All Questions
Tagged with complex-geometry linear-algebra
24 questions
1
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204
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The wedge product of two positive forms is positive
I have previously posted this question on MSE, but still didn't solve it.
Definition. A real $(p, p)$-form $\psi$ on a complex manifold $M^{n}$ is said to be (semi-) positive, if for any $x \in M$, ...
- 195
0
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0
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88
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Separating orthogonal vectors in $\mathbb{C}^2$
Is it possible to partition $\mathbb{C}^2$ into two sets $S$ and $S'$ such that, given any two nonzero orthogonal vectors $\mathbf{v}$ and $\mathbf{w}$ of $\mathbb{C}^2$, one of them lies in $S$ and ...
- 679
1
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0
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86
views
Functional inequality with complex variables
I am interested in considering a function $C(\tau)$ of the complex variable $\tau = t + i\eta$ such that
$C(\tau)$ is analytic for $\Re(\tau)=t>t_0\ge0$
$\exists$ a constant $C_0$ and a function $...
- 33
7
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1
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319
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The set of strongly positive forms is a closed cone
This question comes from the Complex Analytic and Differential Geometry by Demailly. Let $V$ be a $n$ dimensional complex space. Consider the exterior algebra $\Lambda V^* = \oplus \Lambda^{(p,q)}V^*$....
- 130
2
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107
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Two definitions for transverse $(p,p)$ form
Let $V$ be a complex vector space of dimension $n$ and let $V^*$ be its dual. Fix any integer $1\leq p\leq {n-1}.$ A $(p,p)$-form $\alpha\in\bigwedge^{p,p}V^*$ is said to be strictly weakly positive ...
- 295
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108
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Solving a nonlinear equation maybe with Lambert W function
Can you please help me solve the following nonlinear equation?
\begin{equation}
\boldsymbol{z} \odot\left(\boldsymbol{\Gamma}^{\top} \boldsymbol{y}\right)=(\beta)^{\frac{1}{m-1}}\left(\frac{m-1}{...
- 11
2
votes
1
answer
1k
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Derivative of eigenvectors of an Hermitian matrix
In the question "Derivative of eigenvectors of a matrix with respect to its components", Liviu Nicolaescu has provided an answer valid for a real matrix. As outlined in the following, the ...
- 23
3
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0
answers
117
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Geometry of elements with prescribed multiplicity eigenvalues
Let us take $G=\operatorname{Gl}(n,\mathbb{C})$ (considered as a linear algebraic group). Let us take $x \in G$: we know that its orbit $\mathcal{O}_x$ under the conjugation action is isomorphic (as ...
- 1,061
0
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1
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425
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Are anti-linear maps/semi-linear, such as conjugations, linear in other almost complex structures?
I have asked this on mse, but I did not get any responses even after a bounty.
I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much ...
- 247
6
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0
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192
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Bar notation in Bourbaki’s *Lie groups*, Chap. IX
I am looking at Chapter IX (Compact Real Lie Groups), §4, Exercise 8 (translation). Given a complex subspace $\mathfrak p$ in the complexification $\mathfrak g_{\mathbf C}$ of some $\mathfrak g$, they ...
- 31.5k
4
votes
1
answer
607
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Linear Complex Structure and Kahler Angles
I am trying to read Donaldson's paper on symplectic submanifolds
Link
and am getting a bit stuck on some simple linear algebra at the beginning. On the fourth page of the paper (p. 669) there is the ...
- 61
5
votes
0
answers
2k
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What is the Jarlskog invariant, conceptually?
Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity:
$$J_{ij,k\ell} := \operatorname{...
- 32.5k
1
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0
answers
139
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Reference on classifying real subspaces of complex vector spaces (based on restricted complex structure)
Every complex vector space can also been seen as real vector space. If we now choose a real subspace, it may not be a complex subspace (in particular, if it is of odd real dimension).
If the complex ...
- 285
1
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0
answers
151
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Almost complex structure commuting with symplectomorphism
Let $(V,\omega)$ be a symplectic vector space with symplectic form $\omega$. Furthermore, let $\varphi : V \rightarrow V$ be a linear symplectomorphism. Consider the set
$$
\mathcal{I}_{\varphi} := \{ ...
- 49
4
votes
1
answer
690
views
classify antiholomorphic involutions of projective space
On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by
$$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [...
- 533
4
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2
answers
758
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Riemannian metric of hyperbolic plane
I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector space....
- 8,597
-3
votes
1
answer
375
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Opposite complex structure on Kaehler manifold
Let $(M,J)$ be a Kaehler manifold. How can one describe the opposite complex structure? What is the precise definition of the opposite complex structure? Can one describe the opposite complex ...
- 11
8
votes
1
answer
638
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Composite residues with determinant denominators
I am looking for a good reference on composite residues of multi-variable contour integrals (something better and more explicit than Griffiths and Harris or Tsikh). This means I want to evaluate $\...
- 81
2
votes
1
answer
299
views
Division and multiplication that preserve Euclidean norms
I am looking for ways to define
$$\frac{1}{x}\in \mathbb{R}^n\quad \quad and\quad \quad x\cdot y\in \mathbb{R}^n ,$$
where $x,y\in \mathbb{R}^n$ such that
$$\left\|\frac{1}{x}\right\|=\frac{1}{\|...
- 309
4
votes
1
answer
234
views
A trivialization of an almost complex structure
Recently, I have been studying the Carleman Similiarity Principle, which is used to study the regularity and unique continuation of J-holomorphic curves.
Roughly, one takes a solution $ u $ of a ...
- 337
8
votes
1
answer
1k
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Calculating a curvature tensor by polarization
I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson ...
- 4,343
0
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2
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573
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about decomposition of three forms
Patrick D. Baier in his Ph.D. thesis for proving the theorem 2.1.4 used the following non-trivial fact (in chapter 2 on page 14):
Let $0\neq X\in V$ (here $V$ is of dimension 6), $W^\ast = Ann(X)$ ...
user21574
2
votes
1
answer
478
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On linear automorphism on positive definite matrices.
I saw a statement in [Murakami, On automorphisms on Siegel domains] that every linear automorphism $\phi$ on the set of positive definite matrices can be represented as conjugation: i.e. there is a ...
- 537
2
votes
2
answers
164
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Looking for a simple proof that the generalized disc is bounded
So let us define the generalized disc of degree $n$ as
$$
\mathbb{D}_n:=\{w\in M_{n\times n}(\mathbb{C}):w=w^t, I_n-w\overline{w}>0\}.
$$
For a Hermitian matrix $A$, the notation $A>0$ means ...
- 7,609