# Questions tagged [almost-complex]

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

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### Spinors on (euclidean signature) spacetime

Let's consider a spacetime $M$ which is a also spin manifold. In Euclidean signature We have that the frame bundle is a principal $GL(4,\mathbb{R})$ bundle over $M$. Even dimensional spin manifolds ...
• 340
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### Total space of the frame bundle always an almost complex manifold in $dim >2$?

I seem to remember reading that the total space of the frame bundle of a smooth Riemannian manifold always admits an almost complex structure in dimensions greater than 2. However now I can't seem to ...
• 340
64 views

### Understanding an integrable almost complex structure

My notes only say that "Definition An almost complex structure is integrable if it is induced by an underlying complex structure." How do I translate this into a formula? I am not sure why ...
• 3,152
1 vote
52 views

### $I$ is parallel with respect to the Levi-Civita connection if and only if $\omega$ is parallel. $I:$almost non-complex structure

Let $(M, g)$ be a Riemannian manifold, let $I$ be an almost complex structure compatible with $g$ and $\omega$ the corresponding 2-form (that is, $ω^\flat = g^\flat \circ I$). Let $\nabla$ be the ...
• 3,152
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1 vote
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### Understanding the almost complex structure of a complex manifold

I start learning complex manifold by myself and hard to lift my previous intuition of differential geometry over the complex structure. Let $M$ be a real $2m$-dimensional manifold. We define an ...
• 701
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• 765
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### Existence of almost complex structure on smooth even dimensional manifold

I am trying to prove that if $M$ is an even dimensional manifold, and the bundle of linear frames of $TM$ admits a reduction of its structure to group to $GL_n(\mathbb{C})$, then $M$ admit's an almost ...
• 3,431
1 vote
101 views

### Induced almost complex structure of a complex structure

Let $V$ be a complex vector space. A statement in Complex Geometry of Huybrechts p.25 is, that one could define an almost complex structure on the underlying real vector space of $V$ by $v \mapsto iv$....
• 144
1 vote
193 views

### Action of almost complex structure on tensors/forms

I am currently trying to lear about almost complex structures and how they are extended to tensor fields especially differential forms. I have seen some variations but am confused about certain ...
• 165
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### Homotopy type of space of almost complex structures

Let M be an open 2n dimensional manifold that admits a non-degenerate 2 form. By Gromov's h-principle we know that M admits a symplectic form and that the homotopy type of the space symplectic forms ...
• 295
124 views

### Computation involving the Hodge star and exterior covariant derivative on an almost complex manifold

Let $M$ be an almost complex manifold of complex dimension $n$. Let $E$ be a complex vector bundle over $M$ with a Hermitian metric $h$. Let $A$ be a Hermitian connection with respect to $h$. Let $d_A$...
• 2,077
241 views

### Can one define an almost complex structure on any complex vector bundle over an almost complex manifold?

Let $E \to M$ be a complex vector bundle over an almost complex manifold. Is there always an almost complex structure on $E$? I.e. does there always exist some $J : E \to E$ such that $J^2 = -Id$? ...
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### Bundle isomorphisms for $J$-holomorphic tangent bundle

In Chapter 14 of Lectures on Symplectic Geometry by da Silva, she claims that, if $(M,J)$ an almost complex manifold, then there are real bundle isomorphisms \begin{align*}\pi_{1,0}:TM\otimes\mathbb C&...
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• 53
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### 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 ...
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### 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 ...
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### 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 ...
• 1,630
1 vote
154 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 $J^2=-\text{id}$.) ...
• 1,630
1 vote
312 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 ...
• 419
1 vote
105 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 ...
• 4,767
1 vote
141 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 $J'$ ...
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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 ...