Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

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Explicit formula for complex structure on flag manifold/isospectral matrices?

Consider the flag manifold $M = U(n, \mathbb{C})/T^n$, where $T^n$ is the maximal torus of unitary diagonal matrices. Fixing a diagonal matrix $D$ with distinct reals on its diagonal, we can identify ...
ccriscitiello's user avatar
2 votes
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Hodge bundles associated to a family of complex manifolds

I'm reading Voisin's books on Hodge theory. In the first volume she claimed but didn't prove this theorem: Theorem 10.10 (Voisin) Let $\varphi:\chi\rightarrow B$ be a family of compact complex ...
WDDYz's user avatar
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Simply connectedness of leaves of a foliation on an complex manifold

Now I'm searching about leaves of foliation in the following special setting. Let $U,V$ be two holomorphic vector field on $\mathbb{C}^2$ s.t the Lie bracket $[U,V]=UV-VU=0$ and $U$ and $V$ spaned ...
George's user avatar
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4 votes
1 answer
317 views

Residues and blow ups

On a 2-dimensional complex manifold consider two functions which are meromorphic with singularities along two divisors which meet at a point. There is a residue from these meromorphic functions (...
Edwin Beggs's user avatar
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Kähler manifold with negative sectional curvature

Goldberg's theorem states that every almost Kähler manifold of constant curvature is Kähler if and only if the curvature is zero. This seems to contradict the fact that the sectional curvature of the ...
Samir's user avatar
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“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 ...
JustSomeGuy's user avatar
3 votes
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Action of complex Lie group on Dolbeault cohomology

Let $M$ be a compact complex manifold acted holomorphically by a complex Lie group $G$. Let $F$ be a holomorphic $G$-equivariant vector bundle over $M$. Consider the natural representation of $G$ in (...
asv's user avatar
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An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$

It is known that the only spheres that admit an almost complex structures are $\Bbb S^2$ and $\Bbb S^6$ (Borel and Serre, 1953). In particular, $\Bbb S^4$ cannot be given an almost complex structure (...
Random's user avatar
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Kähler metric on the projective space

"Is there a Kähler metric on the complex projective space $\mathbb {P} ^n(\mathbb {C} ) $ different from the Fubini-Study metric?
Samir's user avatar
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Non-Kähler complex structure on $S^2 \times T^4$

Consider $M = S^2 \times T^4$. Then we can construct a non-Kähler complex structure as follows. Let $L$ be a line bundle over $\mathbb{P}^1$ such that there are two sections $s_1, s_2 \in H^0 (L)$ ...
Chicken feed's user avatar
3 votes
1 answer
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Does Hermite-Einstein imply Kähler-Einstein?

Let $M$ be a compact Kähler manifold and let $\nabla$ be its Levi-Civita, or equivalently its Chern, connection. Denoting the vector bundle of complexified one forms of $M$ by $\Omega^1_{\mathbb{C}}$, ...
Didier de Montblazon's user avatar
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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 ...
David Feldman's user avatar
2 votes
1 answer
371 views

Uniformization of $\mathbb{CP}^2-\bigcup C_i$, where $C_i$ are Riemann surfaces intersecting generically

Consider $X=\mathbb{CP}^2-\bigcup C_i$ where $C_i$ are Riemann surfaces intersecting generically. How to compute the fundamental group of this space and what is the universal cover?
0x11111's user avatar
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Integration on algebraic curves

Consider the plane algebraic curve $$f(x, y) = y^4 - (2x - 1)y^2 - (4x - 1) y + x^2 + x + 1 = 0.\tag{1}$$ Its compactification results in a Riemann surface $C_1$ of genus $1$. Hence, it can be ...
mxjia's user avatar
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1 answer
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Common holomorphic forms for two distinct complex structures

Let $S$ be a closed real surface having two complex structures $c_1$ and $c_2$ which are not biholomorphic (so $S$ is a Riemann surface with genus at least 1). Consider $\omega$ a 1-form on $S$ which ...
Dorian's user avatar
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Symmetric group-cocycle descends to symmetric product

Let $C$ be a complex curve with universal covering $\tilde{C}$ (which in my case is the upper half plane). Any group-cocylce $e \in H^1(\pi_1(C^n),H^0(\tilde{C}{}^n,\mathcal{O}^{\times}))$ defines a ...
KuSi's user avatar
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1 answer
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Universal covering of symmetric product

Let $C$ be a 1-dimensional complex manifold whose universal covering is provided by the half-plane $\mathcal{H}=\{z \in \mathbb{C} \mid \operatorname{Im}z>0\}$. The symmetric product $C^{(n)} = C^n ...
KuSi's user avatar
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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 $E_\varepsilon$ ...
kaleidoscop's user avatar
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Hermitian locally symmetric space with nonnegative bisectional curvature

Let $(M,g)$ be an Hermitian locally symmetric space with nonnegative bisectional curvature. Suppose the fundamental group of $M$ is finite, can we prove that $M$ is simply-connected?
Zhiqiang's user avatar
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Do we have uniformization theorems for fractional dimensional spaces?

The Riemann mapping theorem in $\mathbb{R}^2$ is known not to generalize well in higher dimensions and is basically trivial in lower dimensions. I’m interested in how it generalizes for fractional ...
Sidharth Ghoshal's user avatar
1 vote
0 answers
142 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 ...
asv's user avatar
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7 votes
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When $H^{p,q}_{\bar\partial}(X)$ can be seen as a subspace of $H^k(X,\mathbb C)$?

It is known that for a $\partial\bar\partial$-manifold $X$ (a compact complex manifold satisfies the $\partial\bar\partial$-lemma), the Bott-Chern cohomology $H_{BC}^{\bullet,\bullet}:=\frac{\ker \...
Tom's user avatar
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Is there a Cauchy integral formula for complex manifolds?

Is there a Cauchy integral formula for holomorphic functions on complex manifolds?
Mathgrad's user avatar
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2 answers
321 views

Contractible real analytic varieties

If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point? Here a real analytic variety is the set of zeros of a real analytic ...
Brian Lins's user avatar
2 votes
1 answer
160 views

Teichmuller interpretation of unbounded holomorphic quadratic differentials

For a closed Riemann surface $\Sigma$ of genus $g \geq 2$, the space of holomorphic quadratic differentials on $\Sigma$ can be identified with the cotangent space $T_\Sigma^* \mathcal{T}_g$: in other ...
Leo Moos's user avatar
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Complex structure on the product of two real torus

Let $T^{2n}_{\mathbb{R}}$ be a real torus of dimension $2n$, and let $Z_n$ be the space consisting of all possible complex structures on $T^{2n}_{\mathbb{R}}$. It is known that: $$Z_n = \mathrm{GL}(2n,...
Chicken feed's user avatar
2 votes
1 answer
200 views

Is there a maximum principle for CR functions over domains inside CR manifolds?

I am new to this area and I am a bit confused by the literature. Is there a strong maximum principle for CR functions over domains in a CR manifold, please? If so, could someone please state it (...
Malkoun's user avatar
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1 answer
271 views

Given a smooth hyperplane section Y of a variety X there exists a Lefschetz pencil of hyperplane sections of X containing Y

Let $X$ be a variety contained in $\mathbb{P}^N$ and let $Y$ be a smooth hyperplane section of $X$. I have read in page 54 of Voisin's book "Hodge theory and complex algebraic geometry II" ...
Roxana's user avatar
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3 answers
300 views

Classification of surface bundles over surfaces

Can anyone recommend one place or a few places that describe what is known about the classification of (real) surface bundles over (real) surfaces? Now, if the fibre F and the base B are both ...
Daniel Asimov's user avatar
5 votes
1 answer
319 views

Top integer homology of compact analytic variety

Let $V$ be a compact connected complex analytic subvariety (possibly singular) of a complex smooth manifold. Let $n$ denote its complex dimension. Is it true that $H_{2n}(V,\mathbb{Z})\simeq \mathbb{Z}...
asv's user avatar
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3 votes
0 answers
76 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 $...
Doug Liu's user avatar
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3 votes
0 answers
218 views

Kawamata BPF applied to a semi-positive line bundle using Demailly's holomorphic Morse inequalities

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic ...
Misha Verbitsky's user avatar
2 votes
1 answer
112 views

Vertical Fourier decomposition for skew-Hermitian 1-forms

In an arXiv preprint [2108.05125v1], the authors use the following vertical Fourier decomposition (page 7 therein). Let $(M,g)$ be a Riemannian surface and $SM$ be its unit tangent bundle. Denote by $...
Florian R's user avatar
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4 votes
0 answers
122 views

Is there a projective bundle formula for Deligne cohomology?

Given a projective bundle $\mathbb{P}(E) \to X$ on a complex manifold $X$, is there a projective bundle formula for Deligne cohomology? That is, can Deligne cohomology $H_D^n(\mathbb{P}(E),\mathbb{Z}(...
K.M.'s user avatar
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1 vote
0 answers
167 views

Betti numbers of threefolds with trivial canonical class

I am interested in a simply-connected compact complex manifold $M$ of dimension three with trivial canonical class. Note that if it is K"ahler, then it is a Calabi-Yau threefold. Its independent ...
Basics's user avatar
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5 votes
1 answer
431 views

Threefolds with the same Betti numbers and the same Chern numbers

By a threefold, I mean a compact complex manifold of dimension three. My question is a simple one: Are there known INFINITELY many non-homeomorphic threefolds that have the same Betti numbers and the ...
Basics's user avatar
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3 votes
0 answers
210 views

Complex manifold with conjugate complex structure

Let $(M,J)$ be a complex manifold with complex structure $J$. It is clear that $(M,-J)$ is also a complex manifold. Under what condition is $(M,J)$ biholomorphic to $(M,-J)$?
Adterram's user avatar
  • 1,361
7 votes
1 answer
473 views

Do non-projective K3 surfaces have rational curves?

Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K_X \simeq \mathcal{O}_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov ...
AmorFati's user avatar
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3 votes
2 answers
231 views

How do we define the type of a singularity on a cubic surface?

Nine different types of singularities are possible on a cubic surface, according to Wikipedia. How exactly is the "type" of singularity defined? I know that the number corresponding to the ...
mathlander's user avatar
2 votes
1 answer
285 views

Is there a non-singular cubic surface that has a point where four lines intersect?

Every non-singular complex projective cubic surface has $27$ lines. Many such surfaces contain points where three lines intersect (called Eckardt points). There are even surfaces with many Eckardt ...
mathlander's user avatar
4 votes
0 answers
97 views

Modern reference for a theorem by Bott on the Dolbeault cohomology of compact homogeneous manifolds

I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology ...
Max Reinhold Jahnke's user avatar
3 votes
0 answers
83 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 ...
Carlos Martinez's user avatar
2 votes
0 answers
121 views

Is there an extension of Ogg's results to surfaces of Genus 1

The first hints of moonshine appeared around 1974 when Andrew Ogg noticed that quotienting the hyperbolic plane by normalizers of the Hecke Congruence subgroups $\Gamma_{0}(p)$ has genus zero iff p is ...
Sidharth Ghoshal's user avatar
8 votes
2 answers
343 views

Real analytic subvariety in complex manifold which is complex outside of its singular set

Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic?...
Misha Verbitsky's user avatar
11 votes
3 answers
1k views

Is every smooth projective variety contained in a chain of smooth projective varieties of increasing dimension?

Let $X ⊆ \mathbb{P}^n$ be a smooth projective variety (over $\mathbb{C}$). I think we can find a chain of irreducible varieties $X = X_0 ⊆ X_1 ⊆ X_2 ⊆ \cdots ⊆ X_k = \mathbb{P}^n$ whose dimension ...
Carlos Esparza's user avatar
3 votes
2 answers
312 views

Fixed-point free holomorphic involutions

Here is the new version of the question which is more explicit. The older version is below. I am looking for complex projective varieties (in dimensions $2$ and higher) admitting a fixed-point free ...
Mohammad Farajzadeh-Tehrani's user avatar
4 votes
1 answer
228 views

“Logarithmic” form of Kodaira Embedding

Suppose you have a non-compact complex manifold $X$ with a Hodge metric, whose associated Kahler form has integral cohomology class. In the compact case, one would be able to conclude that $X$ is ...
Philip Engel's user avatar
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2 votes
0 answers
52 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 ...
Sébastien Loisel's user avatar
4 votes
2 answers
191 views

Constructions of complex surfaces covered by the ball of $\mathbb{C}^2$

Let $S$ be a compact complex surface. It is well-known that the following two facts are equivalent $c_1^2(S) = 3 c_2(S)$ and $S \neq \mathbb{CP}^2$ The universal cover of $S$ is biholomorphic to the ...
Selim G's user avatar
  • 2,636
2 votes
1 answer
215 views

Extension of a Szegő Kernel to the boundary

Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$. Recall: the Szegő kernel is a kernel of the Szegő projection $P: L^{2}(\partial\...
Naruto's user avatar
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