Questions tagged [almost-complex]
For questions about almost complex structures on vector spaces or manifolds.
93
questions
0
votes
0answers
32 views
$c_1(-, J(-))$ is Riemannian metric
Let $\mathbb{P^nC}$ the complex projective space and $O(1)$ the dual bundle
to the tautological bundle $O(-1)= \{([x], z) \in \mathbb{P^nC} \times
\mathbb{C}^n \ \vert \ z \in \mathbb{C}^* \cdot x \}$...
1
vote
0answers
18 views
Relative version of identity theorem for pseudoholomorphic curves
Let $(M,J)$ be an almost complex manifold and $N\subset M$ a closed almost complex submanifold, i.e. the tangent space $T_xN$ at every point $x\in N$ is invariant under $J$.
Let $\Sigma$ be a ...
3
votes
1answer
79 views
Real $2n$-plane bundle with a complex structure is a complex $n$-plane bundle
I am trying to show that if $\xi=(E,B,\pi)$ is a real $2n$-plane bundle with a complex structure $J:E\to E$ then $\xi$ becomes a complex $n$-plane bundle if we define $(x+iy)v=xv+yJ(v)$ on each fiber.
...
0
votes
1answer
21 views
Fundamental form of almost complex manifold is $(1,1)$-form
Let $M$ be an almost complex manifold with almost complex structure $J$, a compatible Riemannian metric $g$ and fundamental form $\omega$. Consider the eigenspaces $T_p^{1,0}M=\{v\in T_pM\otimes\...
1
vote
1answer
40 views
A complex vector bundle $E$ is a holomorphic vector bundle iff $(\overline{\partial^E})^2=0$ help with proof?
Ok so I am following a set of notes on complex differential geometry and there is a theorem that says the following:
If $E$ is a complex vector bundle over a complex manifold and $\overline{\partial^...
0
votes
0answers
9 views
Is there a reasonable notion of a totally real Grassmannian?
Suppose we are given a $2n$-dimensional real vector space $V$, along with a complex structure $J:V\to V$. For $k\le n$, a $k$-subspace $W\subset V$ is called $J$-totally real if $W\cap JW = 0$. ...
0
votes
0answers
9 views
The integrability of almost complex structure in the sense of Frobenius theorem.
I've tried to distinguish the almost complex structure and complex structure, intuitively.
Without any chance of confusion, I'll assume that the manifolds and the maps are "smooth".
The almost ...
0
votes
0answers
11 views
On Existence of Solutions to a Cauchy-Riemann Equation with Boundary Conditions
Assume $\mathbb{D} $ is the closed unit disk in $\mathbb{C} $ with an almost complex structure $J\in C^{\infty } (\mathbb{D} ;\mathrm{End} (E))$ for $E$ a smooth complex vector bundle of rank $r$ over ...
0
votes
0answers
35 views
Dimension of the complex projective space $\mathbb{C}\mathbb{P}^n$ as almost complex manifold
I have already shown that the complex projective space $\mathbb{C}\mathbb{P}^n$ is a complex manifold by checking the required properties of the transition maps. Since every complex manifold is an ...
0
votes
0answers
24 views
Understanding of Carleman similarity principle
The Carleman similarity principle is stated as follows in my lecture:
What I don't understand is the statement
$u(z)=\Phi(z)f(z)$
since $\Phi(z)$ should be an endomorphism and $u(z)$ just some ...
2
votes
0answers
26 views
Extending an $\omega$-tame complex structure
The following is an execise from Chris Wendl's book Holomorphic Curves in Low Dimensions
Let $(M,\omega)$ be a symplectic manifold and $A\subseteq M$ a closed subset. Let $J_A$ be an $\omega$-...
0
votes
1answer
56 views
What does it mean for an almost complex structure to be compatible with a Riemannian metric?
This is on p. 37 of The Ricci Flow and the Sphere Theorem by Simon Brendle. The author is going to show that a gradient Ricci soliton on $S^2$ has constant scalar curvature.
Here is the setting: Let $...
2
votes
2answers
65 views
Complex and Conformal structure on a trivial tangent bundle of a higher dimensional manifold
Let $M$ be an even dimensional Reimannian manifold with trivial tangent bundle $TM$. Then there is a global orthonormal basis $\{e_1,e_2,...,e_{2n-1},e_{2n}\}$ of $TM$.
Define a bundle map $J :TM \...
1
vote
0answers
57 views
Advantage in considering an almost complex structure
Let $M$ be a real Riemannian manifold of even dimension, what are the advantages of considering an almost complex structure on it rather than only real?
2
votes
1answer
80 views
Question about the (pseudo)-holomorphic map between almost complex manifolds
It may be a very simple question,but I can't figure it out.
Let $(M,I)$ be an almost complex manifold where $I$ is an almost complex structure.Then $-I$ is also an almost complex structure.Then the ...
5
votes
0answers
67 views
Averaging an almost complex structure over compact Lie group action
It is well-known that if a Lie group $G$ acts on a symplectic manifold $(W,\omega)$ preserving $\omega$, then we can easily find a $G$-invariant almost complex structure $J$ by choosing first a $G$-...
2
votes
2answers
134 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
66 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(...
7
votes
1answer
108 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
44 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 ...
4
votes
1answer
138 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
143 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
231 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 $J$:
$$(...
2
votes
0answers
37 views
an exercise about integrability of almost complex structures
i have spent some time over the following problem from a problem sheet of a course on complex geometry.
Let $M=G/H$ be a homogeneous space (where $G$ is a Lie group, $H$ a closed subgroup) and let $I$...
1
vote
1answer
87 views
(Almost) complex manifold problem.
Can some one give me an exemple of an almost compplex manifold that is not a complex manifold and why? I know that an almost complex manifold is of even real dim and is orientable. I also heard that ...
1
vote
0answers
76 views
Local construction of a map on a manifold
Assume $(M,J)$ is a smooth manifold equipped with an almost complex structure. Let $p$ be a point of $M$, and define the linear map $\Theta_p \colon T_pM \to T_pM$ so that $$\Theta_p = -\frac{1}{2}\...
1
vote
1answer
206 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
76 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
181 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
90 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
51 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 $\pi$ ...
5
votes
1answer
276 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 $$\lambda(\xi) = \...
1
vote
0answers
190 views
Complex Vector Bundle vs Holomorphic Vector Bundle vs Almost Complex Structures
I am writing because I am extremely confused with the structure of complex vector bundles. Ok first of all I understand that a complex vector bundle is a just a vector bundle $\pi:E\to X$ such that ...
4
votes
1answer
42 views
Equivalence of tangential and normal stably almost complex structure
Let $M$ be a smooth manifold. $M$ is said to be tangential stably almost complex if $TM \oplus \underline{\mathbb{R}}^k$ can be given a structure of a complex vector bundle, for some $k$. $M$ is said ...
2
votes
2answers
107 views
Integrable almost complex structure conjugated by diffeomorphism
Let $M$ be a smooth manifold and $J$ be an integrable almost complex structure on $M$. Let $f: M\to M$ be a diffeomorphism with $f_{*}:TM\to TM$ its tangent map. Then it is easy to see that $f_{*}Jf_{*...
3
votes
1answer
302 views
Natural complex-linear isomorphism between $ V_J $ and $V^- $
Let $V$ be a real vector space with a linear complex structure $J$ (see Wikipedia).
Denote $V_J$ the complex vector space induced from $V$ by the complex structure $J$.
Also, define the ...
1
vote
1answer
241 views
Nijenhuis tensor in local coordinates
If $(M,J)$ is an almost complex manifold, $\mathcal{N}$ (Nijenhuis tensor) is:
$$\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})$...
2
votes
1answer
270 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
140 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
46 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
118 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
461 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. ...
4
votes
0answers
520 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
27 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
112 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^āK = K Ć \...
4
votes
1answer
1k views
Canonical almost complex structure on symplectic manifold
I'm trying to prove the following (well-known) theorem in symplectic geometry.
Theorem . For $(M, \omega)$ a symplectic manifold with Riemannian metric $g, \exists$ a canonical almost complex ...
0
votes
1answer
473 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 ...
4
votes
1answer
521 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
29 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 $...
