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Questions tagged [almost-complex]

For questions about almost complex structures on vector spaces or manifolds.

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2answers
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Reality check on integrability of almost-complex surface.

Let $(M^2,J)$ be an almost-complex surface. I do not assume integrability of $J$ now. I understand that from the Newlander-Nirenberg theorem, one of the several equivalences for the integrability of $...
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1answer
44 views

Fundamental form of an almost complex manifold

so the setting I am in right now is the following: Let $(M^{2m},h,J)$ be an almost Hermitian manifold with fundamental form $\Omega$. I really would like a local expression for this $\Omega (X,Y)=h(...
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1answer
54 views

An almost complex structure on $M$ is equivalent to a reduction of the structure group of the tangent bundle

Let $M$ be an $2n$-dimensional manifold. Let $\mathcal{F}_{\mathrm{GL}(2n, \mathbb{R})}$ be the frame bundle over $M$. Consider the subgroup $\mathrm{GL}(n, \mathbb{C})\subset\mathrm{GL}(2n, \mathbb{R}...
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0answers
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Definition of an almost complex hyperplane in the projective space $\mathbb{C} P^n$.

Let J be an almost complex Structure in the projective space $\mathbb{C} P^2$. According to Duval a J-line in the almost complex projective space $\mathbb{C} P^2,$ is the almost complex analogue of a ...
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1answer
48 views

Why is the matrix representation of an almost complex structure like this?

Let $M$ be a Riemann surface, $J$ be an almost complex structure (i.e. a 1-1 tensor such that $J^2=-I$ and for any $x \in M,v,w \in T_xM, \{v,Jv\}$ is oriented). Consider a conformal coordinate at a ...
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46 views

Direct proof that a parallel almost complex structure is integrable

Let $(M,g,J)$ be an almost Hermitian manifold nad $\nabla$ be the Levi-Civita connection on $M$. If $\nabla J=0$, it is straightforward to show that the Nijenhuis tensor of $J$ must vanish which ...
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0answers
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Vanishing of the Nijenhuis tensor

The Nijenhuis tensor is defined to be: $$(1):\quad N_J(X,Y)\equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY], $$ for vector fields $X$ and $Y$ on the manifold $M$, equipped with Almost Complex Structure $J$: $$(...
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an exercise about integrability of almost complex structures

i have spent some time over the following problem from a problem sheet of a course on complex geometry. Let $M=G/H$ be a homogeneous space (where $G$ is a Lie group, $H$ a closed subgroup) and let $I$...
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1answer
64 views

(Almost) complex manifold problem.

Can some one give me an exemple of an almost compplex manifold that is not a complex manifold and why? I know that an almost complex manifold is of even real dim and is orientable. I also heard that ...
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0answers
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Local construction of a map on a manifold

Assume $(M,J)$ is a smooth manifold equipped with an almost complex structure. Let $p$ be a point of $M$, and define the linear map $\Theta_p \colon T_pM \to T_pM$ so that $$\Theta_p = -\frac{1}{2}\...
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1answer
102 views

Does every almost complex manifold admit an almost-Kähler structure?

I couldn't find a conclusive answer to this question online. Here is my reasoning. Let $M$ be an almost complex manifold. Then, from what I understand, we can define almost complex structure $J$ on $...
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64 views

Vertical $J$-holomorphic spheres

Let $\pi:M\to N$ be a smooth fiber bundle with fiber $F$ and let $g$ be a Riemannian metric on $M$. Using $g$ we may split the tangent bundle of $M$ as $$TM\cong TF\oplus \pi^{\ast}TN,$$ where $TF$ is ...
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2answers
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Equivalence of two definitions of Almost Complex Structure.

Let $M$ be a smooth manifold of even dimension $2n$. I would like to understand the equivalence of the following two standard definitions for an almost complex structure on $M$. Denote by $TM$ the ...
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$M$ is a complex manifold $\Leftrightarrow \overline{\partial}^2=0$

One way of stating Newlander-Niremberg's theorem is by saying an almost complex manifold $(M,J)$ is complex $\Leftrightarrow \overline{\partial}^2=0$. I'm confused by this, because I can't see why ...
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1answer
47 views

Equivalent definitions of almost complex structures

The definition I have seen for an almost complex structure is the following $$J:TM\to TM$$ which is linear fibre by fibre, such that $J^2 = -\text{Id}$, and such that $\pi(J(X_x)) = x$ where $\pi$ ...
4
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1answer
116 views

Coordinate-free proof of non-degeneracy of symplectic form on cotangent bundle

It's relatively straightforward to provide a coordinate-free definition of the symplectic form on a cotangent bundle; the usual way to do this is to construct the tautological 1-form $$\lambda(\xi) = \...
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0answers
102 views

Complex Vector Bundle vs Holomorphic Vector Bundle vs Almost Complex Structures

I am writing because I am extremely confused with the structure of complex vector bundles. Ok first of all I understand that a complex vector bundle is a just a vector bundle $\pi:E\to X$ such that ...
4
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1answer
34 views

Equivalence of tangential and normal stably almost complex structure

Let $M$ be a smooth manifold. $M$ is said to be tangential stably almost complex if $TM \oplus \underline{\mathbb{R}}^k$ can be given a structure of a complex vector bundle, for some $k$. $M$ is said ...
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2answers
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Integrable almost complex structure conjugated by diffeomorphism

Let $M$ be a smooth manifold and $J$ be an integrable almost complex structure on $M$. Let $f: M\to M$ be a diffeomorphism with $f_{*}:TM\to TM$ its tangent map. Then it is easy to see that $f_{*}Jf_{*...
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1answer
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Natural complex-linear isomorphism between $ V_J $ and $V^- $

Let $V$ be a real vector space with a linear complex structure $J$ (see Wikipedia). Denote $V_J$ the complex vector space induced from $V$ by the complex structure $J$. Also, define the ...
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1answer
139 views

Nijenhuis tensor in local coordinates

If $(M,J)$ is an almost complex manifold, $\mathcal{N}$ (Nijenhuis tensor) is: $$\mathcal{N}(X,Y)=[JX,JY]-J[X,JY]-J[JX,Y]-[X,Y].$$ I am trying to compute it in local coordinates $(x_{1},\cdots,x_{n})$...
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1answer
137 views

Relation between Riemannian metric and Hermitian structure.

Let $M$ be a complex manifold with almost complex structure $J$, and $TM$ it's real tangent space, $TM^{\mathbb{C}}=TM \otimes \mathbb{C}$ its complexified tangent space. Now I'm getting confused ...
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1answer
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Almost complex structure on $\mathbb{S}^{3} \times \mathbb{S}^{5}$

I would like to check, whether the product space $X = \mathbb{S}^{n} \times \mathbb{S}^{m}$ admits an almost complex structure for odd $m,n$. For example, if $m=1$ and $n=3$, then $X = \mathbb{S}^{1} ...
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1answer
41 views

Trying to understand the breaking down of the complexified tangent space

I am reading some notes on the complexified tangent space. I don't understand how do we arrive at that $V^{(1,0)} = \{ X - iJX : X \in V\}$. I mean in one direction we have $JZ = J( v \otimes \alpha) ...
5
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1answer
91 views

On the vector bundles associated to the eigenspaces of an almost complex structure.

Let $(M,J)$ be an almost complex manifold, that is $M$ is a smooth differentiable manifold of even dimension and $J$ is an endomorphism of the tangent vector bundle $TM$ such that $J^2=-\textrm{id}_{...
4
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1answer
254 views

The Definition of an Almost Complex Manifold (Nakahara)

I am having problems making sense of Michio Nakahara's definition of the Almost Complex Structure/Almost Complex Manifold, such as it appears in Geometry, Topology and Physics (2nd Edition). On p. ...
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415 views

Almost complex structure on $S^6$

It is known that the sphere $ S^6$ admits an almost complex structure by identifying $S^6 $ with the space of unit purely imaginary Cayley numbers. I would like to show that this almost complex ...
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0answers
26 views

Subbundle of hermitian maps is isomorphic (as a bundle) to $M\times i\Bbb R$

Let $\pi:E\to M$ be a complex bundle, where $M$ is a $n$-dimensional smooth manifold. We can consider the bundle $\text{Hom}(E,E)=:\text{End}(E)$ of endomorphism of $E$, with fibers $$\text{End}(E)_{...
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0answers
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Reference for complex structures and Hermitian structures

I am reading Complex Geometry: An Introduction by Daniel Huybrechts. Section on complex structures and Hermitian structures deals with Lefschetz operator denoted by $L$. Hodge * -operator. The ...
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Construction of hermitian almost complex structure and invariant $(n,0)$-form

I'm reading a paper and it says the following: Let $K$ be a Lie group and fix an orthonormal framing $\{e_1, ..., e_n\}$ with respect to a left-invariant metric. This gives a splitting $$T^∗K = K × \...
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1answer
322 views

Metric induced almost complex structure on cotangent bundle

I'm trying to understand this question and I was hoping someone could help me. One thing in particular is confusing me. Question: They are starting with a riemannian manifold $(M,g)$ and considering ...
3
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1answer
365 views

Proof of the Wirtinger inequality

This is an exercise from the book "Complex Geometry, An Introduction" by Huybrechts. The statement involves to prove that the restriction of the fundamental form $\omega$ of a vector space $V, I, \...
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0answers
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Local form of complex structure via Bruhat-Whitney

Set-up: Let $(L,g)$ be a real-analytic riemannian manifold. And let $T^*L$ denote the cotangent bundle of $L$. According to Bruhat-Whitney, on a suffciently small neigbourhood of the zero section in $...
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0answers
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Relating two notions of integrable almost complex structure on a complex vector bundle

Suppose $E \to M$ is a complex vector bundle over a complex manifold $M$. As I understand it, there are two possible ways to say what it means that $E$ is a holomorphic vector bundle: $E$ has the ...
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0answers
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complex structure compatible with symplectic form and riemannian metric [duplicate]

Given a non-degenerate symplectic form $\omega$ on a finite dimensional vector space $V$ and a complex structure $J$ (that is, $J\in End(V),~J^2=-Id$). If $J$ is compatible with $\omega$ in the sense ...
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Coordinates for a neighborhood of a totally real submanifold

Motivation: I'm interested in which submanifolds of (half dimension) a complex manifold can be expressed locally in coordinates as the real part of the complex coordinates, and was wondering if the ...
6
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2answers
394 views

Holomorphic vector bundles on almost complex manifolds

Let $M$ be a real manifold with complex structure $J$, making $M$ into an almost complex manifold. I know that the complexification $T_{\textbf{C}}M = TM\otimes \textbf{C}$ of the tangent bundle $TM$ ...
2
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2answers
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If $V$ is finite-dimensional with $J : V \to V$ such that $J^2 = -id$, then $V$ has even dimension [duplicate]

Let $V$ be a $\Bbb R$-vector space, with $J$ being an endomorphism $J: V \to V$ with $J^2=-id$ (identity). I already had to show that $V$ became a $\Bbb C$-vector space with the scalar multiplication:...
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0answers
72 views

Pulling back a Kähler structure on a symplectic submanifold

Let $(K, G, \Omega, J)$ be a Kähler manifold and $(S, \omega)$ be a symplectic manifold. Let $i : S \to K$ be a symplectic embedding. Is it possible to endow $S$ with a Kähler manifold structure, ...
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1answer
103 views

When does contractible space of almost complex structures taming a given symplectic form $\omega$ contain an integrable compatible one?

Given a symplectic form $\omega$ on a compact symplectic manifold $X$, we know there is a contractible homotopy class $\mathcal{J}_{\omega}$ of almost complex structures that tame $\omega$. A subset ...
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0answers
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Fibration induced by an almost complex structure

Let $E \rightarrow M$ be a plane bundle endowed with an almost complex structure $J.$ $J$ induces a natural positive definite inner product in the associated bundle $$End(E)\rightarrow M,$$ denoted by ...
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1answer
261 views

Find type of a differential form on an almost complex manifold

If $M$ is a nearly Kähler manifold (that is, an almost Hermitian manifold on which $\nabla_X(J)X=0$) we have the three-forms $$ A(X,Y,Z)=\langle\nabla_X(J)Y,Z\rangle \quad\text{and}\quad B(X,Y,Z)=\...
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Curvature identity on Nearly Kähler manifolds

Can somemone help me prove the following identity? $$ \| \nabla_X(J)(Y) \|^2 = \langle R_{XY}X,Y\rangle - \langle R_{XY}JX,JY\rangle$$ where $J$ is the almost complex structure, and $R$ the curvature ...
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1answer
199 views

Is any 2$m$-dimensional manifold almost complex?

In Nakahara's book "Geometry, Topology and Physics" (Ch. 8, about the almost complex structure) they write: Note that any 2$m$-dimensional manifold locally admits a tensor field $J$ [type (1,1)] ...
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288 views

Equivalence of (almost) complex structures

Preamble: An almost complex structure on a manifold $M$ is an endomorphism $J : TM \to TM$ such that $J^2 = -1$. An almost complex structure $J$ is said to be integrable if the Nijenhuis tensor, $N_J$,...
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2answers
220 views

Equilateral triangle in complex number [closed]

Let $a$, $b$ and $c$ be the affix of $A$, $B$ and $C$, where $a+b+c=0$ and $a$, $b$ and $c$ are of equal magnitude. Prove that $\triangle ABC$ is an equilateral triangle.
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1answer
281 views

Is there a natural Dolbeault operator on a almost holomorphic vector bundle?

For vector bundles $(\pi: V \rightarrow M )$ over a complex manifold, there is a notion of holomorphicity that can be defined in two equivalent ways : $V$ is a complex manifolds and $\pi:V \...
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0answers
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Parametrizing linear complex structures

So I'm reading this paper by Donaldson on contructing symplectic submanifolds, https://projecteuclid.org/euclid.jdg/1214459407 In section 2, he says the following: On ${\bf C}^n$, we have the ...
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2answers
293 views

smooth vs. analytic in the definition of almost-complex manifolds

Let $A_{\infty}\hspace{-0.03 in}$ be a maximal $C^{\infty}\hspace{-0.02 in}$ atlas on $M\hspace{-0.03 in}$, and with that smooth structure on $M$, suppose $\: j : TM\to TM\:$ is a smooth function ...
7
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1answer
727 views

Does every even-dimensional sphere admit an almost complex structure?

We know that there is an almost complex structure on $S^6$ which is not integrable. Is it always possible to find almost complex structures on $S^{2n}$? In particular does $S^4$ admit one?