# Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

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### Compact Kaehler submanifolds of projectivized Hilbert space

If we take a separable complex Hilbert space $H$, its projective space $PH$ is an infinite-dimensional Kähler manifold in a fairly obvious sense (see below). Suppose $M \subset PH$ is a finite-...

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### First Chern Class of Contact Structure which is not Torsion

Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and ...

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### (Singular) metric associated to the higher cohomology

Suppose $X$ is a smooth complex variety and $L$ is a line bundle with a metric $h_L$, then a section $s \in H^0(X, L)$ gives another metric $\tilde h_L:= e^{-\phi}h_L$ where $\phi=\log \|s\|^2_{h_L}$.
...

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149 views

### Does profinite completion commute with mapping spaces?

Does there exist a prime number $p$ and a smooth complex projective variety $X$ such that $F_{\infty p}\mathrm{Map}(B\mathbb{Z}/p\mathbb{Z}, X)$ is not weakly homotopy equivalent to $\mathrm{Map}(B\...

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### Holomorphic structures on vector bundles over $\mathbb C\mathbb P^2$

It is known that every (topological) complex rank $2$ vector bundle over $\mathbb C\mathbb P^2$ admits holomorphic structures. A proof can be found in the book of Okonek, Spindler, Schneider which is ...

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### $c_2$ of Calabi-Yau three-folds

Let $Y$ be a smooth compact Calabi-Yau three-fold (over $\mathbb C$, with $\pi_1(Y)=0$). Is it true that $c_2(Y)$ is Poincare dual to an effective curve? If not, can one construct a counter-example?
...

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### Counting the number of poles for rational functions in a coordinate ring of a curve

I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in ...

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### Complex manifolds as algebro-geometric objects

A result of Artin states that analytification of proper algebraic spaces over
$\mathbb{C}$ defines an an equivalence of the category of proper algebraic spaces with the category of Moishezon spaces. ...

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### Fuchsian groups of singly branched covers

Let $X/\mathbb{C}$ be an algebraic curve with genus $g \geq 2$. Then by the uniformization theorem, with $X(\mathbb{C})$ viewed as a Riemann surface, it can be realized as the quotient $\mathbb{H}/\...

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### Complex Riemannian metrics over real manifolds

There is a huge literature on complex manifolds and natural metrics over them, but I was unable to find references about Riemannian complex metric on real manifolds (for which we complexify tangent ...

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### Invariant submanifolds tangent to isotypic subrepresentations

Let $G$ be a Lie group acting on a complex manifold $M$. Let $p$ be an isolated fixed point. Let us look at the representation of $G$ on $T_pM$. Suppose $T_pM = \bigoplus V_i^{\oplus n_i}$ where $V_i$ ...

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### GAGA for stacks

I am curious about stacky generalizations of the following GAGA theorem:
If $X, U$ are complex algebraic varieties of finite type, $X$ is proper and $f:X\to U$ is an analytic map then $f$ is ...

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### Closed Kaehler--Einstein surfaces are complex ball quotients

Let $X$ be a closed Kaehler manifold of real dimension 4 endowed with a Kaehler--Einstein metric of negative curvature. Is it true that $X$ is isomorphic, as a Kaehler manifold, to a quotient of a ...

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### Batyrev's theorem in non-algebraic case

Let $X$ and $Y$ be two bimeromorphic closed Kaehler manifolds with trivial real $c_1$. Is it true that $b_n(X)=b_n(Y)$ for $n\geq 0$?

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### Automatic plurisubharmonicity for a non-negative function

I feel confused about a point in this very short paper. On the top of page 3, it is claimed that:
If $S$ is a totally real submanifold in a compact almost complex manifold $(X,J)$, then any function ...

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### SL(2,R) invariant which are not SL(2,C) invariants

Consider four points, $\sigma_i$ i=1,2,3,4 on the line $\mathrm{Im}(z) = 0$ in the complex plane $\mathbb{C}$. Does it exist a rational function of these four points which is $\mathrm{SL}(2,\mathbb{R})...

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### Are there Lorentzian complex manifolds?

Quick and simple...
Is it possible to define complex structures on Lorentzian manifolds? If so, Can you point me to some example(s)?

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### On limits of manifolds

This question should be fairly elementary. I’d just like to check I’m not missing anything.
Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f_{t,s} : M_t\to M_s$,...

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107 views

### Mostow rigidity for complex hyperbolic manifolds

A Riemannian manifold $(X,g)$ is hyperbolic if the sectional curvatures are constant and negative. A theorem of Mostow says that these manifolds are determined by their fundamental group.
Theorem (...

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### How to compute the Kahler potential of a Sasaki metric

The Question
Given Hessian manifold $M$, there is a natural Kahler structures on $TM$. Is it possible to write the Kahler potential of these in terms of the Hessian potential?
Background
To ...

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134 views

### Hodge numbers of compact Ricci-flat Kaehler manifold

Assume that $M$ is a closed connected Ricci-flat Kaehler manifold $M$ of complex dimension $n\geq 3$ with $h^{2,0}(M)=0$. Is is possible that
$h^{n, 0}(M)\neq 1$
$h^{p, 0}(M)\neq 0$ for some $0< p&...

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### $Td_p$ notation of Kotschick

In this paper, notation $Td_p$ is used without explicit definition (it is stated that it is a certain combination of Chern numbers). It is claimed that HRR theorem implies
$$
Td_p(M)=\sum_{q}(-1)^q h^{...

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### Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold?

$\require{AMScd}$
Preliminaries: Let $(X,\omega,J)$ be a closed Kahler manifold. That is, $X$ is a closed $2n$-manifold, $\omega$ is a symplectic form and $J$ is a compatible (integrable) complex ...

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### On extensions of holomorphic mappings with image in a projective algebraic variety

I am reading the paper "Two Theorems on Extensions of Holomorphic Mappings" by PHILLIP A. GRIFFITHS. In Example 2 of the paper, there is a proposition saying that:
Let $N$ be a complex manifold, $S\...

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### Moishezon manifold vs proper complex variety

Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is ...

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### Relation between mirror symmetry, homological mirror symmetry, and SYZ conjecture

I'm very new to mirror symmetry, and have a hard time establishing a broad overview of the subject. In particular I do not understand the precise relation between the following three conjectures:
...

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### Topology of abstract varieties over $\mathbb{C}$

What are the known restrictions on the topology of complex manifolds corresponding to analytifications of smooth proper algebraic varieties over $\mathbb{C}$? I think
they have to have non-zero $b_2$ ...

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### Moishezon manifold with vanishing $b_2$

Does there exist a closed Moishezon manifold with zero second Betti number?

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### Explicit KE metrics

Does there exist an explicit example of a
Ricci-flat, non-flat metric on a closed manifold?
Kaehler--Einstein, non-flat metric on a closed manifold (excluding metrics on homogeneous spaces and ...

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### h-principle for pairs

Let $A,B$ be complex analytic spaces. Suppose that $[A,B]$ satisfies the h-principle: i.e. every class of continuous function $f:A \to B$ up to homotopy, contains a holomorphic representative. Let $C \...

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### Two homeomorphic non-diffeomorphic complex manifolds

Does there exist a closed topological manifold supporting two non-diffeomorphic smooth structures both of which admit a compatible complex structure? Also the same question, but for symplectic ...

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### Kaehler manifold of dimension 6 not homotopy equivalent to a complex submanifold of $\mathbb{C}P^n$

Does there exist a closed Kaehler manifold of real dimension 6 that is not homotopy equivalent to a complex submanifold of $\mathbb{C}P^n$ for some integer $n$?

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### Deformation invariance of homotopy type

Let $\mathscr{X}\to \Delta$ be a flat family of projective varieties over the unit disk so that each fiber $X_t$ has canonical singularities and its canonical sheaf $\omega_{X_t}$ is $\mathcal{Q}$-...

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### Beilinson-Drinfeld quantization and stable bundles

To motivate this question, I'm going to try and explain some background notions. This won't be absolutely necessary for experts, but I want to be vaguely honest about where this question comes from. ...

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### Symplectic form on a Kähler manifold can be not real analytic?

Let $M$ be a Kähler manifold. The complex structure on it naturally gives rise to the real analytic structure. I wonder if there exist Kähler manifolds such that the associated symplectic $2$-form $\...

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### Fundamental Group of small Zariski open set

Let $Y$ be an integral affine variety over $\mathbb{C}$ and $K$ be its function field. How to find a sufficiently small Zariski open set of $Y$ such that it is isomorphic to $K(\pi,1)$? Here $\pi$ is ...

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### The de Rham complex of the octonionic projective spaces

The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic ...

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### Can a birational morphism between two smooth varieties of the same betti numbers exist?

I am considering a birational morphism $f:X\longrightarrow Y$ where $X$ and $Y$ are smooth projective varieties and I want to deform $X$ to another given smooth projective $Z$. It is given that $X$ ...

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### Poincaré metric on the Riemann sphere minus more than two points

If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let ...

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### Complex structures on topological surfaces

I am interested in the number of complex structures on a surface. More precisely, given a genus $g$ surface (topological manifold of real dimension 2) with $n$ punctures $X_{(g,n)}$, how many complex ...

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### Example of variety which is not a complete intersection with respect to any projective embedding

Suppose $X$ is a smooth projective variety. Whether $X$ is a complete intersection or not when viewed as a subvariety of some projective space $\mathbb P^n$ is dependent on the specific choice of the ...

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### On Remmerts reduction

Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...

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### Geometry of Affine Kac-Moody Algebras

I recently asked this question on phys.SE and it was suggested to me to ask it here.
One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric ...

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### 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 ...

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### Interpretation of deformation of complex structure

Let $X$ be a smooth complex analytic space and let $D$ be the unit disk in $\mathbb{C}$. Let $\omega:Y \to D$ be a deformation of complex structures of $X$ in the sense that (1) $\omega^{-1}(0) \simeq ...

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### Some questions about Clemens' paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory

I am reading the paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory written by Herbert Clemens. I have found this paper hard to follow even though I am familiar with the ...

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### Holomorphic map, Instantons of Complex Projective Space and Loop Group

It seems that holomorphic (or rational) maps play a crucial role to relate the following data:
Instanton in 1-dimensional complex Projective Space $$\mathbb{P}^1$$
in a 2 dimensional (2d) spacetime.
...

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### Periods for Irreducible Holomorphic Symplectic Manifolds

Let $f:\mathscr{X}\rightarrow \operatorname{Def(X)}$ be the Kuranishi family of $X$, where $X$ is an irreducible holomorphic symplectic manifold. After shrinking $\operatorname{Def}(X)$, we get that ...

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### Is the complement of an affine open in an abelian variety ample?

Let $U$ be an affine open subscheme of an abelian variety $A$ over $\mathbb{C}$. Is $A-U$ an ample divisor?
If $\dim A =1$ this is true.
If $\dim A = 2$, the complement is a divisor $D_1+\ldots + ...

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### What is the “analytic” analogue of the valuative criterion of properness

Let $X$ be a Hausdorff complex analytic space. Below, let $D$ be the open unit disc in $\mathbb{C}$. Let $D^*$ be the punctured open unit disc.
I am looking for an analogue of the valuative criterion ...