# Questions tagged [almost-complex]

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

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### Symmetric form induced by symplectic form and complex structure is definite

Given a finite dimensional real vector space $V$ of dimension $2n$, then the space of non-degenerate skew-symmetric forms on it is an open subset of skew-symmetric forms, which is diffeomorphic to the ...
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### Commutation of dual with the decomposition given by a complex conjugation

From what I understand, there are some implicit facts used everywhere in defining complex-valued forms on an almost complex manifold $(M,J)$. It boils down to linear algebra. Let $(V,J)$ be a vector ...
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### Does the condition $(\nabla_XJ)Y=(\nabla_YJ)X$ imply $\nabla J=0$?

Let $(M,g,J)$ be an Almost Hermitian manifold and $\nabla$ the Levi-Civita connection. If $$(\nabla_XJ)Y=(\nabla_YJ)X$$ for any $X,Y\in \Gamma(TM)$, can we get $\nabla J=0$, i.e., $(M,g,J)$ is ...
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### Equivalence between two definitions of Complex Tangent Space

I have two definitions of "Complex Tangent Space" over a Complex Manifold $M$ of (complex) dimension $n$. One of them is defining, over the real tangent space $T_p(M)$, the complex structure ...
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### Conditions on space of smooth almost complex structures so that it's a banach manifold

In the following paper by A.Abbondandolo and M. Schwarz https://arxiv.org/pdf/math/0408280.pdf in section $1.6$ we consider the set of smooth almost complex structures $\mathcal{J}$ on $T^*M$ ...
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1 vote
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### Restrictions on coordinate basis of $T_pM$ required for a manifold to admit a Hermitian metric?

I asked this question about relating the Riemannian metric on a manifold $M$ to the Hermitian metric that arises when $M$ is thought of as a complex manifold (i.e. with integrable complex structure). ...
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### An explicit relation betwen Riemannian metric and an associated Hermitian metric?

I am trying to explicitly find the relationship between a Hermitian metric on a complex manifold and the Riemannian metric on the underlying real manifold -- and specifically on how the determinants ...
1 vote
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### Integrable connections and holomorphic structures

Let $X$ be a complex manifold of complex dimension $n$ and let $(E,h)$ be a smooth Hermitian vector bundle over $X$. Then there is a correspondence between unitary connections on $(E,h)$ and almost ...
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### multiplicative extension of the almost complex structure $I$

I was reading Huybrechts complex geometry book,in page 28-29 there is a linear operator defined as follows $\mathbf{I}: \bigwedge^{*} V_{\mathbb{C}} \rightarrow \bigwedge^{*} V_{\mathbb{C}}$ such that:...
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### Global $\omega$-compactible complex structure on symplectic manifold

When we have a symplectic form $\omega$ on an even dimensional linear space, we can consider the complex structure $J$ that is compactible with it, i.e. $\omega(Jv,Jw)=\omega(v,w)$. It is always ...
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### Confusion on the definition of a complex structure

I am reading the notes of Moroianu's Lectures on Kahler Geometry: https://moroianu.perso.math.cnrs.fr/tex/kg.pdf. On page 30 we want to prove that the Levi-Civita connection and Chern connection ...
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### Existence of Complex Frames on a Complex Vector Bundle

$E \rightarrow M$ be a complex vector bundle (of real rank $2r$) with almost complex structure $J:E\rightarrow E \space\space\space(J^2 =-1)$ on it. $U\subset M$ be a trivial neighbourhood. Does ...
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### 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 ...
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### 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 $C^{\infty}(M)$-...
1 vote