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Questions tagged [almost-complex]

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

An almost complex structure on a real vector space $V$ is an endomorphism $J : V \to V$ such that $J\circ J = -\operatorname{id}_V$.

An almost complex structure on a smooth manifold $M$ is a vector bundle endomorphism $J : TM \to TM$ such that $J\circ J = -\operatorname{id}_{TM}$. Note, an almost complex structure on $M$ gives rise to an almost complex structure $J_x : T_xM \to T_xM$ on each tangent space $T_xM$.

Every complex vector space, when identified as a real vector space, has a canonical almost complex structure $J$ given by multiplication of $i$. Similarly, every complex manifold has a canonical almost complex structure. It is a deep theorem of Newlander and Nirenberg that an almost complex structure $J$ on a manifolds comes from a complex manifolds if and only if the Nijenhuis tensor $$\mathcal{N}(X,Y)=[JX,JY]-J[JX,Y]-J[X,JY]-[X,Y]$$ vanishes. See more information here

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204 questions
65
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Does $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there. $\Bbb{CP}^{2n+1} \mathbin{\#} \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism…
user98602
17
votes
5 answers

Almost complex manifolds are orientable

I want to verify the fact that every almost complex manifold is orientable. By definition, an almost complex manifold is an even-dimensional smooth manifold $M^{2n}$ with a complex structure, i.e., a bundle isomorphism $J\colon TM\to TM$ such…
YYF
  • 3,077
13
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1 answer

Identification of the holomorphic tangent space with the real tangent space

I'm reading Griffiths and Harris and just want to check I'm interpreting a passage correctly. Let $M$ be a $n$ dimensional complex manifold. We define $T_p^\mathbb{R}M$ as the tangent space of the underlying smooth manifold, and define…
user2520938
  • 7,503
12
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2 answers

Almost complex structures on spheres

It is fairly well-known that the only spheres which admit almost complex structures are $S^2$ and $S^6$. By embedding $S^6$ in the imaginary octonions, we obtain a non-integrable almost complex structure on $S^6$. By embedding $S^2$ in the…
Michael Albanese
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11
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1 answer

An almost complex structure on real 2-dimensional manifold

Why an almost complex structure on real 2-dimensional manifold is integrable?
Alon
  • 191
11
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1 answer

Nonintegrable almost complex structures

The Newlander-Nirenberg theorem states that any Integrable Almost Complex manifold is a complex manifold. I am looking for natural examples of almost complex structures that are not integrable.
user8621
10
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1 answer

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 $\mathrm{GL}(n, \mathbb{C})\subset\mathrm{GL}(2n, \mathbb{R})$. What I'm trying to prove is: If $M$ has an…
Leonardo Schultz
  • 675
10
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1 answer

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 name "totally real" had any connection to…
Ashley
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9
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2 answers

Topology on the space of compatible almost complex structures in symplectic geometry

I have a few fairly generic questions, with a specific application to symplectic geometry in mind. Let me pose the specific problem first: Let a symplectic manifold $(M,\omega)$ be given. One is naturally led to consider the "space" $\mathcal{J}$ of…
Alex Provost
  • 21,651
9
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$2$ out of $3$ property of the unitary group

I am trying to understand the $2$ out of $3$ property of the unitary group. I have almost got it, but I am not completely sure about the interaction between an inner product and a symplectic form to obtain an almost complex structure. Let $V$ be a…
Michael Albanese
  • 104,029
9
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2 answers

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 Nijenhuis tensor field of $J$ is defined…
Legendre
  • 949
9
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1 answer

Alternative Almost Complex Structures

Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector space by defining $(a+bi)v = av + bJ(v)$. The idea…
Michael Albanese
  • 104,029
9
votes
1 answer

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 $$\lambda(\xi) = \langle D\pi(\xi), \pi'(\xi)\rangle,$$ where…
Thurmond
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9
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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 $X = \mathbb{S}^{1} \times \mathbb{S}^{3}$ -- in this case one can…
hyperkahler
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9
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1 answer

Almost complex structure on $\mathbb CP^2 \mathbin\# \mathbb CP^2$

Why doesn't $\mathbb CP^2 \mathbin\# \mathbb CP^2$ admit an almost complex structure?
Maciej Starostka
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