All Questions
Tagged with analytic-geometry complex-geometry
26
questions
37
votes
2
answers
2k
views
Residues in several complex variables
I am trying to educate myself about the basics of the theory of residues in several complex variables. As is usually written in the introduction in the textbooks on the topic, the situation is much ...
- 1,180
10
votes
1
answer
312
views
Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?
Is there a closed subscheme $D$ in $\mathbb P^2_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map
$$ \phi: X(\mathbb C) \to \mathbb P^...
- 103
8
votes
1
answer
420
views
Easiest proof for showing finite etale (analytic) quotients of algebraic varieties are algebraic
Let $X$ be an algebraic variety over $\mathbb C$. Let $X^{an}\to Y$ be a finite etale morphism with $Y$ a complex analytic space.
I read somewhere that $Y$ algebraizes, ie, $Y=V^{an}$ for some ...
- 81
8
votes
1
answer
498
views
Connectivity of complements of Stein opens
Let $Y$ be an affine open subset of a locally noetherian scheme $X$. Then, $X \setminus Y$ has pure codimension one [EGAIV$_4$, Cor. 21.12.7]. Moreover, if $X$ is proper and of finite type over a ...
- 1,598
6
votes
1
answer
520
views
Intersection theory in analytic geometry
This might be a weird/stupid question, but it came to me a couple of times, and I would like to get an answer for that.
In some papers I read, constantly the authors define some analytic subspaces, ...
- 287
6
votes
0
answers
531
views
Pseudo-effective divisor which is not nef in any birational model
Let $X$ be a smooth complex projective algebraic variety and let $D$ be a $\mathbb{Q}$-Cartier pseudo-effective divisor on $X$. Lets say that $D$ is birationally nef
if there exists a birational ...
- 1,585
5
votes
0
answers
73
views
Subadditivity of multiplier ideals with a pluriharmonic function
I would like to have a reference for the following two facts (if true):
Let $D$ be a nef and big divisor on an algebraic variety $X$ and $h$ a Hermitian metric with minimal singularities on $D$, ...
- 1,585
4
votes
1
answer
250
views
Can an analytic variety extend along a codimension 2 subvariety?
Let $X$ be a smooth, connected, complex analytic variety, and $Y\subset X$ a closed, analytic subvariety of codimension at least 2. Now let $V\subset X\backslash Y$ be a closed, analytic subvariety. ...
- 2,756
4
votes
1
answer
247
views
English reference for Douady/Grauert construction of versal deformations of compact complex spaces
I'm trying to learn about the deformation theory of compact complex spaces. I'm familiar with the case of compact complex manifolds from the paper "On the Locally Complete Families of Complex ...
- 1,721
4
votes
1
answer
562
views
Simple maps: Flat versus locally trivial
In deformation of complex analytic spaces, one usually considers an analytic proper simple surjective map $\varpi: \mathscr{M} \twoheadrightarrow \mathscr{P}$ as an analytic family. However the term ...
- 2,169
3
votes
1
answer
317
views
Is a compact subset of a Stein space admitting a fundamental system of Stein neighbourhoods necessarily holomorphically convex?
Let X be a Stein manifold and let K be a compact subset of X. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Then, it is a result by Rossi that such a ...
- 169
3
votes
0
answers
65
views
Intersection of Stein opens admits a Stein neighborhood basis?
Let $X$ be a Stein manifold, $K$ be a compact subset of $X$. Consider the following conditions:
1.$K$ admit an open neighborhood basis in $X$ whose members are Stein;
2.$K=\cap_{j\ge 1}V_j$, where $...
- 423
3
votes
0
answers
486
views
Regularity of fiber integration between complex analytic spaces
Let $f:X\rightarrow Y$ be a flat surjective morphism between reduced complex analytic spaces. Assume that $Y$ is locally irreducible (i.e. unibranch).
We assume that $X$ (resp. $Y$) is pure-...
- 329
3
votes
0
answers
207
views
Does constructible and analytically open imply Zariski open
Let $U$ be a constructible subset of a complex algebraic variety. Is the following statement true?
If $U$ is open in the analytic topology, then $U$ is open in the Zariski topology on $X$.
- 31
3
votes
0
answers
164
views
Universal covering space of a Zariski open subset of projective space
Let $U$ be a Zariski open subset of $\mathbb P^n_{\mathbb C}$. Assume $U$ is the complement of some divisors.
Have the possible universal covering spaces of $U$ been classified?
Do we know when the ...
- 71
2
votes
2
answers
1k
views
Conformal mappings that preserve angles and areas but not perimeters?
Conformal mappings from $U$ to $V$, both subsets of $\mathbb{C}$, locally preserve angles.
But, in general, such mappings neither preserve areas nor preserve perimeters.
Q. Are there examples of ...
- 149k
2
votes
1
answer
199
views
Two definitions of Teichmüller space: relative isotopy or not?
The definition of Teichmüller space on wikipedia via marked Riemann surfaces say that two markings are equivalent if the map $fg^{-1}$ is isotopic to a holomorphic diffeomorphism.
The definition on ...
- 1,569
2
votes
1
answer
343
views
Is the closure of an open holomorphically convex subset of a Stein space holomorphically convex?
Let X be a Stein manifold and U an open, connected, relatively compact, holomorphically convex subset of X. Is the closure of U in X holomorphically convex?
Also, if X is a Stein space with a finite ...
- 169
2
votes
0
answers
446
views
Descent for complex-analytic spaces
I'm basically interested in knowing the difference between complex spaces and schemes when studying stacks. I'd like to use stacks to study moduli problems in complex analytic geometry.
Citing a ...
- 373
1
vote
0
answers
40
views
Pullback of coherent sheaves on Stein manifolds
Let $i:X\to Y$ be a closed embedding of Stein spaces, $G$ be a coherent $O_Y$-module. Set $I=\ker(i^*:O(Y)\to O(X))$. Then $I$ is an ideal of the ring $O(Y)$. Is that true that $\Gamma(X,i^*G)=G(Y)/...
- 423
1
vote
0
answers
70
views
Representatives of line bundle cohomology over tori
Let $V^n$ a be a $\mathbb{C}$-vector space. For $U\subset V$ a complete lattice, the holomorphic line bundles over $V/U$ are classified (see e.g. `Abelian varieties', D. Mumford) by data $(H,\alpha)$ ...
1
vote
0
answers
212
views
Necessity of cohomological flatness for the Picard functor
Let $f:X\rightarrow S$ be a proper, flat morphism of complex analytic spaces and let $Pic_{X/S}(T)=H^0(T,R^1 {f_T}_*(\mathcal{O}^*_{X_T}))$ be the relative Picard functor. Here $X_T= X\times_S T$.
...
- 373
1
vote
0
answers
171
views
Is there an analytic criterion for quasi-compactness of a scheme?
Let $X$ be a locally finite type scheme over $\mathbb C$.
I'm looking for the analogue of the notion "finite type" for $X^{an}$ and an SGA 1 Exp. XII type of criterion which says that
The scheme $X$...
- 11
0
votes
1
answer
158
views
The intersection number $C\cdot D=\deg(D_{/C})$
Let $S$ be an algebraic complex surface, and $D=[(U_\alpha,f_{\alpha})]$ is a Cartier divisor over $S$, and let $\cal{O}_S(D)$ be the sheaf associated to $D$. And let $C$ be a complex compact curve in ...
- 117
0
votes
0
answers
81
views
Good covering of a (singular) curve in a complex surface
Let $W$ be a $2$-dimensional complex manifold and $C\subset W$ a compact complex curve (possibly singular). I would like to know a reference for the following fact: there exists a collection $\{V_j\}...
- 1,185
-3
votes
1
answer
200
views
Conformal map from a 7-sided polyhedron to a square pyramid
I have a right-angled square pyramid, $A$, whose height and base-length are $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
- 167