Questions tagged [almost-complex]
For questions about almost complex structures on manifolds or vector spaces.
198 questions
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Reduction of structure group of frame bundle under almost complex structure for different metric signatures
I get that there is an isomorphism between $SO(n,m,\mathbb{C})$ for all $n,m$ where $n+m=k$ and $k$ is some fixed integer. What I'm asking is slightly more nuanced. Suppose we have some orientable $k$ ...
- 352
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21
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Local Frame on an Almost Complex Manifold
Let $(M, J)$ be an almost complex manifold of dimension $2n$. For each point $p \in M$, I know that we can find linearly independent tangent vectors $v_1, \ldots, v_n \in T_pM$ such that
$$
v_1, \...
- 1,022
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41
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Find a non-zero holomorphic 1 form on an almost complex manifold
Let $(M,J)$ be an almost complex manifold and $\phi:M\to\mathbb{R}$ a non-constant function which satisfies $dJd(\phi)=0$, where $Jd\phi$ is the 1-form given by $TM\ni v\mapsto d\phi(Jv)$. Prove that $...
- 141
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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 ...
- 352
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226
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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 ...
- 3,508
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62
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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 ...
- 3,508
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174
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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, ...
- 1,825
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1
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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$, ...
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114
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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 ...
- 2,935
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114
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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·...
- 61
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1
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247
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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 ...
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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 $...
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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 $...
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57
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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 ...
- 4,163
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1
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120
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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$....
- 174
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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 ...
- 175
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157
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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
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146
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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$...
- 2,095
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314
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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$?
...
- 2,095
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25
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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 ...
- 111
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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 ...
- 295
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1
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49
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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.$$
I think my instructor made a mistake. As a (...
- 2,692
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0
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42
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Explicit formula for $J[X,Y](f)$?
Let $(M,J)$ be a complex manifold. I'm trying to find an explicit formula for $(J[X,Y])(f)$, like how for the simple commutator we have (by definition even)
$$ [X,Y](f) = X(Y(f)) - Y(X(f)) $$
I'm ...
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129
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How does an almost complex structure on a sphere determine a 1-fold cross product?
In the article "VECTOR CROSS PRODUCTS ON MANIFOLDS", by Alfred Gray, the author states, between theorem 2.8 and corollary 2.9, that
The existence of vector cross product of Type I [1-fold ...
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132
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Symmetric form induced by symplectic form and complex structure is definite
Given a finite dimensional real vector space $V$ of dimension $2n$, then the space of non-degenerate skew-symmetric forms on it is an open subset of skew-symmetric forms, which is diffeomorphic to the ...
- 363
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Does the condition $(\nabla_XJ)Y=(\nabla_YJ)X$ imply $\nabla J=0$?
Let $(M,g,J)$ be an Almost Hermitian manifold and $\nabla $ the Levi-Civita connection. If
$$(\nabla_XJ)Y=(\nabla_YJ)X$$
for any $X,Y\in \Gamma(TM)$, can we get $\nabla J=0$, i.e., $(M,g,J)$ is ...
- 183
3
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316
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Equivalence between two definitions of Complex Tangent Space
I have two definitions of "Complex Tangent Space" over a Complex Manifold $M$ of (complex) dimension $n$. One of them is defining, over the real tangent space $T_p(M)$, the complex structure ...
3
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71
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Conditions on space of smooth almost complex structures so that it's a banach manifold
In the following paper by A.Abbondandolo and M. Schwarz https://arxiv.org/pdf/math/0408280.pdf in section $1.6$ we consider the set of smooth almost complex structures $\mathcal{J}$ on $T^*M$ ...
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1
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137
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$U(2)$ fixes Orthogonal Complex Structures on $\mathbb{R}^4$?
An orthogonal (almost) complex structure on a manifold M equipped with a metric g is a map $$J : TM\mapsto TM , J^2=-1$$ That preserves orientation and satisfies $g(Ju,Jv)=g(u,v)$
By identifying $\...
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99
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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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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 $...
- 1,319
1
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1
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101
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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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71
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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 ...
- 387
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1
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294
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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 ...
- 1,953
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123
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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:...
- 5,190
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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 ...
user388493
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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 ...
- 363
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332
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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 ...
- 315
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1
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373
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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 $...
- 63
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322
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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 ...
- 1,920
3
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1
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412
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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 $...
- 1,920
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100
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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&...
- 29
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49
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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 ...
- 11
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68
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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 $...
- 334
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2
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940
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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 ...
- 909
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1
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1k
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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 ...
- 1,672
1
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1
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170
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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 $J^2=-\text{id}$.) ...
- 1,672
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378
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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 ...
- 439
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0
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132
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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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