Recent MathOverflow Questions

Wishart type matrix

Math Overflow Recent Questions - Fri, 07/07/2017 - 16:01

Assume a positive semi-definite $M\times M$ matrix $A$, not with full rank, and an $M\times N$ matrix $X$, where $M>N$. The elements of $X$ are independent, zero-mean complex Gaussian with variance $1/M$.

My question is simple, what is the distribution of $X^H AX$?

From what I have seen, a matrix of form $X^H X$ is Wishart if the rows of $X$ are correlated, but in my case it is the columns.

Uniqueness of a Solution for a Convex Optimization Problem

Math Overflow Recent Questions - Fri, 07/07/2017 - 15:53

I have the following convex optimization problem: $$\begin{array}{ll} \text{maximize}_{{q_0,q_1}} & \displaystyle\int_{\Omega} q_1^u{q_0}^{1-u}\mathrm{d}\mu\\ \text{subject to} & \displaystyle\int_{\Omega} q_0 = 1,\quad \displaystyle\int_{\Omega} q_1 =1 \\ & f_l \leq {q_0} \leq f_u\\ & g_l \leq q_1 \leq g_u\end{array}$$ where $u\in(0,1) $ and $$\int_{\Omega}f_l \mathrm{d}\mu< 1,\quad\int_{\Omega}g_l \mathrm{d}\mu< 1$$

$$\int_{\Omega}f_u \mathrm{d}\mu> 1,\quad\int_{\Omega}g_u \mathrm{d}\mu> 1$$ Here, $q_0,q_1$ are distinct density functions and $f_l,f_u,g_l,g_u$ are some known positive functions on $\Omega$.

Claim: The solution is unique and it is the same for every $u\in(0,1)$, if $f_u=\infty$ and $g_u=\infty$, i.e., if there are only lower bounds, or $f_l=0$ and $g_l=0$, i.e. if there are only upper bounds. Else, the solution is also unique but it is not the same for all $u$.

Question: Are these claims true, especially the last one?

$\mu$ can be the Lebesgue measure, although I believe the same holds for the counting measure and the discrete sets. The set $\Omega$ can be $\mathbb{R}$ or an interval of real numbers.

I am especially interested in the last ''not'' case and for this case if necessary $g_u$ and $f_u$ can be assumed to be integrable over $\Omega$.

I had previously asked this question at math.stackexchange but with no answers.

Naive equivariant transfer

Math Overflow Recent Questions - Fri, 07/07/2017 - 13:13

Given a group $G$, a $\mathbb Z$-graded cohomology theory $E^*_G$, and a $n$-sheeted covering $p\colon X \to Y$, I would like a transfer map $$p_!\colon E^*_G X \to E^*_G Y$$ satisfying $$\require{cancel}\xcancel{p_! p^* = n \cdot \mathrm{id}.}$$ [this formulation was wrong; see the comments] such that when $n$ is inverted, $p_! p^*$ becomes an isomorphism (what I really need is that $p^*$ is an injection to a direct factor).

I'm aware such transfers exist for arbitrary fibrations $p$ (the number $n$ becomes the Euler characteristic of the fiber) under the additional assumption $E^*_G$ is $RO(G)$-graded, and that the $RO(G)$-grading is necessary for that result, but I only need them for covering maps, and I'd like to avoid the additional hypothesis.

So, are there transfer maps in this generality?

Dynkin Hunt formula

Math Overflow Recent Questions - Fri, 07/07/2017 - 04:26

I have a question about Dynkin Hunt formula.

Last day, I found a formula in this paper enter link description here. The formula is the equation (2.5) in this paper, which is called Dynkin Hunt formula. I know Dynkin formula. Dynkin formula states \begin{align*} R_{\alpha}f(x)=R_{\alpha}^{U}f(x)+E_{x}[e^{-\alpha \tau_{U}}R_{\alpha}f(X_{\tau_{U}})], \end{align*} where $\{X_t,P_x\}$ is a Markov(Hunt) process defined on a topological space $E$ and $R_{\alpha}$ is its resolvent. $R_{\alpha}^{U}f(x)$ is defined by \begin{equation*} R_{\alpha}^{U}f(x)=E_{x}[\int_{0}^{\tau_{U}}e^{-\alpha t}f(X_t)\,dt], \end{equation*} where $\tau_{U}$ is the first exit time from an open set $U$. In other words, $R_{\alpha}^{U}$ is the resolvent of subprocess $(X_{t}^{U},P_x)$ of $(X_t,P_x)$ on $U$.

You can see this formula in Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. The equation (4.1.6) in this book is Dynkin formula.

Question

Assume $\{X_t\}$ has transition probability density function $p(t,x,y)$ and transition density function of $\{X_{t}^{U}\}$ is denoted by $p^{U}(t,x,y)$.

Can we show the Dynkin-Hunt formula \begin{equation*} p(t,x,y)=p^{U}(t,x,y)+E_{x}[1_{\{\tau_{U} \le t\}}p(t-\tau_{U},x,y)] \end{equation*} for every $t>0$, $x,y \in E$?

My attempt

Assume resolvent density $R_{\alpha}(x,y)$ is continuous in $y$. Then, from Dynkin formula, we see \begin{equation*} R_{\alpha}(x,y)=R_{\alpha}^{U}(x,y)+E_{x}[e^{-\alpha \tau_{U}}r_{\alpha}(X_{\tau_{Y}},y)] \end{equation*} for every $\alpha>0,x,y \in E$. Therefore, \begin{align*} \int_{0}^{\infty}e^{-\alpha t}p(t,x,y)\,dt &=\int_{0}^{\infty}e^{-\alpha t}p^{U}(t,x,y)\,dt \\ &+\int_{0}^{\infty}e^{-\alpha t}E_{x}[1_{\{\tau_{U} \le t\}}p(t-\tau_{U},X_{\tau_{U}},y)]\,dt \end{align*} Therefore, for every $x,y \in E$, \begin{equation*} p(t,x,y)=p^{U}(t,x,y)+E_{x}[1_{\{\tau_{U} \le t\}}p(t-\tau_{U},x,y)] \end{equation*} holds for a.e. $t$. Can we refine this equation? That is, can we show \begin{equation*} p(t,x,y)=p^{U}(t,x,y)+E_{x}[1_{\{\tau_{U} \le t\}}p(t-\tau_{U},x,y)] \end{equation*} for every $t>0$, $x,y \in E$?

If you know related works about Dynkin-Hunt formula, please let me know.

What is the correct definition of a context that would enable the axiom of induction to be derived from a construction of the natural numbers? [closed]

Math Overflow Recent Questions - Thu, 07/06/2017 - 15:15

With typical hubris I thought I had a proof of the inference rule for induction starting from a construction of the natural numbers. But I now find on closer inspection that there's a glitch. My original question on mathematics stack exchange, and in particular my answer to the question at the foot of the post, has all of the background. The complete proof, including the glitch, can also be found here. I could give a further explanation now but I might just muddy the waters so it's best just to point the reader to the original post and then just highlight the glitch.

The problem is with the following assertion which can be found in the second subproof of the proof for the main rule:

τ(s(k)) ⊢ T(k)::P(k)

Whilst it is certainly true that T(k) should in some sense belong to the context τ(s(k)), it seems that I can't simply assert that it is a proof of P(k). Or can I?

Update: As far as I understand it the aforementioned judgement τ(s(k)) ⊢ T(k)::P(k) within the subproof is actually fine because the definition of the context should be:

let τ(s(k))=σ,T(k)::P(k)

Quite why this is permissible when (obviously) the judgement T(k)::P(k) hasn't itself been proven is something I am still not one hundred percent sure about. I would appreciate some clarification. There are some rambling comments of mine at the end but I'm not sure these are of much value.

Update: Two things. Firstly, there is a rather glib answer to the remaining question, namely is τ(s(k))=σ,T(k)::P(k) correct? The answer is that if τ(s(k)) were not defined in this way then theorem T(s(k)) would simply not verify. The other thing is a reference:

Hardegree, Metalogic, Mathematical Induction

A part of section 10 would suggest that my proof is wrong:

"This demonstrates that the principle of induction implies the ‘nothing else’ clause. It does not demonstrate the converse – that the ‘nothing else’ clause implies the principle of induction."

He then gives an informal proof (which I don't claim to have taken the time to digest) and finishes:

"Note carefully, that the previous argument is informal. There is no formal proof that corresponds to it."

Oscillation of the summatory Möbius function

Math Overflow Recent Questions - Thu, 07/06/2017 - 10:28

Let the Mertens function $$M(x) = \sum_{n \le x} \mu(n)$$ I assume (perhaps foolishly) that it is known that $M(x)$ changes sign infinitely often. If that's true, the question is a quantitative version :

How many sign changes of $M(x)$ are there between $1$ and $y$ (asymptotically) ?

**ADDITION* GH from MO cites a result which gives a logarithmic number of changes. This, while better than nothing, is not (empirically the truth): for $N=1000000,$ you get around $5500$ sign changes, for $N=10000000,$ around $12000,$ and here is the graph of the total number of sign changes.This looks square-rootish. Now, what is even more interesting is that for a symmetric random walk, the number of returns to the origin is asymptotic to $\frac{2}{\pi} \sqrt{N},$ which is much smaller than this. (see https://math.stackexchange.com/questions/1338097/expected-number-of-times-random-walk-crosses-0-line for deviation). The difference is even more striking, if you remember that a lot of numbers are non-square-free.

Has anyone seen a nice map of multiplicative cohomology theories?

Math Overflow Recent Questions - Thu, 07/06/2017 - 07:28

I have seen lots of descriptions of this map in the literature but never seen it nicely drawn anywhere.

I could try to do it myself but I really lack expertise, hence am afraid to miss something or do it wrong.

Let me just provide some glimpses, and maybe somebody can nicely tie them together.

At the "initial end" there is the stable (co)homotopy, corresponding to the sphere spectrum.

At the "terminal end" there is the rational cohomology (maybe extending further to real, complex, etc.)

From that terminal end, chains of complex oriented theories emanate, one for each prime; the place in the chain corresponds to the height of the formal group attached. Now here I am already uncertain what to place at each spot - Morava $K$-theories? Or $E$-theories? At the limit of each chain there is something, and again I am not sure whether it is $BP$ or cohomology with coefficients in the prime field.

Next, there is complex cobordism mapping to all of those (reflecting the fact that the complex orientation means a $MU$-algebra structure). But all this up to now only happens in the halfplane. There are now some Galois group-like actions on each of these, with the homotopy fixed point spectra jumping out of the plane and giving things like $KO$ and $TMF$ towards the terminal end and $MSpin$, $MSU$, $MSp$, $MString$, etc. above $MU$. Here I have vague feeling that moving up from $MU$ is closely related to moving in the plane from the terminal end (as $MString$, which is sort of "two steps upwards" from $MU$, corresponds to elliptic cohomologies which are "two steps to the left" from $H\mathbb Q$) but I know nothing precise about this connection.

As you see my picture is quite vague and uncertain. For example, I have no idea where to place things like $H\mathbb Z$ and what is in the huge blind spot between the sphere and $MU$. From the little I was able to understand from the work of Devinatz-Hopkins-Smith, $MU$ is something like homotopy quotient of the sphere by the nilradical. Is it correct? If so, things between the sphere and $MU$ must display some "infinitesimal" variations. Is there anything right after the sphere? Also, can there be something above the sphere?

How does connectivity-non-connectivity business and chromatic features enter the picture? What place do "non-affine" phenomena related to algebroids, etc. have?

There are also some maps, like assigning to a vector bundle the corresponding sphere bundle, which seem to go backwards, and I cannot really fit them anywhere.

Have I missed something essential? Or all this is just rubbish? Can anyone help with the map, or give a nice reference?

What function is this?

Math Overflow Recent Questions - Thu, 07/06/2017 - 06:50

Let $0<\lambda\le1$ and consider $$ \Theta:(\Bbb R[X]_0,||\cdot||_{\lambda})\longrightarrow(\mathcal C^{\lambda}[0,1],||\cdot||_{\lambda}) $$ defined as $$ \Theta(p):=\sup_{0\le u\le\cdot}p(u) $$ (in the sense that the polynomial $p$ is sent by $\Theta$ to the function $t\mapsto\sup_{0\le u\le t}p(u)$) where $\Bbb R[X]_0$ denotes the space of one variable polynomials with real coefficients which vanish at $0$, $\mathcal C^{\lambda}[0,1]$ is the space of $\lambda-$ Holder continuous functions $f:[0,1]\to\Bbb R$ and $||\cdot||_{\lambda}$ denotes the usual $\lambda-$Holder norm.

What kind of function is $\Theta$? What property does it have? Does $\Theta$ belong to a family of functions which has a name and for which a theory is developed somewhere? Can someone suggest me some reference?

EDIT: is there a way to prove/disprove $\Theta$ is Lipschitz?

I guess we are near the field of functional analysis but, beyond this, I see only fog.

Many thanks

What is Atiyah's topological formulation of the odd order theorem?

Math Overflow Recent Questions - Thu, 07/06/2017 - 06:50

Here is a quote from an article by Daniel Gorenstein on the history of the classification of finite simple groups (available here).

During that year in Harvard, Thompson began his monumental classification of the minimal simple groups. He soon realized that he didn't need to know that every subgroup of the given subgroup was solvable, but only its local subgroups, and he dubbed such groups N-groups. However, the odd order theorem was still fresh in his mind. One afternoon I ran into him in Harvard Square and noticed he had a copy of Spanier's book on algebraic topology under his arm. "What in the world are you doing with Spanier?" I asked. "Michael Atiyah has given a topological formulation of the solvability of groups of odd order and I want to see if it provides an alternate way of attacking the problem," was his reply.

What is this topological formulation of the solvability of groups of odd order?

Uniformisation for non simple closed curves

Math Overflow Recent Questions - Thu, 07/06/2017 - 02:59

Given a simple closed curve in the plane, there is a homeomorphism from the unit open disk to the interior of the curve. The homeomorphism can be taken conformal, this is the uniformisation theorem.

Is there a version of this result for closed curves that are not simple?

For example, given a $C^1$ closed curve $\gamma: S^1 \rightarrow {\bf R}^2$ with finitely many self-intersections, all of them transverse, is there a continuous map (maybe even conformal) from the unit open disk to the plane, such that the number of preimages of any point in ${\bf R}^2\backslash \gamma(S^1)$ is equal to the absolute value of the number of turns the curve makes around the point?

$\hskip 1in$

When is every orbit closure uniquely ergodic?

Math Overflow Recent Questions - Wed, 07/05/2017 - 14:14

Given a topological dynamical system $(X,T)$ (so that $T$ is a homeomorphism of the compact metric space $X$) and a point $x\in X$ we call the set ${\mathcal O}(x):=\overline{\{T^nx:n\in\mathbb Z\}}$ the orbit closure of $x$.

Question 0: Is there a name for systems with the property that the orbit closure of every point is uniquely ergodic (i.e., supports a unique invariant measure)?

It is well known that nilsystems have this property but not all distal systems do.

Question 1: Is it true that if $(X,T)$ and $(Y,S)$ are uniquely ergodic, then $(X\times Y,T\times S)$ has the property that every orbit closure is uniquely ergodic?

Question 2: What if in addition $(X,T)$ and $(Y,S)$ are distal?

Existence of a unique cyclic and separating vector in a *-representation

Math Overflow Recent Questions - Wed, 07/05/2017 - 12:16

I'm interested in knowing the requirements for a $*$-representation, $\pi_{\omega}$, of a C*-algebra, $\mathbb{C}(\mathcal{G})$, (or equivalently the requirements for the unitary representation, $U_{\mathcal{G}}$, corresponding to $\pi_{\omega}$) to have a unique cyclic and separating vector $\xi$.

P.S. I'm a physicist and I'm new to this field so I would appreciate a more detailed answer. Thank you so much in advance!!

sum over partitions

Math Overflow Recent Questions - Wed, 07/05/2017 - 10:08

Let $\mathcal{P}_{n,k}$ be the set of unordered partitions of a positive integer $n$ into $k$ parts $\mathbf{n}:=(n_1,n_2,\ldots,n_k)$, i.e. such that $n=n_1+n_2+\ldots+n_k$.

Are there known results for the following quantity?

\begin{equation} \sum_{\mathbf{n}\in\mathcal{P}_{n,k}} \prod_{j=1}^k \frac{1}{n_j} \end{equation}

And more in general for quantities such as

\begin{equation} \sum_{\mathbf{n}\in\mathcal{P}_{n,k}} \prod_{j=1}^k f(n_j), \end{equation}

for some function $f$ defined on $\mathbb{N}$?

I am not familiar with results involving partitions so any insight on this will be very appreciated, thank you.

group extension with finite abelian group

Math Overflow Recent Questions - Wed, 07/05/2017 - 10:05

I am a student majoring physics. The question below may be simple but confuse me. Thanks for any suggestion or detailed answer.

Given two finite abelian group $N$ and $A$, the question is how many possible $G$(including abelian and nonabelian $G$) satisfying the following short exact sequence:

$0 \rightarrow N \rightarrow G \rightarrow A \rightarrow 0$.

In another words(to my understanding), how many possible choices of $G$ which are group extension of $N$ by $A$? I am grateful who can give me the detailed consideration of the case: $N=Z_{n_1}\times Z_{n_2}$, and $A=Z_{m_1} \times Z_{m_2}$.

Exponential contraction for the projection on horospheres

Math Overflow Recent Questions - Wed, 07/05/2017 - 09:26

A few years ago, Roberto Frigerio asked for a reference for a geometric property of horospheres, namely exponential decay of the projection onto a horosphere.

My question is: does this exponential decay still holds in a Gromov hyperbolic space ?

The precise statement: Let X be a $\delta$-hyperbolic space, $B$ a horoball and $H$ the corresponding horosphere. Denote by $\pi$ the projection onto $H$. There exists $\alpha>0$ such that the following holds. Assume that $\gamma$ is a path in $X\setminus B$ such that $d(\gamma(t),H)\geq k>0$ for some $\alpha$ and for every $t$. Then, $l(\pi \circ \gamma)\leq e^{-\alpha k}l(\gamma)$, where $l$ is the length of a path.

Is there a reference for such a statement ?

Number of points of elliptic curve over $\mathbb{F}_p$ with CM by $\sqrt{-2}$ when $p\equiv 1\bmod 8$

Math Overflow Recent Questions - Wed, 07/05/2017 - 09:21

I was trying to calculate the number of points of the curve $E:y^2 = x^3 + 4x^2 + 2x$ over $\mathbb{F}_p$ for $p\equiv 1\bmod 8$ (In order to have $\sqrt{-2}\in\mathbb{F}_p$) but I did not succeed. This curve is mentioned in Silverman's Advanced Topics in the Arithmetic of Elliptic curves (Proposition 2.3.1) to have multiplication by $\sqrt{-2}$.

Over these primes $p\equiv 1\bmod 8$ the curve $E$ has full $2$-torsion so $E(\mathbb{F}_p)\cong \mathbb{Z}/(2)\times \mathbb{Z}/(k)$.

In this case my conjecture is that the size will be related to the factorization of $p=(a+b\sqrt{-2})(a-b\sqrt{-2})$ over $\mathbb{Z}[i\sqrt{2}]$, that is $p=a^2 + 2b^2$. Hence, $\#E(\mathbb{F}_p)=p+1\pm 2a$ where $a$ is odd (and the sign I do not know how to choose it yet). Calculating this reminds me to the proof of the Last Entry of Gauss Tagebuch.

I would like to have an elliptic curve with CM by $\sqrt{-2}$ such that I can know the number of points in terms of $p$. Does anybody has a suggestion? or maybe another curve?

Thanks

Non-rational, smooth Fano 3-folds with $\rho(X) >1$

Math Overflow Recent Questions - Wed, 07/05/2017 - 08:59

I am interested to know the list of non-rational smooth Fano 3-folds with Picard number greater than 1 (more precisely families where at least one smooth member is known to be non-rational). It seems that there are not so many.

In the list of Fano 3-folds that I have at hand, it states whether each Fano 3-fold is rational or not in almost all cases. But unfortunately, there is a ? written next to one of the varieties and the authors leave a gap next to 5 other cases.

I am not sure whether these omissions are due to the rationality of these varieties being an open problem, or whether it is due to some complicated situation with some rational and non-rational varieties in the family. Any enlightenment about the state of the art would be greatly appreciated.

The examples I did manage to obtain from this list are (in no particular order):

1.Double cover of $\mathbb{P}^1 \times \mathbb{P}^2$, branched along a divisor of bi-degree $(2,4)$.

  1. Blow up of $V_{1}$ (Del-pezzo 3-fold of degree 1) in the intersection of two divisors from the anti-canonical class.

  2. Blow-up of cubic 3-fold in a line.

  3. Blow-up of cubic 3-fold in a planar cubic curve.

  4. A divisor in $\mathbb{P}^2 \times \mathbb{P}^2$ of bi-degree (2,2).

  5. A blow up of $V_{7}$ ($= Bl_{p} \mathbb{P}^3$) in a divisor in the anti-canonical class (satisfying certain smoothness condition).

  6. Double cover of $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, branched along a smooth divisor of tri-degree (2,2,2).

(1-6. have $\rho(X) = 2$ and 7. has $\rho(X) = 3$.)

Question: Is there any 3-folds missing from the list? i.e. is there a smooth Fano 3-fold $X$ with $\rho(X)$>1, that is known to be non-rational that is not contained in one of these 7 families.

Extending a natural transformation using a distinguished triangle

Math Overflow Recent Questions - Wed, 07/05/2017 - 08:40

$\require{AMScd}$

Let $\mathcal{T}$ be a triangulated category, and $\mathcal{S}$ a full subcategory of $\mathcal{T}$ (which is not triangulated).

Let $F, G: \mathcal{T} \to \mathcal{T}$ be two triangulated functors.

I have a "partial" natural isomorphism: For any $M \in \mathcal{S}$, I am given an isomorphism $\eta_M: F(M) \to G(M)$, such that if $f:M \to N$ is a morphism in $\mathcal{S}$, the diagram $$ \begin{CD} F(M) @>{F(f)}>> F(N)\\ @VVV @VVV \\ G(M) @>{G(f)}>> G(N) \end{CD} $$ is commutative.

Now, I am given two distinguished triangles $$\mathbf{M} = M' \to M'' \to M \to \Sigma M'$$ and $$\mathbf{N} = N' \to N'' \to N \to \Sigma N'$$ with $M',M'',N',N'' \in \mathcal{S}$, but $M,N \notin \mathcal{S}$.

I am also given a morphism of triangles $\mathbf{f} = (f',f'',f): \mathbf{M} \to \mathbf{N}$.

Using the triangles $F(\mathbf{M}), G(\mathbf{M})$, and the isomorphisms $\eta_{M'}, \eta_{M''}$, I can obtain an isomorphism $\eta_M: F(M) \to G(M)$, and similarly an isomorphism $\eta_N:F(N) \to G(N)$.

My question:

Is it possible to choose the isomorphisms $\eta_M, \eta_N$ so that $G(f) \circ \eta_M = \eta_N \circ F(f)$?

Solution of a weakly formulated pde similar to p-Laplacian and Eigenvalue problem

Math Overflow Recent Questions - Wed, 07/05/2017 - 08:11

Let $f:\Omega\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^d$, and $\Omega$ is convex. Let $\{x_i\}_{i=1,2,..N}$ be a set of points in the interior of $\Omega$.

I want to solve this weakly formulated pde:

$$ 0=\frac{A}{N^{d+1}} \sum_i \phi(x_i) f(x_i) |f(x_i)|^{d-1} + \int_\Omega \phi f |f|^{d-1} +Ad\int_\Omega \nabla \phi \cdot \nabla f |\nabla f|^{d-1} $$ holds for all sufficiently smooth $\phi$ such that $\int_\Omega \phi dx = 0$.

The solution $f$ is known to be Holder continuous. From my literature survey, the closely related thing I found which could be useful is the Eigenvalue problem for the p-Laplacian. There are many differences from our problem; one is we don't have any boundary constraints and ours is a weak pde and we have one extra term (first one on RHS) which makes it different from an Eigenvalue problem.

Is the following example of a holomorphic submersion locally trivial?

Math Overflow Recent Questions - Wed, 07/05/2017 - 08:11

I have just came upon the following example. Consider $\pi : M \rightarrow \mathbb C$ the projection on the first coordinate, where

$$M = \mathbb{C}^{2} \setminus (\{(z,w) \in \mathbb{C}^{2} : w=0\} \cup \{(z,w) \in \mathbb{C}^{2} : z=w\} \cup \{(0,1)\}).$$

The example is taken from a question discussed here When is a holomorphic submersion with isomorphic fibers locally trivial?

It is clear that this is a submersion and that the fibers are biholomorphic to $\mathbb{C} \setminus \{0,1\}$. But is this locally trivial? The answer suggested there is a little bit confusing to me: suppose $\pi$ is locally trivial, then extend $\pi$ into the single point $(0,1)$, and then restrict to the fiber. So $\mathbb{C} \setminus \{0\}$ should be mapped onto $\mathbb{C} \setminus \{0,1\}$, contradiction as the latter is hyperbolic.

Can anyone explain a little bit?

Since I was not happy with the previos proof, I have tried my own. Suppose $\pi$ is locally trivial, so $\forall z \in \mathbb{C}$, there exist $U \ni z$ and $\varphi : \pi^{-1}(U) \rightarrow U \times (\mathbb{C} \setminus \{0,1\})$ that is a biholomorphic map.

If $z \neq 0$, then we restrict to the fiber above $z$ and we get that $\varphi_{z} : \mathbb{C} \setminus \{0,z\} \rightarrow \mathbb{C} \setminus \{0,1\}$ is also a biholomorphic map. Now look at $\varphi_{z} \circ F_{z} : \mathbb{C} \setminus \{0,1\} \rightarrow \mathbb{C} \setminus \{0,1\}$, where $F_{z}$ is the mutiplication by $z$. But the number of such maps is finite. Using the fact that $\varphi$ is continuous and that $U \setminus \{0\}$ is connected, we can assume that $\varphi_{z} \circ F_{z}$ is exactly the identity map.

Take a sequence $a_{n}=(\frac{1}{n},2)$ which converges to $(0,2)$. But $\varphi(\frac{1}{n},2)=(\frac{1}{n},2n)$ does not converge to $\varphi(0,2)$ which is $(0,\beta)$, for some $\beta \in \mathbb{C} \setminus \{0,1\}$; contradiction to $\varphi$ being continuous.

Is my proof correct?

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