# Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

287
questions

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### Existence of coframe for Hermitian metric on complex manifold?

I am reading page 28 of the 1994 version of Principles of Algebraic Geometry by Griffith. Let $M$ be a complex manifold of dimension n, Griffith defined a Hermitian metric to be a positive definite ...

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votes

**1**answer

234 views

### Finite self-maps exist on rigid CY3s

Let $X$ be a smooth projective rigid Calabi-Yau threefold.
Question. Does there exist a finite map $X\to X$ of degree $>1$?

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163 views

### Is there a reasonable definition of an octonionic manifold?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\End{End}$
Todorov and Dubois-Violette have recently shown how to understand the structural gauge group of the standard model via octonions.
Q. Is there ...

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144 views

### Of the group of automorphisms of a vector bundle of finite rank

$\DeclareMathOperator\End{End}\DeclareMathOperator\Id{Id}\DeclareMathOperator\AC{AC}\DeclareMathOperator\GL{GL}$This is a question related to a previous question on existence of almost complex ...

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**1**answer

765 views

### An almost complex structure on the real $n$-sphere $S^n$

If $R\mathrel{:=}\mathbb{R}[x_1,\dotsc,x_{n+1}]/(x_1^2+\dotsb+x_{n+1}^2-1)$ and $S^n\mathrel{:=}\operatorname{Spec}(R)$ is the real $n$-spere, a classical result of Borel and Serre says that the only ...

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votes

**1**answer

95 views

### Quasiconformal map from a subset of $\mathbb{C}$ to a polytope

Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such
as a unit disc or rectangle) and a polytope?
Here, I take a polytope to be a two-dimensional surface that could be ...

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71 views

### Change in Connection on a complex Line bundle

Let's say $M$ is a compact Kähler manifold and $L$ is a complex line bundle on $M$. Now let's say $A$ be a connection or equivalently a hermitian metric on $L$. Hence one can have the operators
$\bar\...

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155 views

### Irreducibility of the base and of the general fiber

Let $f:X\rightarrow Y$ be a morphism of scheme over $\mathbb{C}$. Assume that $Y$ and the the general fiber $F_y = f^{-1}(y)$ of $f$ are irreducible.
Does there exists an irreducible component $X'$ of ...

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309 views

### Consequence of the failure of Nagata's conjecture

A modern version of the Nagata's conjecture says that
$$
L_{N,t}:=f_{N}^{*}(-K_{\mathbb{P}^{2}})-t\sum_{j=1}^{N}E_{j}
$$
is Ample for any $t<\frac{3}{\sqrt{N}}$, where $f_{N}:Y_{N}\to \mathbb{P}^{2}...

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169 views

### (1,1)-form that does not come from a divisor

Let $M$ be projective complex manifold. The Lefschetz (1,1)-theorem says that the cycle map
$$
\text{cl}:\operatorname{Pic}(M) \to \text{Hod}^1(M)
$$
is surjective.
Question. Is there an interesting ...

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384 views

### Status of a conjecture of Hirzebruch

I was reading a paper from 1994 which claimed that the following statement was a conjecture of Hirzerbruch:
If a complex surface X is homeomorphic to either $S^2 \times S^2$ or $\mathbb{C}P^2 \# \...

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**1**answer

367 views

### Holomorphic Urysohn Lemma

Let $M,N$ be two disjoint closed holomorphic submanifolds of $\mathbb{C}^n$. Is there a holomorphic map $f:\mathbb{C}^n\to \mathbb{C}$ with $f(M)=0,\;f(N)=1$.

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77 views

### Real part of the Ward correspondence

I am currently very confused about the real side of the Ward correspondence. Recall that the Ward correspondence gives a one-to-one correspondence between:
$M$-trivial holomorphic bundles $E$ on $Z$, ...

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votes

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200 views

### Equivalent definitions of normality for complex algebraic varieties

In Kollár's article The structure of algebraic threefolds: an introduction to Mori's program he gives the following definition of a normal variety:
Definition 5.4. Let $V \subset \mathbb{C}^n$ be an ...

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132 views

### Deform a complex structure fixing marked points

Let $\Sigma$ be a closed orientable surface of genus $g$ with $m$ marked points $x=\{x_1, \ldots, x_m\}$ and $j_0$ denote a complex structure on $\Sigma$. Take a neighborhood $U$ of the isomorphism ...

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296 views

### (Contradiction) All symplectic manifolds are holomorphic

I’m studying symplectic manifolds and almost complex structures. This lead to two propositions:
Proposition 1 (from da Silva’s Lectures on Symplectic Geometry): If $J_0$ and $J_1$ are almost complex ...

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33 views

### complex eigenspace sub-bundle of holomorphic tangent bundle

Let $X$ be a compact K"ahler manifold and let $\varphi: TX\rightarrow TX$ be a self-adjoint endomorphism of the holomorphic tangent bundle, with the property that at each point, the eigenvalues ...

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47 views

### Regarding projective manifolds with decomposable real tangent bundle

Let $X$ be a complex projective manifold. Suppose its real tangent bundle $T_{\mathbb{R}}X$ splits as a direct sum of two sub-bundles of even rank. Does this give any useful information about the ...

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62 views

### Finding $\Omega$ such that the 1-form $\Omega^2 \omega$ is $L^2$ orthogonal to conformal killing vector fields on $S^2$

Consider the space $\mathcal{A}$ of functions $\Omega$ such that $\Omega^2 \gamma_0$ is isometric to the round sphere, where $\gamma_0$ is the round sphere. (so $\Omega^2 \gamma_0$ is of constant ...

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260 views

### Exotic $\mathbb{R}^4$ with a complex structure?

Is there an exotic $\mathbb{R}^4$ admitting an integrable almost complex structure?

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104 views

### Fundamental theorem for real submanifolds into complex space forms

It is a well-known result that Gauss, Codazzi and Ricci equations are necessary and suficiently conditions to guarantee the existence of an isometric immersion of a given $n$-dimensional real ...

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35 views

### Symplectic form on the space of unitary connections $\mathcal{A}(E)$

Let $E\rightarrow X$ be a Hermitian vector bundle over a (Kahler) manifold $X$. The space of unitary connection $\mathcal{A}(E)$ is an affine space modelled over $\Omega^1(X,u(E))$ and is endowed with ...

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1k views

### About a claim by Gromov on proper holomorphic maps

At p. 223 of his paper [G03], Mikhail Gromov makes the following claim:
Let $X$, $Y$ be two complex manifolds (not necessarily compact or Kähler) of the same dimension and having the same even Betti ...

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600 views

### Is every surjective holomorphic self-map on a compact complex manifold finite-to-one?

I have already asked this question on stack exchange, but I didn’t get any answer.
Let $X$ be a compact connected complex manifold.
Let $f:X \to X$ be a surjective holomorphic map. Is it true that $f$...

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130 views

### $f$ is the complexification of a map if $f$ commutes with almost complex structure and standard conjugation. What if we had anti-commutation instead? [closed]

I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures ...

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172 views

### Is there a compact complex surface $X$ with $c_2(X)=7+6n$ and $c_1^2(X)=17+18n$?

As stated in [1], most pairs of positive integers $c_1^2$, $c_2$ satisfying $c_1^2+c_2=0$ $\mod 12$, the BMY inequality and the Noether inequality are actually Chern numbers of compact complex ...

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321 views

### Complexification of realification: Why compute eigenvalues? [closed]

I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures ...

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133 views

### Does the maximum principle hold in this pluriharmonic setting?

Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and ...

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107 views

### Ricci curvature of a Kahler current

Let $M$ be a compact Kahler manifold, with a divisor $D$, $\mathcal{H}_{\omega} = \{\varphi \in C^{\infty}(M - D) \cap C^{0}(M) : \omega_{\varphi} = \omega + \sqrt{-1} \partial \bar \partial \varphi &...

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271 views

### First Chern class and field extensions

Let $X$ be a smooth, complex projective algebraic variety defined over a number field $K$.
Let $D$ be a divisor of $X$ defined over $K$ with the following property:
For any curve $C$ defined over $K$,...

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101 views

### The comparison between Dolbeault cohomology and $L^2$ cohomology

Let $X$ be a complex manifold. Let Dolbeault cohomology and the corresponding $L^2$-cohomology be denoted by $H^{p,q}(X) $
and $H^{p,q}_{(2)}(X)$ respectively.
As is well known, on a compact ...

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219 views

### A non-Kähler compact complex manifold with negative sectional curvature

I am looking for an example of a compact complex manifold with negative sectional (not holomorphic) curvature which is not Kählerian. Can such an example exist?

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135 views

### Real part of a holomorphic section of a vector bundle

Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic ...

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164 views

### Reconstructing the metric on $CP^2$ with special one forms

I know that $(z_1,z_2)$ are the affine\inhomogeneous coordinates on the complex projective space $CP^2$. Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$. I want to rewrite the Fubini Study metric on $...

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160 views

### Kummer surfaces which are not projective

This is a question from an online note. Let $A$ be a two-dimensional $\mathbb C$-torus. And there is an involution on $A$: $A\to A, x\mapsto -x$. The action has 16 fixed points. Let $Y:=A/\{\pm1\}$, ...

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67 views

### Are injective analytic maps between non-archimedean spaces open?

Let $\Omega$ be a non-archimedean complete field, $n\in\mathbb N$ and $f:\Omega^n\to\Omega^n$ be an injective analytic map.
Is the application $f$ open?
In the complex case, this is a consequence of a ...

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**1**answer

124 views

### Koszul-Malgrange Holomorphic structure on a pullback bundle

I'm finding myself a little confused about Koszul-Malgrange holomorphic structures in a certain context.
Suppose $M$ is a complex manifold, $N$ is a smooth manifold with a smooth complex vector bundle ...

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vote

**1**answer

76 views

### How many points of a sequence can we catch with an analytic disc?

Let $X\subset \mathbb{C}^{n}$ be a domain. You can assume that it is nice (e.g. bounded convex balanced ). Let $\{x_n\}$ be a sequence of points that does not have a limit point in $X$.
Let $D$ be the ...

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**1**answer

286 views

### Fubini-Study metric induced by submersion

The Fubini-Study metric $g:=g_{FS}$ is the unique $U(n+1)$-invariant
Riemannian metric on the complex projective space $\mathbb{CP}^{n}$ the complex projective space
which by $U(n+1)$-invariance can ...

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144 views

### Some general questions about deformations

$\newcommand{\spec}[1]{\mathrm{spec}(#1)}$
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\QQ}{\mathbb{Q}}$
$\newcommand{\CC}{\mathbb{C}}$
These days I am reading in Kurke, Pfister, Roczen "...

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192 views

### A density criterion and a submersion map of a Hodge bundle

In Voisin's excellent book 《Hodge theory and complex algebraic geometry II》5.3.4 - a density criterion, there is a important theorem:
Let $X$ be a compact Kähler manifold, $\pi:\mathcal X \rightarrow ...

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131 views

### Indecomposability of local systems and vector bundles

Let $M$ be a (connected) complex manifold, $L$ be a local system on $M$ and $\mathcal{L}$ the vector bundle associated to $L$. If $L$ is indecomposable, does it imply that $\mathcal{L}$ is also ...

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156 views

### Recipe for resolving a coherent sheaf

Let $X$ be a complex manifold and let $V\subset X$ be a subvariety. Let $F\rightarrow V$ be a holomorphic vector bundle over $V$ and let $\mathcal{S}=\Gamma(F)$ be the sheaf of holomorphic section of $...

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62 views

### Geometry of the complex Gauge group

Let $E\rightarrow X$ be holomorphic vector bundle on a complex manifold $X$. Denote by $\mathcal{G}=\Gamma(Aut(E))$ the group of complex smooth automorphisms of $E$.
Is there a way to endow $\...

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86 views

### Opers and global differential operators

This is a follow up question to a previous question of mine and my thought of answer to it.
Given a (compact) Riemann surface $\Sigma$, a $SL(n,\mathbb{C})$-oper is a rank $n$ holomorphic vector ...

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110 views

### Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat

I'm having a hard time proving the following:
If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...

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158 views

### cohomology classes of complex submanifolds

I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be.
For example, say $T^4$ is regarded as a complex ...

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133 views

### Leaves of stable foliation of holomorphic Anosov diffeomorphism

I'm trying to understand the first half of the paper "Holomorphic Anosov systems" by E. Ghys (the journal reference is Inventiones mathematicae volume 119, pages 585–614(1995)). My question is about a ...

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113 views

### Mixed Hodge structures on (infinite) covers of complex varieties?

Let $X$ be a complex variety, and let $\tilde{X}\to X$ be a covering map. Does the singular cohomology $H_\ast(\tilde{X};\mathbb{Z})$ carry a natural mixed Hodge structure?
If the cover is finite, ...

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157 views

### A metric $w$ on a Kahler manifold is extremal if and only if the gradient vector field of the scalar curvature is holomorphic

I am trying to understand the calculation in An introduction to Extremal kahler metrics. On the fourth line of page 55 the author calculated that $\int_{M} - 2 S R^{\bar k j} \partial_{j} \partial_{\...