Is $ mathbb {C} ^ n $ rigid?

To let $ pi: X to S $ be a smooth family of complex manifolds (in the sense of deformation theory), so that $ pi ^ {- 1} (0) cong mathbb {C} ^ n $ and $ S subset mathbb {C} $ is a neighborhood of $ 0 $, is $ pi $ trivial? That is, is $ X cong mathbb {C} ^ n times S $ possibly after shrinking $ S $?

I know a smooth family of compact complex manifolds with $ H ^ 1 (M, mathcal {T} _M) = 0 $ is trivial (where $ M = pi ^ {- 1} (0) $) but I'm not sure if this extends to this non-compact situation.

ag.algebraic geometry – finiteness gives the category from schemes up to $ mathbb {A} ^ 1 $ -homotopy

In algebraic geometry we know that there are geometric conditions in a schema $ X / k $ for finally a lot of rational points, if $ k $ is a number field. For curves there is the Mordell conjecture, and in higher dimensions the long conjecture says that if $ X ( mathbb {C}) ^ {an} $ is hyperbolic $ | X (k) | $ is finite. I would be interested to generalize this train of thought to the category of systems $ mathbb {A} ^ 1 $-Homotopie. Specifically, my question is whether there is a (presumed) geometric condition or not $ (X) $, on $ mathbb {A} ^ 1 $-homotopy class, so the amount of morphism $ text {Hom} _ { mathbb {A} ^ 1 text {-hom}} (( text {Spec} k), (X)) $ in the category of systems up $ mathbb {A} ^ 1 $-Homotopia is finite.

Of course, a naive translation of the folding theorem does not work, for example $ mathbb {A} ^ 1 _ { mathbb {Q}} $ has infinitely many rational points, however $ text {Spec} mathbb {Q} $ Not.

Analogues over finite fields of certain integers, which are multiplicatively defined in $ mathbb Z $

For every polynomial $ f $ Degree $ d $ in the finite field $ mathbb F $ we have that $$ x ^ {| mathbb F | ^ d} -x equiv0 bmod f $$ and thus the polynomial $ x ^ {| mathbb F | ^ d} -x $ is the analogue of the faculty in finite fields. Are there analogues of binomial coefficients that can be represented well? Would it be sensible to look for analogues that can be well represented as the denominator of the Bernoulli numbers? $ B_ {2n} $ which are $ prod _ {(p-1) | 2n} p $?

Reference requirement – Lebesgue measure for solution sets for polynomials in $ mathbb {C}[x]$

Describe $ pmb {a} = (a_1, dots, a_d) $ and look at the crowd
$$ mathcal {E} _d = { pmb {a} in mathbb {R} ^ d: text {every root $ xi $ of $ x ^ d + a_dx ^ {d-1} + cdots + a_2x + a_1 = 0 $ is in $ vert xi vert <1 $} }. $$
In the reference shown below, Fam has demonstrated that the $ d $-dimensional Lebesgue measure met
$$ lambda_d ( mathcal {E} _d) = 2 ^ d prod_ {k = 1} ^ { lfloor frac {d} 2 rfloor} left (1+ frac1 {2k} right) ^ {2k-d}. $$
I would like to suggest a complex version here. Describe $ pmb {c} = (c_1, dots, c_d) $ and look at the crowd
$$ mathcal {S} _d = { pmb {c} in mathbb {C} ^ d: text {every root $ xi $ of $ x ^ d + c_dx ^ {d-1} + cdots + c_2x + c_1 = 0 $ is in $ vert xi vert <1 $} }. $$
Let's ask now:

QUESTION. What is the $ 2d $-dimensional Lebesgue measure
$$ lambda_ {2d} ( mathcal {S} _d)? $$

In contrast: $ lambda_1 ( mathcal {E} _1) = 2 $ while $ lambda_2 ( mathcal {S} _1) = pi $,

Reference.

A. T. Fam.,The volume of the coefficient space stability domain of monic polynomialsProc. IEEE Int. Symp.Circuits and Systems, 2 (1989), pp. 1780-1783.

Differential geometry – visualization of a sub-variety of $ mathbb {C} ^ 3 $

For the function $ F: mathbb {C} ^ 3 to mathbb {C} $. $ (x, y, z) mapsto x ^ 2 + y ^ 2 + z ^ 2 $. $ 0 $ is a regular value. So especially $ F ^ {- 1} (0) subset mathbb {C} ^ 3 $ is an embedded sub-diversity (from dimension 2 over $ mathbb {C} $ or dimension 4 over $ mathbb {R} $). However, I have problems imagining what the set looks like. My ultimate goal is to build a CW structure $ q ( text {im} (F)) $ Where $ q: mathbb {C} ^ 3 to mathbb {C} P ^ 2 $ is the quotient map, but I would like to understand what this sub-diversity looks like first. I tried to write this down in relation to the real and complex parts of the three complex numbers. This gives relationships
$$
sum_ {i = 1} ^ 3 Re (z_i) = sum_ {i = 1} ^ 3 Im (z_i), ; text {and} sum_ {i = 1} ^ 3 Im (z_i) Re (z_i) = 0.
$$

But I don't see how that would help. It is also clear that if $ F (x, y, z) = 0 $, then $ F (kx, ky, kz) = 0 $ for each $ k in mathbb {C} $ but I also don’t know how to use it.

I would be very happy if someone could give me a contribution to the visualization of this set or how I would create a CW structure $ q ( text {im} (F)) $,

Functions – Show that $ (0, 1) $ and $ mathbb {R} $ by the function $ g (x) = frac {1} {2} (1+ frac {x} {1+ | x |}) $ are equivalent

Show that $ (0, 1) $ and $ mathbb {R} $ are synonymous with function $ g (x) = frac {1} {2} (1+ frac {x} {1+ | x |}) $

We have to show the biyectivity of the function $ g $::

Inyectivity: It is done in the usual way

Surjective is where I got bearings. I think the absolute value creates two cases for this part.

Ag.algebraic geometry – The connection of the coarse module scheme from $ mathscr {M} _ * (N) $ to $ mathbb {Z} $

To let $ N ge 1 $ be an integer and $ mathscr {M} _ * (N) $ the stack of elliptic curves with the plane $ Gamma (N), Gamma_0 (N), Gamma_1 (N) $ or $ Gamma _ { text {bal.} 1} (N) $,
(See Katz-Mazur for its definition.)
Then there is the rough module scheme of $ mathscr {M} _ * (N) $, denoted by $ Y = Y _ * (N) $, About $ mathbb {Z} $and it is a coherent regular affine scheme of pure dimension 2 and flat of the finite type of relative dimension 1.
in addition, $ Y times mathbb {Z} (1 / N) $ is smooth over $ mathbb {Z} (1 / N) $,

I've shown this sentence with the exception of connectedness.
I want to show it.

I tried the following:
Since $ Y times mathbb {Q} to Y $ has the dense picture, it is enough to show that $ Y times mathbb {C} $ Is connected.
And since the Euclid topology is finer than the Zariski topology, it is enough to show that the complex diversity $ Z $ caused by $ Y times mathbb {C} $ (i.e. the closed sub-diversity induced by $ Y ( mathbb {C}) subseteq mathbb {A} ^ N ( mathbb {C}) = mathbb {C} ^ N $.) Is connected.

Next it looks like Riemann $ Z cong mathbb {H} / Gamma _ * (N) $, At least for $ * = 0, 1, varnothing $,
(Where $ mathbb {H} $ is the upper half-plane.)
If so, trivial $ Z $ is connected, and such $ Y $ Is connected.

So it is enough to show $ Z cong mathbb {H} / Gamma _ * (N). $
The "rough module card" $ mathscr {M} _ * (N) to Y $ induced $ | mathscr {M} _ * (N) ( mathbb {C}) | $ $ cong Y ( mathbb {C}) cong Z $,
On the other hand, we have a basic set of modular shapes $ | mathscr {M} _ * (N) ( mathbb {C}) | cong mathbb {H} / Gamma _ * (N) $,
So as sets we have $ Z cong mathbb {H} / Gamma _ * (N) $,

It's hard for me to show that this map is holomorphic (or even continuous, which is enough to show) $ Z $ is connected) because I can't understand the topology on $ Z $ "Modules theoretically".
(i.e. the interpretation like "$ U subseteq Z $ is open when elliptic curves in $ U $ are so and so … ")

Thanks a lot!

analytical functions – Prove that $ f (z) = | z | ^ 2 $ is differentiable at $ z_0 = 0 $, but cannot be complexly differentiated at $ z_0 in mathbb {R} setminus {0 } $.

I am trying to answer the following complex analysis question:
"Prove that $ f (z) = | z | ^ 2 $ is differentiable at $ z_0 = 0 $, but not complexly differentiable in $ z_0 in mathbb {R} setminus {0 } $, Derive that $ f (z) $ is not analytical at $ z_0 = 0 $".

We cannot use Cauchy-Riemann and must use a $ Delta Epsilon $ Argument for the first part. Any ideas on how to address this problem?