User đức anh - MathOverflow

Holomorphic objects associated with a compact complex manifold?

2013-03-18T06:54:49Z

Good morning,

I'm just curious about the following. With a compact Kahler manifold, we can associate an Albanese torus. This helps us a lot study the manifold.

My question: Are there other holomorphic objects associated with a compact complex manifold? I'm interested in the objects whose shape is well understood. E.g, an Albanese torus is just a torus, and we know its cohomology, its kahler form etc.

Any help is appreciated. Thanks in advance,

Duc Anh

Is the modification a rational map?

2013-03-17T01:41:20Z

Good morning,

I would like to ask the following question concerning the desingularisation, but I'm not familiar at all with these notions.

We have the following theorem of Hironaka: Let $X\subset \mathbb{CP}^n$ a closed complex projective variety. Then, there exists a modification $f\colon (\tilde{X},E) \to (X, Sing(X))$ such that $\tilde{X}$ is smooth and projective, and the exceptional divisor $E$ is a divisor with normal crossings.

My questions: Are $f$ and $f^{-1}$ rational maps?

Any help is appreciated. Thanks in advance,

Duc Anh

Automorphism group of a compact Kahler manifold

2013-03-11T15:17:37Z

Good evening,

I would like to ask the following questions.

Let $X$ be a compact Kahler manifold. Denote by Aut(X) the group of all the biholomorphisms of $X.$

1) What can we say about this group? E.g, Is it a Lie group?

2) Does there exist a manifold $X$ with $Aut(X)$ trivial?

3) Let $A$ be an analytic subset of $X$ and $x_0\in A$ some point. Does there exist an automorphism $\gamma\in Aut(X)$ with $\gamma(x_0) \not\in A$?

4) Finally, is there a way to produce automorphisms of $X$?

Any help is appreciated. Thanks in advance.

Duc Anh

Properties of the fibers of Albanese map?

2013-03-09T10:58:55Z

Good afternoon,

I encounter the notion of Albanese map $alb$ from a compact Kahler manifold $X$ to its Albanese torus. I would like to know any properties of the fibers of this map, i.e. the set $alb^{-1}(y)$ with $y$ an element of the Albanese torus.

Any help is appreciated. Thanks in advances,

Duc Anh

How did Nochka find weights in his proof of Cartan's conjecture?

2013-01-30T13:42:31Z

Good evening,

I have just read Nochka's proof of Cartan's conjecture (Second Main Theorem of Nevanlinna Theory for linearly degenerate meromorphic curves in $\mathbb{CP}^n$). To prove the conjecture, Nochka introduced weights associated with the family of hyperplanes in N-subgeneral position. These weights are very complicated.

So I wonder how Nochka found these weights. Is there any intuitive explanation on these weights?

Any help is appreciated,

Thanks in advances,

Duc Anh

Cohomology of Complements by an analytic subset?

2013-01-22T06:08:27Z

Good moring,

Let $\Omega$ be a domain in $\mathbb{C}^n$ and $S\subset\Omega$ an analytic subset of codimension 1. What can we say about the cohomology group $H^1(\Omega\backslash S, \mathbb{Z})$? E.g, when $\Omega$ is a ball?

Any help is appreciated. Thanks in advance.

Duc Anh

on the density of hypersurfaces in complex projective spaces

2013-01-10T20:46:45Z

Good morning,

Let $H$ be a hypersurface in a complex projective space $\mathbb{CP}^N.$ Let $d$ be the distance de Fubini-Study on $\mathbb{CP}^N.$

Let $x = [x_0: \ldots :x_N]$ and $y=[y_0:\ldots:y_N]$ two points in $\mathbb{CP}^N.$ Is the following formula true $$d(x,y)^2 = \frac{\sum_{i<j} |x_i \bar{y_j}-x_j\bar{y_i}|^2}{\sum |x_i|^2 \cdot \sum |y_i|^2}?$$
(principal question) Is the following quantity $$\max_{z\in \mathbb{CP}^N} d(z,H)$$ bounded from above by a quantity which depends only on the degree of $H?$ The expected quantity (which depends on the degree of $H$) must converge to $0$ as the degree of $H$ increases to $\infty.$

Any help is appreciated. Thanks in advances.

Duc Anh

Inequality of von Neumann for more than two contractions

2012-05-03T11:12:50Z

Good morning,

I'm doing the Master 2 Practice at the University of Toulouse 3, France, on the spectral Nevanlinna-Pick interpolation, via operator theory. This problem leads to study the symmetrized polydisc $\mathbb{G}_n$ defined as follows

$$\mathbb{G}_n =\{(\sigma_1(\lambda),\ldots, \sigma_n(\lambda))~:~ \lambda\in \mathbb{D}^n\}$$ where $$\sigma_i(\lambda)=\sum_{1\leq j_1<j_2<\ldots<j_i\leq n} \lambda_{j_1}\lambda_{j_2}\ldots\lambda_{j_i}$$ are the elementary symmetric polynomials of $\lambda = (\lambda_1,\ldots,\lambda_n).$

Some authors study these symmetrized polydiscs via operator theory, e.g Jim Agler and Nicholas Young. The main tools they used are the commutant lifting theorem and the inequality of von Neumann. These two authors obtained the following result via this approach : the Caratheodory distance and Kobayashi distance are equal for the symmetrized bidisc $\mathbb{G}_2$. This result is surprising, because the symmetrized bidisc is not biholomorphic to a convex domain (due to Costara), and can not be exhausted by domains biholomorphic to convex domains (due to Edigarian).

However, it's impossible to study the symmetrized polydisc of higher dimension via the commutant lifting/von Neumann's inequality, since these two theorems fail for more than two contractions. I think understanding why these fail will give informations on the symmetrized polydisc and the spectral Nevanlinna-Pick interpolation problem.

Question : I would like to know if there are papers which gather the informations on the failure of the commutant lifting theorem and the inequality of von Neumann for more than two contractions.

Any help is appreciated. Thanks in advance.

Duc Anh

why do we need to study entire curves?

2012-09-16T09:17:17Z

Good afternoon,

I'm just curious about this question, because I see that there are a lot of papers which study the value distribution of an entire curve $f\colon \mathbb{C}\to X,$ with X a complex manifold by using Nevanlinna theory. But I don't know the motivation of this research.

My question : Why do we need to study entire curves? Could anyone tell me some references or notes which mention these motivations? And beside the value distribution, what other aspects of entire curves do we want to know/study?

Thanks in advance,

Duc Anh

Functional calculus for vector-valued holomorphic functions?

2012-06-25T15:51:40Z

Good afternoon,

I would like to ask a question on the functional calculus of several commuting operators. If someone knows some good/standard references, could you please tell me about them.

Firstly, if we have a bounded linear operator on a Hilbert space $T\in \mathcal{B}(\mathcal{H}),$ and a holomorphic function $f\colon \sigma(T) \to \mathcal{B}(\mathcal{C})$ on a neighborhood of the spectrum of $T,$ where $\mathcal{C}$ is another complex Hilbert space, we can define the functional calculus of $T$ for $f$ as follows :

$$f(T) = \frac{1}{2\pi i}\int_{\gamma} f(z)\otimes (z-T)^{-1}dz \in \mathcal{B}(\mathcal{C}\otimes \mathcal{H}),$$ where $\gamma$ is a contour which rounds the spectrum $\sigma(T).$

So how we can define a functional calculus of several commuting operators for vector(operator)-valued holomorphic functions?

In the case of scalar-valued functions, it is the work of Taylor (1970) : http://www.ams.org/mathscinet-getitem?mr=0271741. I have not read all the details of the paper, because in the paper, the author uses a lot of sheaf theory which I don't know well.

So my question is : firstly, can the definition of functional calculus of Taylor be generalized to the case of vector/operator-valued holomorphic functions? Secondly, does there any reference which presents these results?

Any help is appreciated. Thanks in advance.

Duc Anh

on the approximation by holomorphic functions

2012-06-13T18:55:55Z

Good evening,

I have a question on the approximation of holomorphic functions on a space of cartesian product type.

Question: Let $U,V$ be domains in $\mathbb{C}^n$ and $f\in \mathcal{O}(U\times V)$ a holomorphic function on $U\times V.$ Do we always have the following : $f$ can be approximated by holomorphic functions on $U\times V$ of the form $g(z,w) = \sum_{i=1}^N h_i(z)k_i(w)$ where $h_i\in \mathcal{O}(U)$ and $k_i\in\mathcal{O}(V)$ ? (N is arbitrary)

If it is not possible, can this be true if we put some conditions on $U$ and $V$? So what are the conditions?

Any help is appreciated. Thanks in advance.

Duc Anh

Answer by Đức Anh for on the approximation by holomorphic functions

2012-06-15T16:32:14Z

This is the answer : the above statement is true without any conditions on $U$ and $V.$ It is the theorem 1.7.7 in the book of Narasimhan, Analysis on Real and Complex Manifolds. One of my professors has pointed it out for me.

Answer by Đức Anh for on the approximation by holomorphic functions

2012-06-15T10:35:56Z

This is not an answer, but I hope someone will read and explain a little. I think the answer is the theorem 4.2.4, page 107, Eschmeier, Putinar, Spectral Decompostions and Analytic Sheaves. From the theorem, my above statement will be true if $U$ and $V$ are Stein spaces. To understand the statement of the theorem and also its proof, we have to know coherent sheaves, tensor product of two locally convex spaces, etc. So it is very far from my knowledge.

Lebesgue integral with respect to vector measures?

2012-04-12T21:41:58Z

Good evening,

I'm reading some papers of Jim Agler and Nicholas Young, in which they prove a formula of integral representation with respect to a vector measure, but the integration is in the sense of Riemann, not of Lebesgue. And the proofs for Riemann integrals are often long.

Secondly, in the book of Rudin, Functional Analysis, the author doesn't define an integral of Lebesgue's type with respect to the spectral measure. In stead, he always wants the readers to understand the integral with respect to the spectral measure as in the scalar case. Precisely, let $T$ be a normal operator on a hilbert space $H,$ and let $T = \int_{z\in\sigma(T)} z dE(z)$ be the spectral decomposition of $T.$ The integral has to be understood as $\langle Tx,y\rangle = \int_{\sigma(T)}z dE_{x,y}(z)$, where $E_{x,y}$ is the scalar measure defined by $E_{x,y}(\omega) = \langle E(\omega)x,y\rangle$ for all $\omega$ borelian sets of $\sigma(T)$, $x, y \in H$ and $\langle \cdot,\cdot\rangle$ the inner product of $H.$

My questions : Can these integrals be understood in the sense of Lebesgue? What is a good introductory reference for the theory of Lebesgue integral with respect to vector measures (of course, if this theory exists)? What are difficulties when we construct such theory?

Maybe, my questions are not well written, because of my limited english knowledge. I hope you understand my post. Any help is appreciated.

Thanks in advance,

Duc Anh

approximate point spectrum

2012-04-13T23:20:48Z

Good evening,

I have a question concerning the relation between approximate point spectrum and the spectrum of an operator.

Let $T$ be a bounded linear operator of a complex Hilbert space $H.$ The approximate point spectrum of $T$ is the set of all values $\lambda \in \mathbb{C}$ such that there exists a sequence of unit vectors $u_n\in H$ so that $\|(T-\lambda)u_n\|\to 0$ as $n\to \infty.$ We denote this set by $\sigma_{ap}(T)$. We denote the wellknown spectrum of $T$ by $\sigma(T)$.

We know that $\sigma_{ap}(T)$ contains the topological boundary of $\sigma(T)$.

My question : Can we have $\sigma_{ap}(T)\subset\partial\sigma(T)$?

Any help is appreciated. Thanks in advance.

Duc Anh

is a non-invertible operator a boundary point of the group of invertible operators?

2012-04-11T20:26:40Z

Good evening,

I have a question concerning non-invertible operators.

Let $H$ be a Hilbert space and $T$ a non-invertible bounded operator on $H.$ Is it true that $T$ is the limit of some sequence of invertible bounded operators on $H$?

We can see that is true when $H$ is finitely dimensional. So the interesting case is infinitely dimensional. Does anyone know any information about this?

Thanks in advance,

Duc Anh

Long exact sequence for Cech cohomology?

2011-10-13T21:19:32Z

Good evening,

I have two questions concerning Cech cohomology of presheaves.

(1) Let $X$ be a topological space and $0\to\mathcal{F}\to\mathcal{G} \to \mathcal{H}\to 0$ a short exact sequence of sheaves of abelian groups on $X.$ Does there exist a long exact sequence for the Cech cohomology of these sheaves as the case of sheaf cohomology?

(2) Let $\mathcal{F}$ be a presheaf of abelian groups on $X.$ Do we have the equality $\check{H}^p(X,\mathcal{F}) = H^p(X,\mathcal{F}^+)$ where the group on the left is Cech cohomology and the one on the right is the sheaf cohomology, and $\mathcal{F}^+$ is the associated sheaf of the presheaf $\mathcal{F}.$ For example, if $\mathcal{F}$ generates a zero sheaf, is $\check{H}^p(X,\mathcal{F}) = 0$ true for $p>0$ ?

Does anyone know something about these? Thanks in advance.

EDIT : the same question as (2) but for the Cech cohomology : do we have the equality $\check{H}^p(X,\mathcal{F}) = \check{H}^p(X,\mathcal{F}^+)$?

Answer by Đức Anh for Strong topology

2011-08-05T02:03:15Z

For the first question, the strong topology is the polar topology generated by all weakly bounded subsets. The weakly bounded subsets of $E$ are also weakly bounded in $E^{\ast\ast}$ since $E\subset E^{\ast\ast}$ and they have the same dual space $E^{\ast}.$ Therefore $\beta(E^{\ast},E^{\ast\ast})$ is finer than $\beta(E^{\ast},E).$

For the second question, did you try the space of test functions, i.e, infinitely differentiable functions with compact support? More generally, Hausdorff barrelled spaces have the property you want, so you should look for spaces in the class of barrelled spaces.

Refinement mapping

2011-08-03T16:39:45Z

Good evening,

When I read Cech cohomology in Riemann surfaces written by Otto Forster, I see the following proposition : Let $X$ be a topological space and $\mathfrak{F}$ a sheaf of abelian groups on $X.$ Let $\mathfrak{A}$ and $\mathfrak{B}$ be open coverings of $X$ such that $\mathfrak{B}$ is finer than $\mathfrak{A}.$ Then the induced mapping by the refinement mapping $t^{\mathfrak{A}}_{\mathfrak{B}}\colon H^1(\mathfrak{A}, \mathfrak{F})\to H^1(\mathfrak{B},\mathfrak{F})$ is injective.

So, my question : Is this induced mapping injective at the higher orders of cohomology?

Compact Riemann surfaces and Algebraic Functions

2011-07-15T19:32:27Z

Good evening,

In Riemann surfaces by Otto Forster there is the following theorem : Let $X$ be a Riemann surface and $P(T)=T^n+c_1T^{n_1}+\ldots + c_n\in\mathcal{M}(X)[T]$ an irreducible polynomial of degree $n,$ where $\mathcal{M}(X)$ is the set of all meromorphic functions on $X.$ Then there exist a Riemann surface $Y,$ a branched holomorphic n-sheeted covering $\pi : Y\to X$ and a meromorphic function $F$ on $Y$ such that $(\pi^{\ast}P)(F) = 0$.

We call $Y$ the algebraic function defined by the polynomial $P(T).$ (I don't restate the uniqueness of this Riemann surface).

My question : If $X$ is a compact Riemann surface, can we consider it as an algebraic function defined by some irreducible polynomial $P(T)\in\mathcal{M}(\mathbb{P}^1)[T]$?

I'm thinking of meromorphic functions on $X,$ which we can consider them as holomorphic mappings $X\to\mathbb{P}^1,$ having the smallest positive degree,i.e the inverse image of each point of $\mathbb{P}^1$ contains the smallest number of points. But I'm not sure.

existence of analytic continuation

2011-07-11T05:07:29Z

Good morning,

I have just started reading Riemann surfaces. I would like to ask a question, maybe it is naive.

Let $X$ be a Riemann surface and $\phi\in\mathcal{O}_{a,X}$ a holomorphic function germ at $a$ of $X.$ Let $u : [0,1]\to X$ be a curve, i.e a continuous mapping. Does it exist always an analytic continuation of $\phi$ along the curve $u$?

Comment by Đức Anh

2013-03-18T18:50:54Z

thank you very much for the informations.

Comment by Đức Anh

2013-03-18T09:47:31Z

Thank you. I'm interested in the both cases : compact general complex manifolds, and Kahler manifolds. I need some objects to orient my studies. If you know anything, please give me some informations.

Comment by Đức Anh

2013-03-18T07:22:29Z

Thank you. I hear about it for the first time.

Comment by Đức Anh

2013-03-17T02:12:01Z

Thank you very much.

Comment by Đức Anh

2013-03-11T18:19:25Z

interesting informations! Thank you very much.

Comment by Đức Anh

2013-03-11T18:17:19Z

Thank you all very much for useful informations. I should try another approach to my problems.

Comment by Đức Anh

2013-03-11T18:13:55Z

Thank you very much. I'll try to read these results.

Comment by Đức Anh

2013-03-11T15:57:08Z

Thank you very much. So the automorphism group may be bad (?) Btw, I would like to explain my motivation. Consider an entire curve $f\colon \mathbb{C}\to X.$ There is some place in $X$ where we can control the image of the curve. So, is there any way to perturbate the curve, or to move the curve in a reasonable way?

Comment by Đức Anh

2013-03-11T15:54:37Z

Thank you very much. At this moment, I'm only interested in a general compact Kahler manifold. My motivation comes from the following: I want to move a litte bit (in a reasonable way, e.g. in 3) ) the image of an entire curve $f\colon \mathbb{C}\to X,$ but I don't know any tool. So my questions are very vague.

Comment by Đức Anh

2013-03-09T17:56:55Z

@Felipe Voloch: I wonder about the singularity of the fibers. Are they non-singular varieties or not? Is there an Albanese torus for a singular projective variety? Do you know any information?

Comment by Đức Anh

2013-03-09T17:40:21Z

Thank you. But could you please give me some links/references?

Comment by Đức Anh

2013-01-22T12:39:15Z

thank you very much

Comment by Đức Anh

2013-01-22T06:40:32Z

thank you very much. So what can we say about the homology of $S$? Do you know any reference?

Comment by Đức Anh

2013-01-11T17:42:13Z

Thank you. I thought that since I've received a downvote. But I will continue to ask some more questions.

Comment by Đức Anh

2013-01-11T02:07:35Z

thank you very much. I have seen how naive my question is. Thank you also for recommanding the paper.