Questions tagged [complex-manifolds]
For questions about or involving complex manifolds.
354
questions
3
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0
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45
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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 ...
- 549
2
votes
0
answers
148
views
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 ...
- 21
2
votes
0
answers
75
views
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 ...
- 83
4
votes
1
answer
326
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 (...
- 1,213
0
votes
0
answers
142
views
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 ...
- 43
0
votes
0
answers
146
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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 ...
- 371
3
votes
0
answers
157
views
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 (...
- 21.1k
3
votes
0
answers
186
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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 (...
- 1,087
0
votes
1
answer
79
views
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?
- 43
4
votes
0
answers
113
views
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 $s_1, s_2 \in H^0 (L)$ ...
- 619
3
votes
1
answer
342
views
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 $\Omega^1_{\mathbb{C}}$, ...
3
votes
0
answers
118
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 ...
- 17.5k
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?
- 493
0
votes
0
answers
129
views
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 ...
- 89
1
vote
1
answer
88
views
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 ...
- 331
2
votes
0
answers
131
views
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 ...
- 53
0
votes
1
answer
249
views
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 ...
- 53
3
votes
0
answers
161
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 $E_\varepsilon$ ...
- 1,268
1
vote
0
answers
31
views
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?
- 881
3
votes
1
answer
140
views
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 ...
- 2,283
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 ...
- 21.1k
7
votes
1
answer
287
views
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 \...
- 341
0
votes
0
answers
140
views
Is there a Cauchy integral formula for complex manifolds?
Is there a Cauchy integral formula for holomorphic functions on complex manifolds?
- 293
7
votes
2
answers
324
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 ...
- 585
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 ...
- 4,968
3
votes
0
answers
161
views
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,...
- 619
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 (...
- 5,011
3
votes
1
answer
272
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" ...
- 519
5
votes
3
answers
302
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 ...
- 2,578
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}...
- 21.1k
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 $...
- 463
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 ...
- 9,095
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 $...
- 215
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}(...
- 91
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 ...
- 1,821
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 ...
- 1,821
3
votes
0
answers
211
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 $(M,-J)$?
- 1,361
7
votes
1
answer
474
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 ...
- 1,349
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 ...
- 179
2
votes
1
answer
286
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 ...
- 179
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 ...
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 ...
- 166
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 ...
- 2,283
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?...
- 9,095
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 ...
- 1,111
3
votes
2
answers
315
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 ...
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 ...
- 1,493
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 ...
- 847
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 ...
- 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\...
- 63