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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 ...
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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 ...
R. Rankin's user avatar
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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 ...
some_math_guy's user avatar
1 vote
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$I$ 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 ...
some_math_guy's user avatar
4 votes
1 answer
159 views

A manifold with an almost-complex structure that admits a holomorphic frame is integrable

Let $(M,I)$ be an almost-complex manifold such that every point $m\in M$ has a neighborhood $U\ni m$ and holomorphic functions $f_1, \dots, f_n:U\to\mathbb C$ such that $\text{span}(df_1(m), \dots, ...
Derivative's user avatar
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2 votes
1 answer
72 views

Integral Stiefel-Whitney classes and stable almost complex structure

Question: Let $\xi$ be a real vector bundle over a space $B$. Suppose $\xi$ admits stable almost complex structure, i.e., $\xi\oplus \epsilon^k$ admits almost complex structure for some $k\geq 0$, ...
Random's user avatar
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Symplectic forms on $\mathbb{C}P^3$ with trivial first Chern class

Question Does there exist a symplectic form $\omega$ on $\mathbb{C}P^3$ with first Chern class $c_1(\mathbb{C} P^3, \omega) = 0$? Context Let $(M,\omega)$ be a symplectic manifold. An almost complex ...
Derived Cats's user avatar
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Hyperkähler structure of Quaternion

I think there was an issue with the question I asked earlier. I want to prove that the quaternion is a hyperkahler manifold. I know that there is a natural metric $\rho$ on that given by $\rho(a,a)=a·...
ymm's user avatar
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1 answer
137 views

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 ...
N00BMaster's user avatar
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Decompositions of exterior products

Let $V$ be a real vector space equipped with an almost complex structure $J$ and denote the complexification by $V_\mathbb C$. The complexified vector space splits into the eigenspaces $V^{1,0}$ and $...
Nathaniel Johonson's user avatar
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Show that any almost complex structure is induced by at most one complex structure.

Show that any almost complex structure is induced by at most one complex structure. Suppose $X$ is a complex manifold of dimension $n$ and $M$ is the underlying $2n$ dimensional real manifold. If $...
Danlo's user avatar
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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 ...
Chris's user avatar
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1 vote
1 answer
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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 $v \mapsto iv$....
jr01's user avatar
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1 answer
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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 ...
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1 answer
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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 ...
cr1t1cal's user avatar
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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 $d_A$...
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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 $J^2 = -Id$? ...
Paul Cusson's user avatar
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Arriving at an expression of the $(0,1)$ part of the LC connection as a pseudoholomorphic operator on a complex manifold

Let $(M, J, h)$ be a complex Hermitian manifold with Levi-Civita connection $\nabla$. If we see $\nabla$ as an operator $\nabla: \Gamma(TM) \rightarrow \Gamma(\Omega^1(M)) \otimes \Gamma(TM)$, and $\...
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On the almost complex structure [closed]

We say that $J$ is an almost complex structure on a 2n-dimensional manifold M is a map $J:TM\rightarrow TM$ such that $J^2=−id_{TM}.$ What is the difference between an almost complex structure and a ...
J.abdou's user avatar
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1 answer
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Existence of metric-compatible almost complex structure on $S^2$

I'm trying to understand this answer, since I'm stuck on the same proof as OP. $2)$ A two-dimensional surface admits an almost complex structure if and only if it is orientable. If $X$ is an oriented ...
Leroy Od's user avatar
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2 votes
1 answer
49 views

Does an orthogonal matrix always conmutes or anticonmutes with a matrix $J$ such that $J^2=-I$?

I was asked by my professor to show that if $A$ is an orthogonal matrix and $J$ is a matrix such that $J^2=-I$ ($I$ is the identity matrix) then $$AJ=JA.$$ I think my instructor made a mistake. As a (...
James Garrett's user avatar
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Explicit formula for $J[X,Y](f)$?

Let $(M,J)$ be a complex manifold. I'm trying to find an explicit formula for $(J[X,Y])(f)$, like how for the simple commutator we have (by definition even) $$ [X,Y](f) = X(Y(f)) - Y(X(f)) $$ I'm ...
rosecabbage's user avatar
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1 answer
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How does an almost complex structure on a sphere determine a 1-fold cross product?

In the article "VECTOR CROSS PRODUCTS ON MANIFOLDS", by Alfred Gray, the author states, between theorem 2.8 and corollary 2.9, that The existence of vector cross product of Type I [1-fold ...
Eduardo Toffolo's user avatar
1 vote
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110 views

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 ...
Jack Fres's user avatar
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3 votes
1 answer
195 views

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 ...
Geom Zari's user avatar
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272 views

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 ...
Marcos Martínez Wagner's user avatar
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67 views

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 $T^*M$ ...
Someone's user avatar
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2 votes
1 answer
130 views

$U(2)$ fixes Orthogonal Complex Structures on $\mathbb{R}^4$?

An orthogonal (almost) complex structure on a manifold M equipped with a metric g is a map $$J : TM\mapsto TM , J^2=-1$$ That preserves orientation and satisfies $g(Ju,Jv)=g(u,v)$ By identifying $\...
mcwiga's user avatar
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1 answer
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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&...
boink's user avatar
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3 votes
0 answers
119 views

Transform Fourier of paths in symplectic vector spaces

In the proof of Lemma 4.4.4 in "J-holomorphic Curves and Symplectic Topology" by McDuff and Salamon, They state the two following claims: Let $(V,\omega)$ be a symplectic vector space, and $...
Rei Henigman's user avatar
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1 vote
1 answer
94 views

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). ...
nonreligious's user avatar
3 votes
0 answers
66 views

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 ...
nonreligious's user avatar
1 vote
1 answer
207 views

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 ...
topolosaurus's user avatar
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4 votes
1 answer
101 views

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:...
yi li's user avatar
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2 votes
1 answer
63 views

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 ...
user avatar
6 votes
1 answer
212 views

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 ...
61plus's user avatar
  • 353
3 votes
1 answer
309 views

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 ...
Prajwal Samal's user avatar
2 votes
1 answer
296 views

Extension of compatible almost complex structures from a closed set

Suppose that $(M,\omega)$ is a symplectic manifold, and $N \subset M$ is a closed submanifold of $M$. If $J_N: TM|_N \rightarrow TM|_N$ is an $\omega$-compatible almost complex structure, defined on $...
Cinlef89's user avatar
2 votes
1 answer
296 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 ...
blancket's user avatar
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3 votes
1 answer
312 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 $...
blancket's user avatar
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2 votes
0 answers
99 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&...
Kai Xu's user avatar
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1 vote
0 answers
44 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 ...
mb28025's user avatar
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2 votes
1 answer
59 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 $...
GoogleME's user avatar
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7 votes
2 answers
769 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 ...
Legendre's user avatar
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4 votes
1 answer
842 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 ...
user302934's user avatar
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1 vote
1 answer
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 $J^2=-\text{id}$.) ...
user302934's user avatar
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1 vote
0 answers
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 ...
Timmathy's user avatar
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1 vote
0 answers
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 ...
Someone's user avatar
  • 4,767
1 vote
1 answer
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 $J'$ ...
blancket's user avatar
  • 1,780
2 votes
1 answer
134 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 ...
miraunpajaro's user avatar
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