Questions tagged [almost-complex]
For questions about almost complex structures on vector spaces or manifolds.
2
votes
2answers
99 views
Reality check on integrability of almost-complex surface.
Let $(M^2,J)$ be an almost-complex surface. I do not assume integrability of $J$ now. I understand that from the Newlander-Nirenberg theorem, one of the several equivalences for the integrability of $...
1
vote
1answer
44 views
Fundamental form of an almost complex manifold
so the setting I am in right now is the following:
Let $(M^{2m},h,J)$ be an almost Hermitian manifold with fundamental form $\Omega$.
I really would like a local expression for this $\Omega (X,Y)=h(...
6
votes
1answer
54 views
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}...
1
vote
0answers
28 views
Definition of an almost complex hyperplane in the projective space $\mathbb{C} P^n$.
Let J be an almost complex Structure in the projective space $\mathbb{C} P^2$.
According to Duval a J-line in the almost complex projective space $\mathbb{C} P^2,$ is the almost complex analogue of a ...
3
votes
1answer
48 views
Why is the matrix representation of an almost complex structure like this?
Let $M$ be a Riemann surface, $J$ be an almost complex structure (i.e. a 1-1 tensor such that $J^2=-I$ and for any $x \in M,v,w \in T_xM, \{v,Jv\}$ is oriented). Consider a conformal coordinate at a ...
2
votes
0answers
46 views
Direct proof that a parallel almost complex structure is integrable
Let $(M,g,J)$ be an almost Hermitian manifold nad $\nabla$ be the Levi-Civita connection on $M$. If $\nabla J=0$, it is straightforward to show that the Nijenhuis tensor of $J$ must vanish which ...
2
votes
0answers
81 views
Vanishing of the Nijenhuis tensor
The Nijenhuis tensor is defined to be:
$$(1):\quad
N_J(X,Y)\equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY],
$$
for vector fields $X$ and $Y$ on the manifold $M$, equipped with Almost Complex Structure $J$:
$$(...
2
votes
0answers
29 views
an exercise about integrability of almost complex structures
i have spent some time over the following problem from a problem sheet of a course on complex geometry.
Let $M=G/H$ be a homogeneous space (where $G$ is a Lie group, $H$ a closed subgroup) and let $I$...
1
vote
1answer
64 views
(Almost) complex manifold problem.
Can some one give me an exemple of an almost compplex manifold that is not a complex manifold and why? I know that an almost complex manifold is of even real dim and is orientable. I also heard that ...
1
vote
0answers
62 views
Local construction of a map on a manifold
Assume $(M,J)$ is a smooth manifold equipped with an almost complex structure. Let $p$ be a point of $M$, and define the linear map $\Theta_p \colon T_pM \to T_pM$ so that $$\Theta_p = -\frac{1}{2}\...
1
vote
1answer
102 views
Does every almost complex manifold admit an almost-Kähler structure?
I couldn't find a conclusive answer to this question online. Here is my reasoning.
Let $M$ be an almost complex manifold. Then, from what I understand, we can define almost complex structure $J$ on $...
3
votes
0answers
64 views
Vertical $J$-holomorphic spheres
Let $\pi:M\to N$ be a smooth fiber bundle with fiber $F$ and let $g$ be a Riemannian metric on $M$. Using $g$ we may split the tangent bundle of $M$ as $$TM\cong TF\oplus \pi^{\ast}TN,$$ where $TF$ is ...
1
vote
2answers
122 views
Equivalence of two definitions of Almost Complex Structure.
Let $M$ be a smooth manifold of even dimension $2n$. I would like to understand the equivalence of the following two standard definitions for an almost complex structure on $M$.
Denote by $TM$ the ...
4
votes
0answers
74 views
$M$ is a complex manifold $\Leftrightarrow \overline{\partial}^2=0$
One way of stating Newlander-Niremberg's theorem is by saying an almost complex manifold $(M,J)$ is complex $\Leftrightarrow \overline{\partial}^2=0$.
I'm confused by this, because I can't see why ...
2
votes
1answer
47 views
Equivalent definitions of almost complex structures
The definition I have seen for an almost complex structure is the following
$$J:TM\to TM$$ which is linear fibre by fibre, such that $J^2 = -\text{Id}$, and such that $\pi(J(X_x)) = x$ where $\pi$ ...
4
votes
1answer
116 views
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 $$\lambda(\xi) = \...
1
vote
0answers
102 views
Complex Vector Bundle vs Holomorphic Vector Bundle vs Almost Complex Structures
I am writing because I am extremely confused with the structure of complex vector bundles. Ok first of all I understand that a complex vector bundle is a just a vector bundle $\pi:E\to X$ such that ...
4
votes
1answer
34 views
Equivalence of tangential and normal stably almost complex structure
Let $M$ be a smooth manifold. $M$ is said to be tangential stably almost complex if $TM \oplus \underline{\mathbb{R}}^k$ can be given a structure of a complex vector bundle, for some $k$. $M$ is said ...
1
vote
2answers
73 views
Integrable almost complex structure conjugated by diffeomorphism
Let $M$ be a smooth manifold and $J$ be an integrable almost complex structure on $M$. Let $f: M\to M$ be a diffeomorphism with $f_{*}:TM\to TM$ its tangent map. Then it is easy to see that $f_{*}Jf_{*...
2
votes
1answer
96 views
Natural complex-linear isomorphism between $ V_J $ and $V^- $
Let $V$ be a real vector space with a linear complex structure $J$ (see Wikipedia).
Denote $V_J$ the complex vector space induced from $V$ by the complex structure $J$.
Also, define the ...
1
vote
1answer
139 views
Nijenhuis tensor in local coordinates
If $(M,J)$ is an almost complex manifold, $\mathcal{N}$ (Nijenhuis tensor) is:
$$\mathcal{N}(X,Y)=[JX,JY]-J[X,JY]-J[JX,Y]-[X,Y].$$
I am trying to compute it in local coordinates $(x_{1},\cdots,x_{n})$...
1
vote
1answer
137 views
Relation between Riemannian metric and Hermitian structure.
Let $M$ be a complex manifold with almost complex structure $J$, and $TM$ it's real tangent space, $TM^{\mathbb{C}}=TM \otimes \mathbb{C}$ its complexified tangent space.
Now I'm getting confused ...
5
votes
1answer
111 views
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} ...
0
votes
1answer
41 views
Trying to understand the breaking down of the complexified tangent space
I am reading some notes on the complexified tangent space. I don't understand how do we arrive at that $V^{(1,0)} = \{ X - iJX : X \in V\}$.
I mean in one direction we have $JZ = J( v \otimes \alpha) ...
5
votes
1answer
91 views
On the vector bundles associated to the eigenspaces of an almost complex structure.
Let $(M,J)$ be an almost complex manifold, that is $M$ is a smooth differentiable manifold of even dimension and $J$ is an endomorphism of the tangent vector bundle $TM$ such that $J^2=-\textrm{id}_{...
4
votes
1answer
254 views
The Definition of an Almost Complex Manifold (Nakahara)
I am having problems making sense of Michio Nakahara's definition of the Almost Complex Structure/Almost Complex Manifold, such as it appears in Geometry, Topology and Physics (2nd Edition).
On p. ...
3
votes
0answers
415 views
Almost complex structure on $S^6$
It is known that the sphere $ S^6$ admits an almost complex structure by identifying $S^6 $ with the space of unit purely imaginary Cayley numbers.
I would like to show that this almost complex ...
1
vote
0answers
26 views
Subbundle of hermitian maps is isomorphic (as a bundle) to $M\times i\Bbb R$
Let $\pi:E\to M$ be a complex bundle, where $M$ is a $n$-dimensional smooth manifold. We can consider the bundle $\text{Hom}(E,E)=:\text{End}(E)$ of endomorphism of $E$, with fibers
$$\text{End}(E)_{...
1
vote
0answers
98 views
Reference for complex structures and Hermitian structures
I am reading Complex Geometry: An Introduction by Daniel Huybrechts.
Section on complex structures and Hermitian structures deals with
Lefschetz operator denoted by $L$.
Hodge * -operator.
The ...
0
votes
0answers
32 views
Construction of hermitian almost complex structure and invariant $(n,0)$-form
I'm reading a paper and it says the following:
Let $K$ be a Lie group and fix an orthonormal framing $\{e_1, ..., e_n\}$ with respect to a left-invariant metric. This gives a splitting
$$T^∗K = K × \...
0
votes
1answer
322 views
Metric induced almost complex structure on cotangent bundle
I'm trying to understand this question and I was hoping someone could help me. One thing in particular is confusing me.
Question: They are starting with a riemannian manifold $(M,g)$ and considering ...
3
votes
1answer
365 views
Proof of the Wirtinger inequality
This is an exercise from the book "Complex Geometry, An Introduction" by Huybrechts. The statement involves to prove that the restriction of the fundamental form $\omega$ of a vector space $V, I, \...
1
vote
0answers
27 views
Local form of complex structure via Bruhat-Whitney
Set-up: Let $(L,g)$ be a real-analytic riemannian manifold. And let $T^*L$ denote the cotangent bundle of $L$.
According to Bruhat-Whitney, on a suffciently small neigbourhood of the zero section in $...
3
votes
0answers
97 views
Relating two notions of integrable almost complex structure on a complex vector bundle
Suppose $E \to M$ is a complex vector bundle over a complex manifold $M$. As I understand it, there are two possible ways to say what it means that $E$ is a holomorphic vector bundle:
$E$ has the ...
1
vote
0answers
93 views
complex structure compatible with symplectic form and riemannian metric [duplicate]
Given a non-degenerate symplectic form $\omega$ on a finite dimensional vector space $V$ and a complex structure $J$ (that is, $J\in End(V),~J^2=-Id$). If $J$ is compatible with $\omega$ in the sense ...
8
votes
0answers
157 views
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 ...
6
votes
2answers
394 views
Holomorphic vector bundles on almost complex manifolds
Let $M$ be a real manifold with complex structure $J$, making $M$ into an almost complex manifold. I know that the complexification $T_{\textbf{C}}M = TM\otimes \textbf{C}$ of the tangent bundle $TM$ ...
2
votes
2answers
91 views
If $V$ is finite-dimensional with $J : V \to V$ such that $J^2 = -id$, then $V$ has even dimension [duplicate]
Let $V$ be a $\Bbb R$-vector space, with $J$ being an endomorphism $J: V \to V$ with $J^2=-id$ (identity).
I already had to show that $V$ became a $\Bbb C$-vector space with the scalar multiplication:...
3
votes
0answers
72 views
Pulling back a Kähler structure on a symplectic submanifold
Let $(K, G, \Omega, J)$ be a Kähler manifold and $(S, \omega)$ be a symplectic manifold. Let $i : S \to K$ be a symplectic embedding. Is it possible to endow $S$ with a Kähler manifold structure, ...
3
votes
1answer
103 views
When does contractible space of almost complex structures taming a given symplectic form $\omega$ contain an integrable compatible one?
Given a symplectic form $\omega$ on a compact symplectic manifold $X$, we know there is a contractible homotopy class $\mathcal{J}_{\omega}$ of almost complex structures that tame $\omega$. A subset ...
2
votes
0answers
50 views
Fibration induced by an almost complex structure
Let $E \rightarrow M$ be a plane bundle endowed with an almost complex structure $J.$ $J$ induces a natural positive definite inner product in the associated bundle $$End(E)\rightarrow M,$$ denoted by ...
5
votes
1answer
261 views
Find type of a differential form on an almost complex manifold
If $M$ is a nearly Kähler manifold (that is, an almost Hermitian manifold on which $\nabla_X(J)X=0$) we have the three-forms
$$ A(X,Y,Z)=\langle\nabla_X(J)Y,Z\rangle \quad\text{and}\quad B(X,Y,Z)=\...
1
vote
0answers
130 views
Curvature identity on Nearly Kähler manifolds
Can somemone help me prove the following identity?
$$ \| \nabla_X(J)(Y) \|^2 = \langle R_{XY}X,Y\rangle - \langle R_{XY}JX,JY\rangle$$
where $J$ is the almost complex structure, and $R$ the curvature ...
5
votes
1answer
199 views
Is any 2$m$-dimensional manifold almost complex?
In Nakahara's book "Geometry, Topology and Physics" (Ch. 8, about the almost complex structure) they write:
Note that any 2$m$-dimensional manifold locally admits a tensor field $J$ [type (1,1)] ...
4
votes
0answers
288 views
Equivalence of (almost) complex structures
Preamble:
An almost complex structure on a manifold $M$ is an endomorphism $J : TM \to TM$ such that $J^2 = -1$. An almost complex structure $J$ is said to be integrable if the Nijenhuis tensor, $N_J$,...
-3
votes
2answers
220 views
Equilateral triangle in complex number [closed]
Let $a$, $b$ and $c$ be the affix of $A$, $B$ and $C$, where $a+b+c=0$ and $a$, $b$ and $c$ are of equal magnitude. Prove that $\triangle ABC$ is an equilateral triangle.
6
votes
1answer
281 views
Is there a natural Dolbeault operator on a almost holomorphic vector bundle?
For vector bundles $(\pi: V \rightarrow M )$ over a complex manifold, there is a notion of holomorphicity that can be defined in two equivalent ways :
$V$ is a complex manifolds and $\pi:V \...
1
vote
0answers
39 views
Parametrizing linear complex structures
So I'm reading this paper by Donaldson on contructing symplectic submanifolds, https://projecteuclid.org/euclid.jdg/1214459407
In section 2, he says the following: On ${\bf C}^n$, we have the ...
1
vote
2answers
293 views
smooth vs. analytic in the definition of almost-complex manifolds
Let $A_{\infty}\hspace{-0.03 in}$ be a maximal $C^{\infty}\hspace{-0.02 in}$ atlas on $M\hspace{-0.03 in}$, and with that smooth structure on $M$,
suppose $\: j : TM\to TM\:$ is a smooth function ...
7
votes
1answer
727 views
Does every even-dimensional sphere admit an almost complex structure?
We know that there is an almost complex structure on $S^6$ which is not integrable. Is it always possible to find almost complex structures on $S^{2n}$? In particular does $S^4$ admit one?
