Questions tagged [almost-complex]

For questions about almost complex structures on manifolds or vector spaces, or the complexification of vector spaces.

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47 views

Riemannian metric of a Kähler manifold

In p.52 of Morgan's book on Seiberg-Witten equations, there is the following paragraph: $(\cdots)$ Assume that $X$ is a complex manifold with a Kahler metric. This means that $X$ has a Riemannian ...
3
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1answer
37 views

A complex orthonormal basis induces a real orthonormal basis

Let $V$ be a real inner product space with an orthogonal almost complex structure $J:V\to V$. Then we can view $V$ as a complex vector space using $J$. Choose a complex basis $\{e_1,\dots,e_n\}$ for $...
2
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0answers
56 views

The identification of $iX$ with $JX$

I am reading the book, Lectures on Kahler Geometry, by Andrei Moroianu. Here a link, https://books.google.com.hk/books?id=oqmroUc9E8YC&printsec=frontcover&hl=zh-CN&source=gbs_ge_summary_r&...
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29 views

Canonical isomorphism from $\Lambda^{(p,q)} (V \otimes \mathbb{C})$ to $\Lambda^{p+q} (V \otimes \mathbb{C})$

I'm having trouble understanding this 'natural' isomorphism when discussing complex differential forms. Let $(M, J)$ be an almost complex manifold. Then its complexified cotangent bundle decomposes as ...
2
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1answer
42 views

Equivalent definitions of almost quaternionic structures

I came across this thread today when I was reading about almost quaternionic structures. I was wondering if there exists an argument similar to the answer to the thread above that can show that the $...
3
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1answer
57 views

Motivation for the Nijenhuis Tensor

I'm learning about complex and almost complex structures on smooth manifolds, in particular the Newlander-Nirenberg theorem. Recall that for an almost complex structure $J$ on a smooth manifold, the ...
0
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1answer
71 views

Relation between symplectic manifolds and (almost) complex manifolds

I'm a beginner in symplectic geometry, and I recently learned that every symplectic manifold has an almost complex structure. I am curious about the converse. Does every almost complex manifold have a ...
1
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1answer
24 views

The action of $\text{GL}(V)$ on the set of complex structures of $V$ is transitive

Let $J(V)$ be the set of all complex structures on a finite-dimensional real vector space $V$. (A complex structure on $V$ is by definition a linear isomorphism $J:V\to V$ such that $J^2=-\text{id}$.) ...
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0answers
40 views

Orientation and almost complex structures

Let $(M,\omega)$ be a symplectic manifold. Define $J(M)=\{\text{smooth almost complex structures compatible with the orientation of M}\}$. That specific definition gives me a few things to think. How ...
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0answers
19 views

Regular Value theorem for Complex submanifolds of an Almost complex submanifold

Suppose we have an almost complex manifold $(M,J)$ and $f:M\rightarrow \mathbb{C}$ a smooth function such that $0$ is a regular value, and that $(\bar \partial f)_p=0 $ for any $p\in M$. Then I would ...
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1answer
40 views

Almost complex structure on a contractible manifold

Let $M$ be a contractible manifold with an almost complex structure $J:TM\to TM$. Suppose $J':TM\to TM$ is another almost complex structure. Since $M$ is contractible, so is $TM$, hence $J$ and $J'$ ...
2
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1answer
34 views

Almost complex structure invariant by hamiltonian action

I was reading J.Evans lectures, in particular: Not necesary to read to understand question. At some point, he claims that given a compact symplectic manifold $(M,\omega)$ and a hamiltonian action from ...
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0answers
37 views

Tensor product of complexification

I am working with the definition of complexification $V_\mathbb{C}= V \otimes_\mathbb{R} \mathbb{C}$ where V is a real vector space, in order to better understand forms and vectors of type (p,q) when $...
2
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0answers
52 views

Proof that Nijenhuis Tensor is a Tensor

I have spent probably more time than I should attempting to verify this fact which is a question in Da Silva's "Lectures on Symplectic Geometry." I attempted to show it is $C^{\infty}(M)$-...
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36 views

$E$ is orientable given condition

Suppose $(E, M, \pi)$ is a vector bundle and $J:(E, M,\pi)\rightarrow (E, M,\pi)$ is a homomorphism such that $J\circ J=-id$. Then $E$ is orientable. What I thinking is applying this fact: vector ...
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1answer
110 views

An almost complex structure on the real 2-sphere $S^2$

If $R:=\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ and $S^2:=Spec(R)$ is the real 2-spere, a classical result of Borel and Serre says that the only real spheres with an almost complex structure is $S^2$ and $S^...
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1answer
70 views

Proving $\Bbb{H\otimes_R C }\cong M_2(\Bbb{C})$ versus proving $\Bbb{H\otimes_R C }\cong_{\mathbb R} M_2(\Bbb{C}).$

1- Is there a difference between saying I want to prove this $\Bbb{H\otimes_R C }\cong M_2(\Bbb{C})$ and I want to prove this $\Bbb{H\otimes_R C }\cong_{\mathbb R} M_2(\Bbb{C})$? I see the OP in the ...
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0answers
26 views

Symplectic manifolds and almost complex structures

On a symplectic vector space $(V,\omega)$ with an inner product $g$, one can construct a canonical almost complex structure using polar decomposition. If $(M,\omega)$ is a symplectic manifold with a ...
4
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3answers
256 views

What's the bijection between scalar/inner products and (certain) almost complex structures (on $\mathbb R^2$)?

Asked on maths overflow here. What's the bijection between (equivalence classes of) scalar products (I guess 'scalar product' is the same as 'inner product') and a.c.s. (almost complex structure/s) ...
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0answers
60 views

The question about Newlander-Nirenburg theorem and almost complex manifold

While studying the definition of almost complex structure, I have thought that a counterexample of almost complex structure but not complex structure. To ask a more complete question, I write the ...
5
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1answer
105 views

Is every plane in $\mathbb{R}^4$ a line in $\mathbb{C}^2$?

Every complex line, that is, one-dimensional complex affine space, in $\mathbb{C}^2$ is a real plane in $\mathbb{R}^4$. Is the converse true? That is, is every real plane in $\mathbb{R}^4$ a complex ...
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1answer
84 views

Eigenvalues of Complex Stucture in a Complexified Vector Space

For context, this question comes from my reading of Kobayashi+Nomizu's differential geometry book, volume 2 (pages 116-117). Given a real vector space $V$ with $\mathrm{dim}(V)=2n$, a complex ...
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1answer
72 views

Almost complex structure on real differentiable manifolds

An almost complex structure on a real differentiable manifold $M$ is a tensor field $J \in \Gamma(\mbox{End}(TM))$ satisfying $J^{2}=-I$, where $I$ is the identity tensor field. The pair $(M,J)$ is ...
4
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1answer
75 views

Why did I get $J \bar{v} = \sqrt{-1}\bar{v}$ ??

Q. I cannot deduce the equation $J \bar{v}= -\sqrt{-1}\bar{v}$ for $v$ satisfying $Jv= \sqrt{-1}v$. Details Let $V$ be a vector space on $\mathbb{R}$ equipped with a complex structure $J$. Using $J$, ...
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1answer
24 views

Two equivalent conditions of Pseudo-holomorphic maps

Let $\varphi: (S,J_S) \to (M,J_M)$ be a map from a complex manifold equipped with a integrable complex strucure $J_S$ to a smooth manifold equipped with a almost complex structure $J_M$. Using $J_M$, ...
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1answer
62 views

Defining the natural almost complex structure on a complex manifold.

The definition of an almost complex structure is as follows. If $X$ is a differentiable manifold and $TX$ is its tangent bundle, then the endomorphism $I: TX \to TX$ defines an almost complex ...
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0answers
113 views

Chern classes and almost complex submanifolds

Suppose $M$ is a manifold endowes with an almost complex structure, such that the Chern classes $c_i(M)$ of $M$ are defined. Can we say that the Poincare dual to $c_i(M)$ can be represented by an ...
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1answer
155 views

The first chern class of $O(1)$ is positive

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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0answers
25 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 ...
3
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1answer
135 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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1answer
58 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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1answer
86 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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0answers
23 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 $W\cap JW = 0$. ...
0
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1answer
91 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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2answers
103 views

Finding an almost complex structure (aka anti-involution) given an involution

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 ...
1
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1answer
90 views

For any two almost complex structures on infinite-dimensional space: Do they give isomorphic vector spaces? Are they similar?

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 ...
1
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1answer
50 views

If subspace $A$ is the fixed points of an involution $\sigma$, then is $K(A)$ the fixed points of $-\sigma$?

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 ...
1
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0answers
96 views

Formula for anti-involutive (almost complex structures) and involutive maps on any $V^2$ in terms of linear maps on $V$ based on $\mathbb R^2$

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 ...
0
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1answer
160 views

Bijection for involutive maps and $\mathbb R$-subspaces given almost complex structure (anti-involutive)? Formula for conjugation?

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 ...
3
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1answer
263 views

What exactly is the relationship between the concepts of conjugate complex vector space and conjugations/real structures?

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 ...
1
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2answers
117 views

Existence of subspaces such that almost complex structures restrict to almost complex structures

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 ...
0
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1answer
71 views

Restriction of complexification is almost complex but not conversely?

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 ...
2
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0answers
172 views

Are anti-linear/semi-linear maps, such as conjugations, linear in other almost complex structures?

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 ...
1
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1answer
96 views

Questions on the complexification of a map

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 ...
3
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1answer
226 views

Eigenvalues and eigenspaces of almost complex structures under each other

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 ...
0
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1answer
220 views

Complexification of realification: Unnecessary computation of eigenvalues?

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 ...
1
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1answer
132 views

In what way is $(L_{\mathbb R})^{\mathbb C}$ more like $L \bigoplus \overline L$ than like $L \bigoplus L = L^2$?

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 ...
2
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1answer
149 views

$f$ is the complexification of a map if $f$ commutes with structure $J$ and conjugation $\chi$. What is the relationship between $J$ and $\chi$?

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 ...
3
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2answers
421 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?

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 ...
1
vote
1answer
281 views

Complexification of a map under nonstandard complexifications of vector spaces

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 ...