Questions tagged [complex-geometry]
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
3,129
questions
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On some curves of real values of a rational function
For given parameters $a_{1},\dots,a_{k}\in\mathbb{R}$, define the rational function $\phi:\mathbb{C}\to\mathbb{C}$ as
$$\phi(z)=\frac{1}{z}-a_{1}z-a_{2}z^{2}-\dots-a_{k}z^{k}.$$
The domain of its real ...
10
votes
0
answers
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How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?
Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry?
What do I mean by complex geometry? ...
8
votes
0
answers
347
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Symplectic invariance of Hodge numbers?
Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$.
My ...
1
vote
0
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148
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Calabi-Yau with nodes
Suppose $X$ is a singular projective irreducible complex variety of dimension 3, and its singular loci are finite number of nodes, and its smooth locus $X_1$ is a Calabi-Yau quasi-projective variety, ...
4
votes
0
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235
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Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical
Let start with a definition
Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...
1
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0
answers
132
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Poincaré inequality for holomorphic line bundles
Let $M$ be a Riemann surface of genus >1, $g$ be an Hermitian metric on $M$. Let $E$ is a holomorphic negative line bundle over $M$, for example, the holomorhic tangent bundle of $M$. Let $h$ be an ...
3
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0
answers
209
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local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$
Let $X$ be a rational threefold (over the field of complex numbers) with terminal singularities. It is well-known that $X$ has only finitely many singular points $x_1,x_2, \ldots,x_n$.
To be more ...
8
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0
answers
229
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A Hartogs-type criterion for flatness
Let $U$ be a smooth affine connected variety over $\mathbb C$ and let $V\subset U$ be an open whose complement is of codimension at least two.
Now, let $Y$ be a smooth quasi-affine connected variety ...
1
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2
answers
202
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Smooth, irreducible surface with real part containing two projective planes
Let $X$ be a smooth and irreducible projective variety over $\mathbb{R}$ of dimension two. I am looking for an instance of such a variety where two distinct connected components of $X(\mathbb{R})$ are ...
4
votes
1
answer
351
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Quantifying the monotonicity property of the hyperbolic metric
Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} &...
8
votes
1
answer
281
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Complex manifolds with spanning sets of holomorphic tensor fields
This question is an extension of this one. Consider a complex manifold $(M^{2n}, J)$. Fix $1 \leq p \leq n-1$, and suppose that the space of holomorphic sections of $\Lambda^{p,0}$ spans $\Lambda^{p,...
6
votes
1
answer
934
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Confusion surrounding the Koszul-Malgrange theorem
I recently had the need to appeal to some complex geometry in my research and have been trying to unravel the various relationships surrounding the Koszul-Malgrange theorem.
According to nlab, the ...
7
votes
2
answers
403
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Are square tiled surfaces dense in the moduli space of translation surfaces?
I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt.
At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense....
1
vote
1
answer
174
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Kernel of projection formula
For a closed embedding of compact complex manifolds
$$
\iota : Y \hookrightarrow X
$$
and any $\alpha \in H^*(X,\mathbb Q)$, we have trivially:
$$
\iota^*(\alpha)=0\quad \Rightarrow \quad\iota_*\iota^*...
5
votes
1
answer
640
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Complex manifolds with spanning sets of holomorphic vector fields
I want to understand compact complex manifolds $(M^{2n}, J)$ with the following property: there exists a collection $\{X_i\}_{i=1}^L$ of holomorphic vector fields (sections of $(T^{1,0}_{\mathbb C} M)$...
6
votes
1
answer
567
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Chern-Einstein metrics on complex Hermitian manifolds
Metric on a Riemannian manifold $(M,g)$ is Einstein, if for some function $\lambda\colon M\to \mathbb R$
$$
Ric(g)=\lambda g.
$$
It is well know, that such $\lambda$ is, in fact, a constant.
The ...
-1
votes
1
answer
94
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Zariski open set in orthogonal grassmanian [closed]
I am confused about the following question.
Consider $\mathbb C^4$ endowed with nondegenerate symmetric bilinear form $J:=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&1&0&...
0
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0
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547
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A question about invariance of plurigenera
Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So deformation invariance of plurigenera says that the ...
7
votes
0
answers
457
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Kähler quotients of affine varieties and GIT
Let $X\subseteq \Bbb C^n$ be a smooth affine variety and $G=K_{\Bbb C}$ a complex reductive group acting linearly on $\Bbb C^n$ preserving $X$ (where $K$ is a maximal compact subgroup of $G$). Suppose ...
1
vote
1
answer
299
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A necessary condition for existence of Ricci flat metric on pair (X,D)
Let $X$ be a complex compact manifold with simple normal crossing divisor $D$. Is the condition $K_X +D = 0$ necessary for the existence of Ricci-flat metric?
8
votes
0
answers
908
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Fubini-Study form on weighted projective spaces
As it is known, $\mathbb CP^n$ with Fubini-Study symplectic form can be get by the symplectic reduction of $\mathbb C^{n+1}$ with a symplectic form $\sum_{i=0}^n dz_i\wedge d\bar z_i$ by a hamiltonian ...
0
votes
1
answer
520
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Canonical metric on moduli space of singular Calabi-Yau varieties
Let $\pi:X\to Y$ be a surjective holomorphic map with connected fibers
and let fibers are singular Calabi-Yau varieties (i.e. numerical dimension is zero) then is it possible to construct canonical ...
7
votes
1
answer
458
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Embed a bordered Riemann surface into punctured Riemann surfaces?
Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ ...
9
votes
1
answer
771
views
Is the analytification functor part of a geometric morphism of topoi?
Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras.
A complex analytic space for our purpose is a locally ringed space locally ...
7
votes
0
answers
201
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Biholomorphic neighborhoods of the boundary of Stein domains
Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
1
vote
0
answers
431
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Is the category of mixed Hodge modules bi-filtered?
Let $X$ be a smooth complex algebraic variety and let $MHM(X)$ be the category of mixed Hodge modules on $X$, as defined in (Saito, "Mixed Hodge Modules", 1990), (Peters-Steenbrink, "Mixed Hodge ...
12
votes
2
answers
935
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Bott Chern cohomology via currents
Let $X$ be a compact complex manifold. Is the space of $(p,p)$ $d$-closed currents modulo $\partial\bar{\partial}$-exact ones naturally isomorphic to the Bott-Chern cohomology (made in the same way ...
2
votes
1
answer
335
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Existence of non-constant solutions for this equations
This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...
5
votes
1
answer
949
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On the complexification of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $...
1
vote
0
answers
140
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Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic
In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...
2
votes
1
answer
461
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Local deformations of Lagrangian submanifolds in holomorphic symplectic manifold and their intersections
Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are ...
3
votes
1
answer
298
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Do some kind of maximum principle exist on complex manifold?
Consider a holomorphic function $u:C\to C^n$, as $|u|^2 $is sub-harmonic, it satifies a maximum principle.
Do some general kind of complex manifold enjoy such property? Say, square of some distance ...
1
vote
1
answer
102
views
Triviality of a circle fibration induced by an almost complex structure
Let $E→M$ be a plane bundle endowed with an almost complex structure $J.$
$J$ induces a natural positive definite inner product in the associated bundle $End(E)→M$,denoted by $<,>$. More ...
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votes
1
answer
194
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Is it easy to see that a cubic surface $V$ in $CP^3$ has no holomorphic 2-forms? [closed]
More specifically, what facts do you need to know to conclude $H^2(V) = H^{1,1}$? In general, are there hypersurfaces in $CP^n$ without holomorphic $k$-forms for some $k$?
4
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0
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348
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Narasimhan-Simha Hermitian metric vs Weil-Petersson metric
What is relation between Weil-Petersson metric on holomorphic fibre space $f:X\to Y$ of compact complex manifolds $X,Y$ . (let fibres are Calabi-Yau manifolds)
And Ricci curvature of Narasimhan-Simha ...
1
vote
1
answer
568
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Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?
It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type:
https://en.wikipedia.org/wiki/...
2
votes
0
answers
168
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Question regarding a lemma in Principles of Algebraic Geometry
My question is regarding the lemma on page 81 of the book by Griffiths and Harris.
The lemma says the following:
A $\bar{\partial}$-closed form $\psi\in Z^{p,q}_{\bar{\partial}}(M)$ is of minimal ...
4
votes
1
answer
553
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What is the relation between holomorphic blow-up and symplectic blow-up?
McDuff has shown us exactly how the symplectic blow-up procedure along a symplectic submanifold affects the symplectic structure in the ambient space, i.e., if $\omega$ is the original symplectic form,...
11
votes
1
answer
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Restriction of the Picard group of a surface to a curve
In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious:
For a general (smooth) surface $S$ in $\mathbb{P}^3$ ...
18
votes
3
answers
3k
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What are parabolic bundles good for?
The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...
3
votes
1
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483
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A conjecture from Jean Varouchas on Kahler varieties
Conjecture: Let $\pi: X\to X'$ be a proper flat surjective morphism of complex spaces.
If $X$ is Kahler, is $X'$ Kahler?
This conjecture when $X$ and $X'$ are smooth solved by Jean Varouchas from ...
2
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0
answers
524
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Tian's approach for solving the conjecture of invariance of plurigenera in Kahler setting
Let $f:X\to Y$ be a smooth holomorphic fibre space whose fibres $f^{-1}(y)$ have pseudoeffective canonical bundles. suppose that
$$\frac{\partial \omega(t)}{\partial t}=-Ric_{X/Y}(\omega(t))-\omega(...
6
votes
0
answers
227
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Deformation of Complex Spaces
I am trying to learn about deformations of complex spaces from the paper of Palamodov. I am particularly interested in the relative tangent cohomology.
Is there any other modern reference to this ...
2
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0
answers
93
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Complex Structure Moduli of Elliptic Fibrations
Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration.
I know how it is done for the generic Weierstrass ...
1
vote
2
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630
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Contact and CR Examples
What is an example of a manifold such that:
(A) It is both a contact manifold and a CR manifold
(B) It is a contact manifold but not a CR manifold
(C) It is not a contact manifold but not a CR ...
8
votes
1
answer
507
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Connectivity of complements of Stein opens
Let $Y$ be an affine open subset of a locally noetherian scheme $X$. Then, $X \setminus Y$ has pure codimension one [EGAIV$_4$, Cor. 21.12.7]. Moreover, if $X$ is proper and of finite type over a ...
8
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2
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520
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Inequality on Kähler classes
Let $X$ be a compact Kähler manifold of complex dimension $n$, and let
$\omega_1, \omega_2$ be Kähler classes on $X$. Denote the Lefschetz
operator of a Kähler class $\omega$ by $\Lambda_{\omega}$. ...
1
vote
0
answers
182
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horizontal lift along fibres
Let $f:X\setminus D\to Y$ be a smooth family of Kahler manifolds, where $D=\{\sigma=0\}$ is a divisor on $X$. Taking a local
coordinate $(s_1,...,s_d)$ of
$Y$
and a local coordinate $(z_1,...,z_n)$ of ...
4
votes
1
answer
247
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Smooth algebraic stacks with precisely two $\mathbb C$-objects
In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one $\...
5
votes
1
answer
569
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Is the automorphism group of a Calabi-Yau variety an arithmetic group
Let $X$ be a smooth projective variety over the complex numbers with trivial canonical bundle. Suppose that $X$ is Calabi-Yau.
Is the automorphism group of $X$ an arithmetic group?
What if $X$ is a ...