All Questions
43 questions
3
votes
2
answers
130
views
Vector bundles over a Stein space are projective
It is a "well known" fact that
locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules
(see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
- 1,109
6
votes
0
answers
200
views
Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
- 1,286
3
votes
1
answer
173
views
Can we define $\partial\bar{\partial}(\log|z_1|^2)\wedge \partial\bar{\partial}(\log|z_2|^2)$ as a current?
In complex analysis, by Poincare-Lelong theorem, we have
$$
\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}(\log|z|^2)=T_{z=0}
$$
as currents, where
$$
T_{z=0}(\eta)=\int_{z=0}\eta.
$$
Now suppose we have ...
- 9,009
21
votes
0
answers
2k
views
Cartan–Oka vanishing in one variable without $\overline{\partial}$?
This is a literature question, about possible proofs of some very basic results in complex analysis.
Some key facts about holomorphic functions are proved via reduction to smooth functions, using $\...
- 21.3k
3
votes
1
answer
174
views
first order quasilinear partial differential equations
I am interested in understanding complex first-order quasilinear partial differential equations. In the real setting there is a huge literature dealing with such equations but in the complex setting, ...
- 35
5
votes
1
answer
519
views
Branched covers of the sphere branched over few points
Let $X$ be a compact Riemann surface of genus $g\geq 2$. By the Riemann-Roch theorem, $X$ is a branched cover of the sphere, branched over finitely many branched values. What is the smallest number of ...
- 6,548
3
votes
1
answer
214
views
Holomorphic vector fields with a non-degenerate isolated zero
Let $v$ be a holomorphic vector field defined in a neighbourhood of $0$ on $\mathbb C^n$ with an isolated zero at $0$. Let $\sum_{i,j}{a_{ij}}z_i\frac{\partial}{\partial z_j}$ be the linear term of $...
- 14.3k
6
votes
2
answers
240
views
Continuity of a differential of a Banach-valued holomorphic map
Originally posted on MSE.
Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...
- 5,529
2
votes
0
answers
154
views
Algebra of meromorphic functions on a Riemann surface
Let $C$ be compact Riemann surface of high genus. Let $p$ be a general point on $C$ and $z$ a local coordinate around $p$.
Given a meromorphic function on $C$, regular outside $p$, we can look at its ...
- 2,384
6
votes
1
answer
260
views
Do analytic functionals form a cosheaf?
Let $X$ be a complex-analytic manifold.
Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
- 890
8
votes
0
answers
277
views
Cohomology of complex manifold vs cohomology of its complex submanifold
Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that
$$H^i(Z, A|_Z)=0 \mbox{ for any } ...
- 21.8k
5
votes
1
answer
153
views
An estimate on deviation of two smooth tangent $J$-holomorphic curves
Take $\mathbb C^2$ with coordinates $(z,w)$. Suppose that $J$ is a $C^{\infty}$ almost complex structure on $\mathbb C^2$ such that the line $w=0$ is $J$-holomorphic and $J(0,0)$ is given by $(z,w)\to ...
- 14.3k
4
votes
1
answer
150
views
Linearisation of complex $S^1$ actions at fixed points
Let $(U,x)$ be an open complex $n$-manifold (say an $n$-ball) with an action of $S^1$ by holomorphic transformations that fix $x$. How to prove that there is a neighbourhood $U_1\subset U$ of $x$ ...
- 14.3k
4
votes
0
answers
157
views
Analytic maps $\varphi: \mathbb C^n\to \mathbb C^n$ with degenerate differentials
Let $B^n\subset \mathbb C^n$ be a unit ball with center $p$ . Let $\varphi: B^n\to \mathbb C^n$ be a complex analytic map such that $d\varphi$ has rank at most $n-1$ at $p$. I would like to know if ...
- 14.3k
2
votes
2
answers
417
views
Stoilow Theorem
I want to see the precise statement and a proof for a theorem of Stoilow on "inner" functions (I do not know what this exactly means, I suppose it is an open map with other natural properties). A ...
- 85
7
votes
1
answer
832
views
Monodromy representations and branched covers
I need to use the following result (that I'm pretty sure is true):
Theorem. Let $Y$ be a compact complex manifold and $B \subset Y$ be a connected submanifold of codimension one. Then isomorphism ...
- 66.3k
4
votes
3
answers
687
views
Finite covers of punctured Riemann surfaces
Let $X$ be a compact Riemann surface, i.e. compact smooth complex analytic (hence automatically algebraic) curve. Let $A\subset X$ be a finite subset, and $X_0:=X\backslash A$.
Let $Y_0$ be a smooth ...
- 21.8k
1
vote
0
answers
201
views
Applications of Iss'sa's theorem on homomorphisms between algebras of meromorphc functions
In Remmert's book Funktionentheorie II, the following theorem, apparently due to Hironaka under the pseudonym Iss'sa, is proved:
Let $U,V \subseteq \mathbb{C}$ be open subsets. Every $\mathbb{...
- 21
3
votes
1
answer
305
views
The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold
Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...
- 7,426
5
votes
1
answer
353
views
Quantifying the monotonicity property of the hyperbolic metric
Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} &...
- 6,548
1
vote
0
answers
217
views
Homeomorphism of fibers of holomorphic maps
EDIT (after the comment by Jason Starr): Let $X$ be a complex algebraic (or, more generally, analytic) variety, possibly singular and non-compact. Let $f\colon X\to D^*$ be a proper algebraic morphism ...
- 21.8k
2
votes
1
answer
149
views
Higher dimensional analogue of Ahlfors covering surface theory
It is well known that Ahlfors covering surface theory in one dimensional is very powerful in dealing with many problems. I wonder whether there exists some generalization of this theory into higher ...
- 21
9
votes
2
answers
873
views
Implicit Function Theorem on Singular Varieties
Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function
$$f\colon X \to Y $$
and assume that it is bijective at the level of $...
- 2,384
5
votes
1
answer
366
views
A weak analytic version of the valuative criterion of properness
EDIT: Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that
(a) $f$ is injective on points;
(b) $f$ is local imbedding near each point $x\in ...
- 21.8k
1
vote
0
answers
237
views
Proper monomorphisms in complex analytic spaces
In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ...
- 163
5
votes
1
answer
372
views
Special Kähler normal coordinates around a point
Let $(M,\omega)$ be a compact Kähler manifold and suppose there are holomorpic vector fields vanishing at a point $p$. As a consequence we have a group $G_{p}$ of biholomorpisms fixing $p$. Let $T_{p}$...
- 149
1
vote
0
answers
156
views
Exactness of the relative de Rham complex restricted to subschemes
I think that the statement below about relative de Rham complex is true (am I wrong?) If it is the case, a reference would be very helpful. (I admit that the statement sounds somewhat technical and ...
- 21.8k
0
votes
1
answer
304
views
Hilbert scheme of a closed subscheme
Let $X$ be a complex algebraic variety. Its Hilbert scheme represents the functor $G$ from schemes to sets given by $$G(S)=\{Z\subset X\times S|\, Z \mbox{ is a closed subscheme, flat and proper over }...
- 21.8k
1
vote
1
answer
252
views
Hilbert scheme of an infinitesimal neighborhood of a subvariety
Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the ...
- 21.8k
2
votes
1
answer
373
views
Does the "Ohsawa-Takegoshi theorem without bounds" have a name?
There are many theorems which now could be called "The Ohsawa-Takegoshi" theorem. Of these, the most basic is roughly the following:
Let $\Omega \subset \subset \mathbb{C}^n$ be a psuedoconvex ...
- 12.1k
4
votes
2
answers
700
views
Basic questions on the Hilbert scheme/ Douady space
Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More precisely,...
- 21.8k
9
votes
1
answer
3k
views
Complex geometry text/research introduction for the analyst
To give some background, I am mainly an analyst trained in harmonic/functional and do work on geometric pde's and spectral multipliers. Of late, I am trying to learn more about (research level) ...
- 141
4
votes
1
answer
229
views
Flat family with special fiber $\mathbb{C}\mathbb{P}^1$
Let $C=Spec \mathbb{C}[t]/(t^{n+1})$. Let $X$ be an algebraic (or complex analytic) scheme, flat over $C$ with the structure morphism $f\colon X\to C$. Assume that the special fiber is isomorphic to $\...
- 21.8k
2
votes
1
answer
518
views
When flatness of a morphism implies smoothness?
EDIT: Let $f\colon X\to C$ be a flat proper morphism of complex algebraic (or analytic) varieties. Assume the special fiber over a point $p\in C$ is smooth.
Is it true that there exists a ...
- 21.8k
1
vote
0
answers
436
views
A question related to the Grauert semi-continuity theorem
Let $f\colon X\to Y$ be a proper holomorphic map of complex analytic manifolds. Assume $f$ to be submersive for simplicity, but probably it is not important. Let $\mathcal{F}$ be a coherent sheaf on $...
- 21.8k
1
vote
1
answer
1k
views
A generalization of the Grauert direct image theorem
EDIT: Let $f\colon X\to Y$ be proper holomorphic submersive map of complex analytic manifolds. Let $\mathcal{F}$ be the sheaf of holomorphic sections of a holomorphic vector bundle over $X$. Assume ...
- 21.8k
0
votes
1
answer
148
views
Meromorphic extension of local defining equations of a complex submanifold
let $M$ be a smooth compact complex manifold of dimension $m$ and $N\subset M$ a smooth complex submanifold of dimension $1\leq n \leq m-2$. Covering $N$ with well chosen open sets of $M$ we can ...
- 1,727
8
votes
1
answer
734
views
Local polynomial form of holomorphic functions
It is well-known that a germ of a holomorphic function of $n\geq 2$ variables, with at most an isolated singularity (i.e. the singularity of the analytic variety defined by the zero locus of the ...
- 5,428
4
votes
1
answer
532
views
why do we need to study entire curves?
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 ...
- 758
5
votes
1
answer
668
views
Same betti numbers as $\Bbb{CP}^n$
I am sure that there is an answer out there for the following question. If one is given an n dimensional Kahler manifold $X$ with Betti numbers that are the same as in the case of $\Bbb{CP}^n$, then ...
- 1,201
8
votes
4
answers
1k
views
Monge Ampere equations
I am a graduate student trying to understand complex Monge-Ampere equations(mostly on complex manifolds with or without boundary, but also in C^n), but I can't put my hand on any monograph/textbook ...
- 1,201
13
votes
4
answers
1k
views
"Simple" Kahler manifolds
I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$...
- 4,343
8
votes
1
answer
638
views
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