Questions tagged [almost-complex]

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

Filter by
Sorted by
Tagged with
1 vote
0 answers
32 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 ...
user avatar
  • 127
0 votes
0 answers
26 views

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 ...
user avatar
3 votes
1 answer
73 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 ...
user avatar
  • 93
2 votes
0 answers
104 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 ...
user avatar
2 votes
0 answers
52 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 $T^*M$ ...
user avatar
  • 4,175
2 votes
1 answer
78 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 $\...
user avatar
  • 23
3 votes
1 answer
39 views

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&...
user avatar
  • 1,232
0 votes
0 answers
79 views

$\nabla\omega=0$ if and only if $(M,g)$ is Kähler

Given an almost complex manifold $(M,g)$ with an almost complex structure $J$, we know that $(M,g)$ is Kähler if and only if $J$ is integrable and $\mathrm{d}\omega=0$, where $\omega$ is the Kähler ...
user avatar
  • 141
3 votes
0 answers
79 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 $...
user avatar
1 vote
1 answer
49 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). ...
user avatar
3 votes
0 answers
44 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 ...
user avatar
1 vote
0 answers
41 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 ...
user avatar
  • 1,703
4 votes
0 answers
54 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:...
user avatar
  • 2,928
2 votes
1 answer
48 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
  • 1,104
6 votes
1 answer
100 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 ...
user avatar
  • 333
3 votes
1 answer
138 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 ...
user avatar
0 votes
1 answer
76 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 $...
user avatar
2 votes
1 answer
123 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 ...
user avatar
  • 1,710
3 votes
1 answer
68 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 $...
user avatar
  • 1,710
2 votes
0 answers
71 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&...
user avatar
  • 29
1 vote
0 answers
37 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 ...
user avatar
  • 11
2 votes
1 answer
47 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 $...
user avatar
  • 128
5 votes
2 answers
185 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 ...
user avatar
  • 707
0 votes
1 answer
174 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 ...
user avatar
  • 2,034
1 vote
1 answer
44 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 $J^2=-\text{id}$.) ...
user avatar
  • 2,034
1 vote
0 answers
66 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 ...
user avatar
  • 328
1 vote
0 answers
32 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 ...
user avatar
  • 4,175
1 vote
1 answer
59 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 $J'$ ...
user avatar
  • 1,710
2 votes
1 answer
52 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 ...
user avatar
  • 1,426
1 vote
0 answers
63 views

Tensor product of complexification

I am working with the definition of complexification $V_\mathbb{C}= V \otimes_\mathbb{R} \mathbb{C}$ where V is a real vector space, in order to better understand forms and vectors of type (p,q) when $...
user avatar
  • 1,703
3 votes
0 answers
110 views

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 $C^{\infty}(M)$-...
user avatar
1 vote
1 answer
209 views

An almost complex structure on the real 2-sphere $S^2$

If $R:=\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ and $S^2:=Spec(R)$ is the real 2-spere, a classical result of Borel and Serre says that the only real spheres with an almost complex structure is $S^2$ and $S^...
user avatar
  • 1
1 vote
1 answer
82 views

Proving $\Bbb{H\otimes_R C }\cong M_2(\Bbb{C})$ versus proving $\Bbb{H\otimes_R C }\cong_{\mathbb R} M_2(\Bbb{C}).$

1- Is there a difference between saying I want to prove this $\Bbb{H\otimes_R C }\cong M_2(\Bbb{C})$ and I want to prove this $\Bbb{H\otimes_R C }\cong_{\mathbb R} M_2(\Bbb{C})$? I see the OP in the ...
user avatar
1 vote
0 answers
60 views

Symplectic manifolds and almost complex structures

On a symplectic vector space $(V,\omega)$ with an inner product $g$, one can construct a canonical almost complex structure using polar decomposition. If $(M,\omega)$ is a symplectic manifold with a ...
user avatar
4 votes
3 answers
383 views

What's the bijection between scalar/inner products and (certain) almost complex structures (on $\mathbb R^2$)?

Asked on maths overflow here. What's the bijection between (equivalence classes of) scalar products (I guess 'scalar product' is the same as 'inner product') and a.c.s. (almost complex structure/s) ...
user avatar
2 votes
1 answer
154 views

Question about the Newlander-Nirenberg theorem for almost complex manifolds

While studying the definition of almost complex structure, I have thought about almost complex structures which are not complex structures. To ask a more complete question, I write the following ...
user avatar
5 votes
1 answer
111 views

Is every plane in $\mathbb{R}^4$ a line in $\mathbb{C}^2$?

Every complex line, that is, one-dimensional complex affine space, in $\mathbb{C}^2$ is a real plane in $\mathbb{R}^4$. Is the converse true? That is, is every real plane in $\mathbb{R}^4$ a complex ...
user avatar
  • 13.3k
2 votes
1 answer
150 views

Eigenvalues of Complex Stucture in a Complexified Vector Space

For context, this question comes from my reading of Kobayashi+Nomizu's differential geometry book, volume 2 (pages 116-117). Given a real vector space $V$ with $\mathrm{dim}(V)=2n$, a complex ...
user avatar
0 votes
1 answer
94 views

Almost complex structure on real differentiable manifolds

An almost complex structure on a real differentiable manifold $M$ is a tensor field $J \in \Gamma(\mbox{End}(TM))$ satisfying $J^{2}=-I$, where $I$ is the identity tensor field. The pair $(M,J)$ is ...
user avatar
  • 432
4 votes
1 answer
83 views

Why did I get $J \bar{v} = \sqrt{-1}\bar{v}$ ??

Q. I cannot deduce the equation $J \bar{v}= -\sqrt{-1}\bar{v}$ for $v$ satisfying $Jv= \sqrt{-1}v$. Details Let $V$ be a vector space on $\mathbb{R}$ equipped with a complex structure $J$. Using $J$, ...
user avatar
1 vote
1 answer
25 views

Two equivalent conditions of Pseudo-holomorphic maps

Let $\varphi: (S,J_S) \to (M,J_M)$ be a map from a complex manifold equipped with a integrable complex strucure $J_S$ to a smooth manifold equipped with a almost complex structure $J_M$. Using $J_M$, ...
user avatar
2 votes
1 answer
114 views

Defining the natural almost complex structure on a complex manifold.

The definition of an almost complex structure is as follows. If $X$ is a differentiable manifold and $TX$ is its tangent bundle, then the endomorphism $I: TX \to TX$ defines an almost complex ...
user avatar
  • 1,769
6 votes
0 answers
121 views

Chern classes and almost complex submanifolds

Suppose $M$ is a manifold endowes with an almost complex structure, such that the Chern classes $c_i(M)$ of $M$ are defined. Can we say that the Poincare dual to $c_i(M)$ can be represented by an ...
user avatar
1 vote
1 answer
255 views

The first chern class of $O(1)$ is positive

Let $\mathbb{P^nC}$ the complex projective space and $O(1)$ the dual bundle to the tautological bundle $O(-1)= \{([x], z) \in \mathbb{P^nC} \times \mathbb{C}^n \ \vert \ z \in \mathbb{C}^* \cdot x \}$...
user avatar
  • 43
2 votes
0 answers
28 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 ...
user avatar
4 votes
1 answer
195 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. ...
user avatar
  • 1,710
0 votes
1 answer
117 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\...
user avatar
2 votes
1 answer
133 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^...
user avatar
1 vote
0 answers
27 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 $W\cap JW = 0$. ...
user avatar
  • 1,834
0 votes
1 answer
234 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 ...
user avatar