# Questions tagged [almost-complex]

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

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### $c_1(-, J(-))$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...