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.
2
votes
1answer
29 views
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 $V_i$ ...
14
votes
1answer
763 views
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 ...
2
votes
0answers
59 views
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 ...
9
votes
0answers
137 views
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 $n\geq 0$?
2
votes
0answers
73 views
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 ...
1
vote
0answers
98 views
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})...
1
vote
0answers
61 views
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)?
5
votes
1answer
275 views
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 $f_{t,s} : M_t\to M_s$,...
1
vote
1answer
97 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 (...
6
votes
1answer
114 views
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 ...
1
vote
1answer
129 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&...
4
votes
1answer
127 views
$Td_p$ 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^{...
4
votes
2answers
182 views
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 ...
1
vote
1answer
43 views
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\...
8
votes
1answer
169 views
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 ...
8
votes
2answers
237 views
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:
...
2
votes
0answers
102 views
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$ ...
6
votes
1answer
137 views
Moishezon manifold with vanishing $b_2$
Does there exist a closed Moishezon manifold with zero second Betti number?
3
votes
0answers
106 views
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 ...
5
votes
0answers
93 views
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 \...
9
votes
2answers
440 views
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 ...
5
votes
0answers
175 views
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$?
2
votes
0answers
145 views
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}$-...
7
votes
1answer
125 views
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. ...
3
votes
2answers
248 views
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 $\...
2
votes
0answers
99 views
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 ...
19
votes
0answers
221 views
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 ...
2
votes
0answers
147 views
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$ ...
12
votes
1answer
469 views
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 ...
0
votes
1answer
139 views
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 ...
0
votes
0answers
137 views
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 ...
1
vote
0answers
33 views
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-...
9
votes
0answers
93 views
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 ...
4
votes
0answers
46 views
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 ...
1
vote
0answers
71 views
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 ...
1
vote
1answer
200 views
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 ...
2
votes
0answers
118 views
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.
...
3
votes
0answers
89 views
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 ...
12
votes
1answer
301 views
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 + ...
8
votes
1answer
323 views
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 ...
14
votes
3answers
610 views
How to find a conformal map of the unit disk on a given simply-connected domain
By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of ...
6
votes
2answers
120 views
Embedding open connected Riemann Surfaces in $\mathbb{C}^2$
This question arises in the context of a question asked on MSE: Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$. Part of the answer to that question is the question above which is ...
1
vote
0answers
49 views
Openness of regular mappings and the conjugacy of Cartan subalgebras
In the book Lie Algebras of finite and affine type by Roger Carter, in chapter 3, the conjugacy of Cartan subalgebras of a finite-dimensional Lie algebra over $\mathbb{C}$ is established via a ...
3
votes
0answers
57 views
Is a domain of a holomorphic flow pseudoconvex?
Let $Z$ be a holomorphic vector field on $\mathbb{C}^n$. I would like to know whether (it seems that it is) the domain $D_\phi \subset \mathbb{C} \times \mathbb{C}^n$ of a maximal flow $\phi: D_\phi \...
26
votes
3answers
970 views
References for Riemann surfaces
I know this question has been asked before on MO and MSE (here, here, here, here) but the answers that were given were only partially helpful to me, and I suspect that I am not the only one.
I am ...
7
votes
0answers
160 views
Diffeomorphism type of Ricci-flat four manifolds
Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows:
A) Is there a classification of the possible homeomorphism types of ...
3
votes
1answer
103 views
The ample cone of a surface with an algebraic $\mathbb C^*$ action
Let $X$ be a compact complex protective surface that admits a nontirvial algebraic $\mathbb C^*$-action. It seems to me, that the ample cone of $X$ is polyhedral with finite number of faces. I wonder ...
6
votes
1answer
253 views
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
0answers
119 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 ...
5
votes
1answer
431 views
Are there enough meromorphic functions on a compact analytic manifold?
Let $X$ be a compact complex analytic manifold, $D\subset X$ an irreducible smooth divisor, given as zeroes of a global meromorphic function $f\in {\mathfrak M} (X)$. Are there enough other ...
