Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

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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 ...
asv's user avatar
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3 votes
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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 ...
asv's user avatar
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8 votes
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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 } ...
asv's user avatar
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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 ...
Bilateral's user avatar
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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 ...
Sumanta's user avatar
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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: $0 \to \mathbb Z \to \mathcal O_X \to \mathcal O_X^* \to 0$ ...
Asvin's user avatar
  • 7,648
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 $n$-...
Sergio Charles's user avatar
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 $\mu_X, \mu_Y$...
Mozhgan Mirzaei's user avatar
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{...
Tony's user avatar
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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 ...
Flavius Aetius's user avatar
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 $\sim$ ...
D_S's user avatar
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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....
Florian R's user avatar
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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^...
Kevin's user avatar
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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 ...
sdn's user avatar
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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 ...
Evangelion045's user avatar
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 ...
BnPrs's user avatar
  • 195
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 \...
Saal Hardali's user avatar
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1 vote
0 answers
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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 ...
tst's user avatar
  • 483
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, \...
Pan's user avatar
  • 167
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 $d>0$ ...
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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 ...
user avatar
22 votes
1 answer
1k views

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 ...
diverietti's user avatar
  • 7,852
2 votes
0 answers
178 views

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, ...
user avatar
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 $\...
Lars Pettersen's user avatar
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 ...
Lars Pettersen's user avatar
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 ...
asv's user avatar
  • 21.1k
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(\...
user avatar
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 ...
Ali Taghavi's user avatar
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 ...
Joaquín Moraga's user avatar
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)=...
C.F.G's user avatar
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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 ...
user36931's user avatar
  • 1,331
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 $\...
Fallen Apart's user avatar
  • 1,605
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 $\mathbb{R}$?
wuzx's user avatar
  • 517
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 $G$-...
Porto Vino's user avatar
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 $\mathbb{S}^n$ ...
Eduardo Longa's user avatar
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: ...
Saal Hardali's user avatar
  • 7,549
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 $Y$ ...
Misha Verbitsky's user avatar
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 $S$ ...
Dima Sustretov's user avatar
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 $...
asv's user avatar
  • 21.1k
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 ...
erz's user avatar
  • 5,385
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 ...
asv's user avatar
  • 21.1k
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 $...
asv's user avatar
  • 21.1k
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 ...
user104109's user avatar
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 ...
Andrew McHugh's user avatar
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), ...
Dubious's user avatar
  • 1,237
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 $M$ ...
Jaikrishnan's user avatar
  • 1,149
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.
Jaikrishnan's user avatar
  • 1,149
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 ...
Jaikrishnan's user avatar
  • 1,149
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 ...
user avatar
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 ...
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