Questions tagged [almost-complex]
For questions about almost complex structures on manifolds or vector spaces.
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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 ...
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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 ...
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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 ...
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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 ...
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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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$U(2)$ 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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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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$\nabla\omega=0$ 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 ...
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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 $...
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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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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 ...
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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 ...
- 1,703
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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:...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 $...
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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&...
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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 ...
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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 $...
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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 ...
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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 ...
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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 $J^2=-\text{id}$.) ...
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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 ...
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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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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 $J'$ ...
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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 ...
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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 $...
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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 $C^{\infty}(M)$-...
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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^...
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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 ...
user888948
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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 ...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $J$, ...
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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 $J_M$, ...
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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 ...
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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 ...
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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 $O(-1)= \{([x], z) \in \mathbb{P^nC} \times
\mathbb{C}^n \ \vert \ z \in \mathbb{C}^* \cdot x \}$...
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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 ...
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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.
...
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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\...
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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^...
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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 $W\cap JW = 0$. ...
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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 ...
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