Questions tagged [complex-manifolds]
For questions about or involving complex manifolds.
351
questions
6
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1
answer
313
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Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$
Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.)
Is it true that for any ...
3
votes
0
answers
151
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 ...
8
votes
0
answers
275
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 } ...
1
vote
1
answer
120
views
Minimal complex surfaces with pseudo-effective canonical bundles
A complex line bundle $L$ over a complex surface $X$ is said to be pseudo-effective if it admits a (possibly singular) Hermitian metric $h$ whose curvature is positive semi-definite in the sense of ...
1
vote
0
answers
436
views
Path lifting property of holomorphic unbranched map
Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on ...
3
votes
0
answers
313
views
Finding the torsion of the Neron Severi group in the first homology group
Let X be a variety over $\mathbb C$. I will implicitly identify this with us complex analytification. Consider the exponential sequence:
$0 \to \mathbb Z \to \mathcal O_X \to \mathcal O_X^* \to 0$
...
7
votes
1
answer
630
views
How does one complexify a real $n$-dimensional Riemannian manifold $(M,g)$?
If $V$ is a real vector space, then the complexification of $V$ is formally defined as $V^{\mathbb{C}}=V\otimes_{\mathbb{R}}\mathbb{C}$. Is there an analogous complexification operation for a real $n$-...
7
votes
0
answers
247
views
When is Radon-Nikodym derivative induced by a proper map of manifolds bounded?
Let $X,Y$, be compact complex manifolds, and let $f:X\to Y$ be a smooth, proper (i.e. for each $y\in Y$, $f^{-1}(y)$ is a compact set) and surjective map. Choose metrics on $X,Y$ and let $\mu_X, \mu_Y$...
5
votes
0
answers
130
views
Criterium for algebraicity of an analytic map
Let $X$ and $Y$ be algebraic varieties over $\mathbb{C}$. Let $f:X^{an}\to Y^{an}$ be a holomorphic map.
Is the following statement correct?
If there is an algebraic variety $V$ over $\mathbb{...
5
votes
0
answers
215
views
Lagrangian foliation for a holomorphic symplectic manifold
I am interested in gathering as many examples as possible for Lagrangian foliations of holomorphically symplectic manifolds $(X, \omega)$, where $X$ is a $2n$-dimensional complex manifold equipped ...
4
votes
1
answer
359
views
Definition of quotient manifolds, and $\Gamma \backslash \mathscr H$ as a quotient manifold
I have just encountered some subtlety with quotient manifolds and now I don't think I understand some things as well as I thought I did.
Let $X$ be a real or complex analytic manifold, and $\sim$ ...
5
votes
1
answer
819
views
Clarification on Beltrami Differentials
I have troubles with the theory of existence of quasi-conformal homeomorphisms realizing Beltrami coefficients. Let $X$ be a (compact) Riemann surface and $f \colon X \rightarrow \mathbb{C}$ be smooth....
1
vote
1
answer
137
views
Upper bound of the dimension of automorphism group of compact Kähler manifolds
It is well-known that the dimension of the isometry group of an $n$-dimensional compact Riemannian manifold is no larger than $\frac{1}{2}n(n+1)$, which is attained precisely by $S^n$ and $\mathbb{R}P^...
2
votes
1
answer
185
views
Saddle-point method for complex functional of real field
If I have a path integral of the form
$$\int D\phi e^{i S}$$
where the action S is a complex functional of a real-valued field $\phi$, say
$$S=\int dt ( \phi^2 + i \phi )$$
what is the correct way ...
6
votes
0
answers
250
views
Holomorphic vector fields and derivations
Let $M$ be a complex manifold and $U\subset M$ a domain.
Question: Is every derivation of the complex algebra of holomorphic functions $\mathcal{O}(U)$ induced by a holomorphic vector field defined ...
3
votes
0
answers
160
views
Duality of Mixed Hodge Structures without compactness
Let $X$ be a smooth separated algebraic variety over $\mathbb{C}$ and $Z \subset X$ a subvariety of codimension $p$. There are no compactness assumptions. I am looking for an isomorphism of mixed ...
10
votes
1
answer
376
views
Is every endomorphism of the sheaf of holomorphic functions on a disk a differential operator?
Let $D= \{z\in \mathbb{C}:|z| < 1\}$ be the unit disk. And consider the sheaf of holomorphic functions $\mathcal{O}_{D}$.
Question (?) : Is there a sheaf endomorphisms $\phi : \mathcal{O}_D \to \...
1
vote
0
answers
34
views
First return map in complex 2DOF Hamiltonian systems
The standard way to construct the first return map around a periodic orbit in real 2DOF Hamiltonian systems is the following:
We choose a periodic orbit and a point on it.
We restrict the system on ...
7
votes
1
answer
451
views
Elliptic operator on compact Hermitian manifold
Let $X^n$ be a compact complex manifold, and $\omega$ be a Hermitian metric on $X$.
Define an operator $P:=i\Lambda_\omega \bar{\partial} \partial$ on the space of the smooth function $C^\infty(X, \...
4
votes
0
answers
299
views
About the exponential sequence
For a complex analytic space $X$, we have the exponential sequence
$$0\to\mathbf{Z}(1)_X\to\mathcal{O}_X\to\mathcal{O}_X^{\times}\to 1$$
the last map being the exponential $\text{exp}$.
For $d>0$ ...
0
votes
1
answer
211
views
What is a sufficient condition for summability of formel power series? [closed]
There are several kind of summability , i accrossed differents conditions for applying for example Borel summation or laplace transform which let me mixed and confused , really i don't know if i have ...
22
votes
1
answer
1k
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Relationship between the signs of different notions of curvature in complex geometry
Let $(X,\omega)$ be a complex hermitian manifold, and call $\Theta$ its Chern curvature tensor. Out of this we can consider different notions of curvature, namely the holomorphic bisectional curvature ...
2
votes
0
answers
178
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Stein subspaces of polydiscs and balls
Let $D$ be a either an open polydisc or an open ball in $\mathbf{C}^n$.
(1) Let $\mathcal{O}$ be the $\mathbf{C}$-algebra of holomorphic functions on $\mathbf{C}^n$, resp. $D$, and let $f_1,\ldots, ...
1
vote
1
answer
653
views
de Rham closed harmonic form on a Kähler manifold
For a compact Kähler manifold, we say that a form is primitive if it is contaned in the kernel of the dual Lefschetz operator, or the co-Lefschetz operator. For all examples I know, a primitive form $\...
4
votes
0
answers
149
views
Hermitian and Kähler Metrics for Projective Stiefel Manifolds
Let $V(n,r)$ denote the complex Stiefel manifold of orthonormal complex
$r$-frames in complex $\mathbb{C}^n$-space. Each $V(n,r)$ admits a canonical $U(1)$-action, and the quotient with respect to ...
4
votes
1
answer
276
views
Shrinking the boundary of a Riemann surface
Let $X$ be a compact Riemann surface with boundary. Let us shrink each connected component of the boundary into a point. We get a closed topological surface $Z$ with several marked points (which came ...
2
votes
0
answers
200
views
Is every monotone Lagrangian Hamiltonian isotopic to minimal Lagrangian?
Assume we have a closed Lagrangian submanifold $L$ in Kaehler-Einstein manifold of positive scalar curvature (for instance, complex projective space). Dazord has proved that 1-form $\alpha=\omega(\...
2
votes
1
answer
221
views
Real diffeomeorphism preserving the space of Holomorphic vector fields
Assume that $M$ is a complex manifold.
Let $G$ be the group of all (real) smooth diffeomorphisms $\phi$ of $M$ such that $\phi^* (X)$ is a holomorphic vector field for all holomorphic ...
3
votes
0
answers
104
views
Restriction of a singular metric with minimal singularities
Let $X $ be a smooth complex algebraic variety and $L $ a pseudo-effective line bundle on $X $, consider $h $ to be a singular Hermitian metric with minimal singularities on $L$ and $|A|$ be the ...
3
votes
1
answer
157
views
Almost Hermitian manifolds of constant curvature
Let $(M,g)$ be a $2n$-dimensional almost Hermitian manifold ($n\geq 2$) with a almost complex structure $\cal J$ (not necessary integrable). i.e.,
$${\cal J}^2=-I,\quad\qquad g({\cal J} X,{\cal J} Y)=...
2
votes
0
answers
83
views
Is the affine hypersurface defined by ((xz+1)^2/z)-((yz+1)^3/z)+xyz=1 a "known" hypersurface?
The hypersurface $\lbrace (x,y,z) \in \mathbb{C}^3:\frac{(xz+1)^2}{z}-\frac{(yz+1)^3}{z}=1 \rbrace$ is a well-known example of a contractible hypersurface in $\mathbb{C}^3$. See for instance, Example ...
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 $\...
12
votes
2
answers
2k
views
Can the group of holomorphic automorphisms of an open subset of the complex plane be isomorphic to the additive group of real numbers?
We can construct open sets in the complex plane $\mathbb{C}$ whose automorphism group is isomorphic to $\mathbb{Z}$, but is there an open set whose automorphism group is isomorphic to $\mathbb{R}$?
1
vote
0
answers
160
views
On $G$-gerbes over the punctured disk
Let $G$ be a finite (not necessarily abelian) group and let $\mathcal{X}\to D^*$ be a $G$-gerbe over the punctured disk $D^*$.
Is there a finite etale cover $D^*\to \mathcal{X}$?
I think of $G$-...
1
vote
0
answers
64
views
Largest elementary neighbourhood
The real projective space $\mathbb{R}P^n$ can be defined as the quotient space of $\mathbb{S}^n$by the equivalence relation that identifies antipodal points. The largest open set of $\mathbb{S}^n$ ...
6
votes
2
answers
1k
views
Proof of the holomorphic Frobenius theorem in Voisin's book on Hodge theory (Theorem 2.26)
I'm trying to understand the proof of the holomorphic version of the Frobenius integrability theorem given in p. 51-52 of Voisin's text "Hodge Theory and Complex Algebraic Geometry I".
Statement: ...
5
votes
2
answers
351
views
fibers of birational contraction for complex manifolds - are they Moishezon?
Let $X$ be a smooth complex manifold and
$\phi:\; X \mapsto Y$ a proper holomorphic
map which is birational ("birational contraction"),
and $Z= \phi^{-1}(y)$ its fiber in a point $y$.
The variety $Y$ ...
8
votes
1
answer
287
views
Families of curves on compact complex surfaces and algebraicity
Let $S$ be a compact complex manifold of dimension $2$ and assume that there exists a two-dimensional family of curves on $S$. Is it true then that the algebraic dimension of $S$ is $2$, i.e. that $S$ ...
6
votes
0
answers
202
views
A topological property of flat morphisms
Let $f\colon X\to Y$ be a faithfully flat morphism of smooth projective varieties over $\mathbb{C}$. Assume that the generic fiber of $f$ is smooth. Then there exists a non-empty Zariski open subset $...
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 ...
6
votes
0
answers
170
views
A relation of convergence in Hilbert scheme to convergence in sense of currents
Let $\{X_i\}$ be a sequence of closed irreducible $k$-dimensional subvarieties of $\mathbb{C}\mathbb{P}^n$ of degree $d$ (they may be assumed to be smooth if necessary). Assume that this sequence ...
0
votes
0
answers
142
views
Limit of a sequence of smooth varieties in Hilbert scheme
Let $\{Z_i\}_{i=1}^\infty$ be a sequence of smooth irreducible $k$-dimensional submanifolds of $\mathbb{C}\mathbb{P}^n$ which converges to a closed subscheme $Z$ in the sense of the Hilbert scheme of $...
8
votes
2
answers
314
views
Vanishing of Aronhold S-invariant on the cubic forms on $H^2(X, \mathbb Q)$
I am considering several examples of compact complex threefolds $X$ such that $rk H^2(X)=3$.
Note that we have a cubic form on $H^2(X, \mathbb Q)$ which comes from the cup product.
I calculated the ...
2
votes
1
answer
314
views
Are there compact complex manifolds with non-constant pluriclosed functions?
What are some examples of a (connected) compact complex manifold with a non-constant global complex valued function, $f$, such that
$\partial {\bar{\partial}} f = 0$.
In other words, what are examples ...
2
votes
0
answers
1k
views
Chern Classes: two approaches
The following question is closely related to this one.
Let $X$ a non singular projective variety over $\mathbb C$, and let $\mathscr E$ a locally free sheaf of rank $r$ (an algebraic vector bundle), ...
3
votes
0
answers
60
views
Tautly embedding a ball quotient
Let $\mathbb{B}^n$ denote the unit ball in $\mathbb{C}^n$. Let $\Gamma$ be a fixed-point free, discrete subgroup of the automorphism group of $\mathbb{B}^n$. Let $M := \mathbb{B}^n / \Gamma$. Then $M$ ...
5
votes
1
answer
163
views
Natural classes of compact complex manifolds whose universal covers are bounded domains
What are some general classes of compact complex manifolds whose universal covers are bounded domains? One class I know are the Kodaira fibered surfaces.
4
votes
0
answers
144
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 ...
14
votes
1
answer
876
views
Reference for symplectic structures on schemes?
My original goal was to read the PTVV paper Shifted Symplectic Structures https://arxiv.org/pdf/1111.3209v4.pdf. I was quickly humbled!
Being told the theory ought to generalize symplectic structures ...
28
votes
7
answers
7k
views
Elementary Proof of Riemann-Roch for Compact Riemann Surfaces
I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps ...