M,\pi^*\omega)$ is a compact symplectic manifold. Suppose we know that $(\tilde M,\pi^*\...

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### Proving algebraicity of compact Riemann surfaces without Chow's theorem

I am trying to write a report for a complex analysis class where I prove Riemann-Roch and apply it to prove algebraicity of compact Riemann surfaces. While writing this, I found that Riemann-Roch ...

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### Example of usual Laplacian does not respect bidegree for general hermitian manifolds

We know that the Kähler identity $\Delta=2\Delta_{\partial}=2\Delta_{\bar{\partial}}$ on a Kähler manifold $(X,g)$ implies that the usual Laplacian $\Delta:=dd^*+d^*d$ respects the bidegree, i.e. for ...

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### Holomorphic mapping on a manifold approximating a constant map

Let $X,Y$ be complex manifold, $Y$ Stein. It sounds quite reasonable to formulate the following claim: given $y_0\in Y$, for every $\epsilon>0$ and $M\subset X$ compact, there exists an holomorphic ...

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### holomorphic functions on Stein manifolds reaching maxima in a given point on the boundary (Shilov boundary)

Let $M$ be a Stein manifold with smooth, strictly
pseudoconvex boundary, and $x$ a point on its
boundary. Is there a holomorphic function $f$ on
$M$, smooth on the boundary, with strict
maximum of $|f|...

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1
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### Interesting examples of direct image bundles

Let $\pi : W \rightarrow Y$ be a holomorphic fibration of complex manifolds. Let $L\rightarrow W$ be a holomorphic line bundle on its total space and denote by
$$E^k_q := R^q \pi_*L^k$$
the direct ...

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### A Riemann surface is automatically paracompact

[A question I remember from many years ago.]
Definition
A Riemann surface is a connected complex manifold $X$ of complex dimension one. This means that $X$ is a connected Hausdorff space that is ...

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### Artin vanishing for Stein manifolds and restriction maps

In the setting of complex Stein manifolds $X$ of complex dimension $d$, the theorem of Andreotti--Frankel implies the vanishing of the singular cohomology group $H^i(X,\mathbb Z)=0$ for $i>d$. With ...

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### How to judge whether an orbifold is good

My own case comes from dynamic system on compact complex manifolds. To be precise, let $M$ be a compact complex 3-dimentional manifold, $W^c$ a holomorphic foliation of M with 1-dimentional uniformly ...

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### Family of Dolbeault operators on complex vector bundles over $\mathbb{CP}^1$

Let $\pi \colon E \rightarrow \mathbb{CP}^1$ be a complex vector bundle. It is a well-known fact that a Dolbeault operator on $\pi\colon E \rightarrow \mathbb{CP}^1$ gives a holomorphic structure on $...

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### Approximation of a holomorphic function vanishing at a submanifold by polynomials

Suppose $M$ is a complex affine algebraic manifold in ${\mathbb C}^n$ (I mean, a set of common zeroes of a system of polynomials on ${\mathbb C}^n$, which is at the same time a smooth manifold). ...