49
votes
Putnam 2020 inequality for complex numbers in the unit circle
Darij, such stuff is usually Gauss-Lucas in disguise and this case is no exception, though one needs to use once the version for polar derivative $D_1f(z)=(1-z)f'(z)+nf(z)$ of a polynomial $f$ of ...
- 61.9k
24
votes
Accepted
Putnam 2020 inequality for complex numbers in the unit circle
Here is a detailed and self-contained proof for general $n$, which also covers the "equality" case. It is based on Fedja's post, but it only uses (a variant of) the Gauss-Lucas theorem once.
...
- 105k
21
votes
Laurent series in several complex variables
A good condition is that the exponent vectors lie in a union of
finitely many translates of a fixed pointed
polyhedral cone $\mathcal{C}$ (with vertex at the origin). ``Pointed''
means that $\mathcal{...
- 50.8k
18
votes
Accepted
Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology
On differential forms, take complex conjugate to turn $\partial$ into $\bar\partial$, and holomorphic functions into conjugate holomorphic.
All of the proofs about differential forms then go through ...
- 26.3k
18
votes
Functions of several complex variables: book recommendations?
I know this is somewhat late answer for the original question, but here goes. I've taught Several Complex Variables twice at Oklahoma State, and I wrote a hopefully easy to read textbook for the ...
16
votes
Accepted
Laurent series in several complex variables
The easiest way to deal with series like $\sum_{n=0}^\infty z^n w^{-n}$ is with iterated Laurent series. This series is an element of the ring $\mathbb{Z}((w))[[z]]$: power series in $z$ whose ...
- 17k
11
votes
Accepted
Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)?
No.
Working on projective space, consider a composition $$ \mathcal O \to \mathcal O \oplus \mathcal O/\mathcal O(-1) \to \mathcal O $$ where $\mathcal O/\mathcal O(-1)$ is the constant sheaf on a ...
- 148k
10
votes
Accepted
Holomorphic Sard's theorem?
No. We can enumerate $\mathbb{Q}[i]$ as $\{a_0,a_1,a_2,\dotsc\}$ and then choose a holomorphic function $f\colon\mathbb{C}\to\mathbb{C}$ with $f(n)=a_n$ and $f'(n)=0$ for all $n$. This follows from ...
- 56.9k
8
votes
Accepted
zeros of holomorphic function in n variables
The conjecture is obviously false even for $n=2$. Check $f(z,w)=(w-z^2)(z-(w+1)^2)$. Write $z=x+iy$ and $w=u+iv$. Given $x$ and $u$, I can make first term zero unless $u>x^2$. I can make the second ...
- 2,263
8
votes
Accepted
Computing Dolbeault cohomology of some simple domains
$\def\CC{\mathbb{C}}\def\cO{\mathcal{O}}$Here is a computation of the Dobault cohomology of $X:=B(\infty) \setminus B(0) = \CC^2 \setminus \{ (0,0) \}$. I think that balls of finite radius should be ...
- 156k
8
votes
Accepted
Do we have the Oka coherence theorem for finite group actions?
This would be true. You need two facts:
Grauert's theorem that coherent sheaves are preserved by proper direct images. This implies
$\pi_*\mathcal{O}_{\mathbb{C}^n}$ is coherent.
Sub modules of ...
- 35.2k
8
votes
Accepted
Complex analytic function $f$ on $\mathbb{C}^n$ vanish on real sphere must vanish on complex sphere
The usage of Nullstellensatz-like results should be more subtle here. A similar idea was discussed in a recent question: A version of Hilbert's Nullstellensatz for real zeros. A solution based on ...
- 3,599
7
votes
Accepted
Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?
The series $f(x,y)=y+xy+x^2y+x^3y+\dots$ converges to $0$ when $y=0$, and converges to $y/(1-x)$ when $|x|<1$. This function is not continuous at $(x,y)=(1,0)$.
- 55.9k
6
votes
Accepted
Are there such things as non-trivial entire semigroups?
Set $f_t(z)=\phi(t,z)$.
Notice that entire functions $z\mapsto f_t(z)$ all commute with each other.
I. N. Baker proved that if $f$ is a non-affine entire function with a repelling fixed point
then the ...
- 91.8k
6
votes
Accepted
How to get a Stein space which has homotopy type of suspension of another Stein space
I'm not sure what you mean by "suspension" of $V$ here. The notion of suspension I have in mind (doubling the cone of $V$ over its base) doesn't yield a manifold, and even if it did it would ...
- 10.9k
6
votes
$n$-th root of meromorphic functions of several complex variables
If $\Omega$ is simply connected, this is true.
Let $\tilde \Omega$ be the space of pairs $(z, g)$ where $z \in \Omega$ and $g$ is the germ of an $n$th root of $f$ at $z$. If $\mathrm{div}(f) = nD$, ...
- 2,338
5
votes
Complex manifold with boundary
If $M$ is real analytic then Élie Cartan proved that, in suitable holomorphic coordinates, $M$ is cut out by the imaginary part of $z$. I learned this from the paper https://hal.archives-ouvertes.fr/...
- 8,828
5
votes
Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology
If $X$ is a complex manifold and $E\to X$ is a holomorphic vector bundle, only $\bar\partial_E$ can be defined naturally, i.e., it depends only on the complex structures of $X$ and $E$. The $\...
- 641
5
votes
Existence of plurisubharmonic functions on complex manifolds
I don't think it's true. Take for instance $Y$ to be a non-Kahler compact complex surface which has no compact complex curves, for instance an Inoue surface. If $p\in Y$ is some arbitrary point, set $...
- 641
5
votes
Analogue of Grauert's upper semi-continuity for Bott–Chern cohomology
The semi-continuity is true for elliptic complexes:
if $(C, d_t)$ is a continuous family of elliptic complexes,
parametrized by $t\in \mathbb R$, the cohomology of $(C, d_t)$
is semicontinuous in $t$. ...
- 9,237
5
votes
$\frac{\partial f}{\partial \overline{z}}=0$ in distributional sense implies $f$ is holomorphic
The trick is to replace the Cauchy integral formula $f_n(z)=\frac1{2i\pi}\int_{|s-a|=r} \frac{f_n(s)}{s-z}ds$ on a circle
by one on an annulus
$$f_n(z)=\int_r^R \psi_{r,R}(t)\frac1{2i\pi}\int_{|s-a|=t}...
- 3,403
5
votes
Zeroes of entire function on $\mathbb C^n$
This follows immediatelly from the following paper:
The Zero Set of a Real Analytic Function
- 2,071
4
votes
Accepted
Understanding Remmert-Stein extension theorem
I am posting my comment as an answer. I assume that you are taking the union of $\text{Zero}(y-ax)$ as $a$ ranges over an infinite set of integers. The set $V$ is not a complex analytic subvariety ...
4
votes
Is the projection of a pseudoconvex domain necessarily pseudoconvex?
Cezar Joita showed that any domain is the projection of a pseudoconvex domain. See his paper On projections of pseudoconvex domains Math Zeit vol 233 (2000) 625-631 .
- 4,131
4
votes
Accepted
Practically calculating the domain of a power series for function of several complex variables
The usual Cauchy-Hadamard formula has a generalization to several variables.
The numbers $r_1,\ldots,r_n$ are called conjugate radii of convergence if the
series converges in the open polydisk $B(r_1,\...
- 91.8k
4
votes
Accepted
Hartogs's extension theorem
I think what you would like is actually true:
Let $P'=F(P), H'=F(H)=f(H)\subseteq \mathbb C^n$. Consider the holomorphic map $g=f^{-1}:H'\to H$. Since $f$ is injective, $g$ cannot have essential ...
- 42.9k
4
votes
Accepted
Space of holomorphic embeddings of open unit ball in ${\mathbb C}^n$
The space of holomorphic embeddings with Jacobi matrix = 1 at zero is contractible due to the standard construction $f_t=(1/t)f(zt)$, and $f_0$ defined as a limit will be equal to the identity map ($t$...
- 1,842
4
votes
Proper analytic embedding of $\overline{\Bbb C}$ minus a Cantor set into $\Bbb C^2$
You should be able to find anwers to your question in a well witten book by Franc Forstnerič: Stein Manifolds and Holomorphic Mappings -The Homotopy Principle in Complex Analysis, https://www.springer....
- 589
4
votes
Accepted
Non-constant holomorphic map onto a smooth curve
Let me show that such a map usually doesn't exist (even if we don't remove any additional curves). Consider the case when $\Gamma$ is a curve of degree $\ge 4$. Then one can slightly perturb $\Gamma$ ...
- 28.9k
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