Questions tagged [almost-complex]

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

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32 views

$c_1(-, J(-))$ is Riemannian metric

Let $\mathbb{P^nC}$ the complex projective space and $O(1)$ the dual bundle to the tautological bundle $O(-1)= \{([x], z) \in \mathbb{P^nC} \times \mathbb{C}^n \ \vert \ z \in \mathbb{C}^* \cdot x \}$...
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18 views

Relative version of identity theorem for pseudoholomorphic curves

Let $(M,J)$ be an almost complex manifold and $N\subset M$ a closed almost complex submanifold, i.e. the tangent space $T_xN$ at every point $x\in N$ is invariant under $J$. Let $\Sigma$ be a ...
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79 views

Real $2n$-plane bundle with a complex structure is a complex $n$-plane bundle

I am trying to show that if $\xi=(E,B,\pi)$ is a real $2n$-plane bundle with a complex structure $J:E\to E$ then $\xi$ becomes a complex $n$-plane bundle if we define $(x+iy)v=xv+yJ(v)$ on each fiber. ...
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21 views

Fundamental form of almost complex manifold is $(1,1)$-form

Let $M$ be an almost complex manifold with almost complex structure $J$, a compatible Riemannian metric $g$ and fundamental form $\omega$. Consider the eigenspaces $T_p^{1,0}M=\{v\in T_pM\otimes\...
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40 views

A complex vector bundle $E$ is a holomorphic vector bundle iff $(\overline{\partial^E})^2=0$ help with proof?

Ok so I am following a set of notes on complex differential geometry and there is a theorem that says the following: If $E$ is a complex vector bundle over a complex manifold and $\overline{\partial^...
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9 views

Is there a reasonable notion of a totally real Grassmannian?

Suppose we are given a $2n$-dimensional real vector space $V$, along with a complex structure $J:V\to V$. For $k\le n$, a $k$-subspace $W\subset V$ is called $J$-totally real if $W\cap JW = 0$. ...
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9 views

The integrability of almost complex structure in the sense of Frobenius theorem.

I've tried to distinguish the almost complex structure and complex structure, intuitively. Without any chance of confusion, I'll assume that the manifolds and the maps are "smooth". The almost ...
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11 views

On Existence of Solutions to a Cauchy-Riemann Equation with Boundary Conditions

Assume $\mathbb{D} $ is the closed unit disk in $\mathbb{C} $ with an almost complex structure $J\in C^{\infty } (\mathbb{D} ;\mathrm{End} (E))$ for $E$ a smooth complex vector bundle of rank $r$ over ...
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35 views

Dimension of the complex projective space $\mathbb{C}\mathbb{P}^n$ as almost complex manifold

I have already shown that the complex projective space $\mathbb{C}\mathbb{P}^n$ is a complex manifold by checking the required properties of the transition maps. Since every complex manifold is an ...
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24 views

Understanding of Carleman similarity principle

The Carleman similarity principle is stated as follows in my lecture: What I don't understand is the statement $u(z)=\Phi(z)f(z)$ since $\Phi(z)$ should be an endomorphism and $u(z)$ just some ...
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26 views

Extending an $\omega$-tame complex structure

The following is an execise from Chris Wendl's book Holomorphic Curves in Low Dimensions Let $(M,\omega)$ be a symplectic manifold and $A\subseteq M$ a closed subset. Let $J_A$ be an $\omega$-...
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1answer
56 views

What does it mean for an almost complex structure to be compatible with a Riemannian metric?

This is on p. 37 of The Ricci Flow and the Sphere Theorem by Simon Brendle. The author is going to show that a gradient Ricci soliton on $S^2$ has constant scalar curvature. Here is the setting: Let $...
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65 views

Complex and Conformal structure on a trivial tangent bundle of a higher dimensional manifold

Let $M$ be an even dimensional Reimannian manifold with trivial tangent bundle $TM$. Then there is a global orthonormal basis $\{e_1,e_2,...,e_{2n-1},e_{2n}\}$ of $TM$. Define a bundle map $J :TM \...
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57 views

Advantage in considering an almost complex structure

Let $M$ be a real Riemannian manifold of even dimension, what are the advantages of considering an almost complex structure on it rather than only real?
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1answer
80 views

Question about the (pseudo)-holomorphic map between almost complex manifolds

It may be a very simple question,but I can't figure it out. Let $(M,I)$ be an almost complex manifold where $I$ is an almost complex structure.Then $-I$ is also an almost complex structure.Then the ...
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67 views

Averaging an almost complex structure over compact Lie group action

It is well-known that if a Lie group $G$ acts on a symplectic manifold $(W,\omega)$ preserving $\omega$, then we can easily find a $G$-invariant almost complex structure $J$ by choosing first a $G$-...
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134 views

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
66 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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108 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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44 views

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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138 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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143 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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231 views

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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37 views

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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87 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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76 views

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
206 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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76 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
181 views

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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90 views

$M$ 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
51 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 $\pi$ ...
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1answer
276 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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190 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 ...
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1answer
42 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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107 views

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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302 views

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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241 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
270 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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140 views

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
46 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) ...
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1answer
118 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}_{...
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1answer
461 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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520 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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27 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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112 views

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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32 views

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
1k views

Canonical almost complex structure on symplectic manifold

I'm trying to prove the following (well-known) theorem in symplectic geometry. Theorem . For $(M, \omega)$ a symplectic manifold with Riemannian metric $g, \exists$ a canonical almost complex ...
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1answer
473 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 ...
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1answer
521 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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29 views

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 $...