All Questions
Tagged with complex-manifolds cv.complex-variables
32 questions with no upvoted or accepted answers
16
votes
0
answers
519
views
Gabriel's theorem for complex analytic spaces
Let $X,Y$ be noetherian schemes over $\mathbb{C}$.
Then, it is known that
$$
\text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y,
$$
by P. Gabriel(1962).
Are there some results in the case of ...
- 889
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
8
votes
0
answers
964
views
Etymology of the O-notation for algebras of holomorphic functions
The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...
- 1,111
7
votes
0
answers
129
views
holomorphic functions on Stein manifolds reaching maxima in a given point on the boundary (Shilov boundary)
Let $M$ be a Stein manifold with smooth, strictly
pseudoconvex boundary, and $x$ a point on its
boundary. Is there a holomorphic function $f$ on
$M$, smooth on the boundary, with strict
maximum of $|f|...
- 9,237
5
votes
0
answers
189
views
Extension of holomorphic maps to smooth family of holomorphic maps
Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a ...
- 1,409
4
votes
0
answers
197
views
Approximation of a holomorphic function vanishing at a submanifold by polynomials
Suppose $M$ is a complex affine algebraic manifold in ${\mathbb C}^n$ (I mean, a set of common zeroes of a system of polynomials on ${\mathbb C}^n$, which is at the same time a smooth manifold). ...
- 7,426
4
votes
0
answers
229
views
Real part of a holomorphic section of a vector bundle
Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic ...
- 163
4
votes
0
answers
70
views
Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains
Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and ...
- 83
4
votes
0
answers
148
views
Universal cover of Kodaira surface
From an earlier question, the universal cover of a Kodaira fibered surface is a bounded domain in $\mathbb{C}^2$. It is also not the polydisk or the ball. Can we say more about the structure of the ...
- 1,159
3
votes
0
answers
119
views
Basic obstruction to anything like holomophic symmetric functions of infinitely many variables?
The totality of all holomorphic functions on the unit disk forms some sort of infinite-dimensional complex manifold, where the coefficients of the Taylor expansion might serve as coordinates for the ...
- 17.6k
3
votes
0
answers
179
views
Topology of level sets for meromorphic function
Let $F$ be a meromorphic function on $\mathbb{C}$.
I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ ...
- 1,352
3
votes
0
answers
85
views
Can a punctured ball $(B\setminus\{0\})\subset\mathbb{C}^n$ be foliated by complete leaves?
Recently Antonio Alarcón proved that in the case of the unit ball $B\subset\mathbb{C}^n$ for $n\geq 2$ every smooth closed complex submanifold of dimension $q\leq n$, $V\subset\mathbb{C}^n$ defines a ...
- 166
3
votes
0
answers
167
views
Does the maximum principle hold in this pluriharmonic setting?
Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and ...
- 5,215
3
votes
0
answers
235
views
Chern number of projection-Topological magic in physics
I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...
- 31
3
votes
0
answers
169
views
Cohomology of a tubular neiborhood of submanifold vs cohomology of the formal neighborhood
Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$.
Is ...
- 21.8k
3
votes
0
answers
94
views
Isotropy symmetric holomorphic functions
Let $G$ be a bounded homogeneous domain in $\mathbb{C}^{n}$ and let $z\in G$.
Assume that $f$ is a holomorphic function on $G$, which is isotropy symmetric, i.e. $f\circ \varphi=f$ for any ...
- 5,529
3
votes
0
answers
90
views
Two questions on homogeneous domains
Let $G$ be a domain in $\mathbb{C}^{n}$ and let $Aut(G)$ be the group of biholomorphic selfmaps of $G$. $G$ is called:
(1) homogeneous if $Aut(G)$ acts transitively on $G$, i.e. for any $z,w\in G$ ...
- 5,529
2
votes
0
answers
53
views
Approximating an infinite family of holomorphic functions by polynomials in relative error
I think I just proved a theorem I haven't found in the literature, and I think it must generalize. I therefore have two questions. First, if this is in the literature, what is it called? Second, what ...
- 981
2
votes
0
answers
253
views
cohomology classes of complex submanifolds
I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be.
For example, say $T^4$ is regarded as a complex ...
2
votes
0
answers
99
views
What's the behavior of the laplacian of dbar operator w.r.t. a singular metric of a holomorphic line bundle?
What's the behavior of the laplacian of dbar operator w.r.t. a singular metric of a holomorphic line bundle or other holomorphic vector bundle over a complex manifold ?Do we have anything similar ...
- 391
1
vote
0
answers
155
views
Top cohomology of the canonical class of a compact non-Kähler manifold
Let $X$ be a complex compact manifold of complex dimension $n$. Let $K_X$ denote its canonical class. Is it true that the cohomology group
$$H^n(X,K_X)$$
is one dimensional?
Remark. If $X$ is Kähler ...
- 21.8k
1
vote
0
answers
112
views
Quasi-plurisubharmonic function with polynomial decay
Let $(M, \omega)$ be a compact Kähler manifold. An $\omega$-quasi-plurisubharmonic function on $M$ is an upper semi-continuous function $\varphi : M \to \mathbb{R} \cup \{ - \infty \}$ such that $\...
- 651
1
vote
0
answers
234
views
Riemann's bilinear relations
I am reading the paper [1], which states
Haupt showed that a vector with complex entries $(w_1, \cdots, w_g, z_1, \cdots, z_g)$ is the period row of some holomorphic differential with respect to a ...
- 11
1
vote
0
answers
137
views
Does $\mathfrak{m}_z/\mathfrak{m}_z^2\cong\overline{\mathfrak{m}_z}/\overline{\mathfrak{m}_z}^2 $ on all complex manifolds?
Let $M$ be a complex manifold with its sheaf $\mathcal{O}_M$ of holomorphic functions.
Fix a point $z\in M$ and denote by $\mathcal{O}_z$ the stalk of $\mathcal{O}_M$ at $z.$
Cosider ideals $\...
- 1,615
1
vote
0
answers
217
views
Homeomorphism of fibers of holomorphic maps
EDIT (after the comment by Jason Starr): Let $X$ be a complex algebraic (or, more generally, analytic) variety, possibly singular and non-compact. Let $f\colon X\to D^*$ be a proper algebraic morphism ...
- 21.8k
1
vote
0
answers
131
views
Hurwitz's theorem for a system of functions
First, let me define a notation of $H(G_1\times G_2 \times \ldots \times G_m)$.
We say that $f\in H(G_1\times G_2 \times \ldots \times G_m)$ if $$f:G_1\times G_2 \times \ldots \times G_m \rightarrow \...
- 11
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
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
0
answers
234
views
glue together a sequence of holomorphic forms
hallo,
my problem is the following: i have a finite sequence of holomorphic $k-$forms $\alpha_{k}$, each defined on open subsets $U_{k} \subset M$, where $M$ is a complex $n$-dimensional manifold, ...
- 19
0
votes
0
answers
144
views
Function of several complex variables with prescribed zeros
I accidentally stumbled upon a problem of complex analysis in several variables, and I have a hard time understanding what I read, it might be related to the Cousin II problem but I cannot say for ...
- 1,352
0
votes
0
answers
39
views
Contraction of an inclusion with respect to Kobayshi hyperbolic metric
Suppose that $X = \mathbb{C}^n - \Delta_X$ and $Y = \mathbb{C}^n - \Delta_Y$, where $\Delta_X$ and $\Delta_Y$ are unions of hyperplanes in $\mathbb{C}^n$ such that $\Delta_Y \subset \Delta_X$, $\...
- 41
0
votes
0
answers
200
views
“Holomorphic” bump function
I was wondering in what sense can I construct a holomorphic “bump function”? Now, of course we cannot really construct a holomorphic bump function in the usual sense, but I have a much rougher idea in ...
- 502