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
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Hilbert scheme of a closed subscheme
Let $X$ be a complex algebraic variety. Its Hilbert scheme represents the functor $G$ from schemes to sets given by $$G(S)=\{Z\subset X\times S|\, Z \mbox{ is a closed subscheme, flat and proper over }...
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Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$
I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial ...
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Hilbert scheme of an infinitesimal neighborhood of a subvariety
Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the ...
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Characterization of certain analytic vector fields on $S^{2}$
Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:
$X$ has a finite number of singularities on $S^{2}$
The equator is invariant under flow of $X$
3.$g_{*}X=\pm X$ where $g$ ...
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Does the "Ohsawa-Takegoshi theorem without bounds" have a name?
There are many theorems which now could be called "The Ohsawa-Takegoshi" theorem. Of these, the most basic is roughly the following:
Let $\Omega \subset \subset \mathbb{C}^n$ be a psuedoconvex ...
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Uniruled degenerations of abelian varieties
Suppose I have a smooth projective variety $X$ over $\mathbb{C}$ with $K_X$ semiample, and consider the fiber space $f:X\to Y$ given by $|\ell K_X|$, for some $\ell>0$ large, where $Y$ is a normal ...
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All Kähler metrics on a complex manifold?
Let $M$ be a complex manifold of complex dimension 2. What do we know about the set all Kähler metrics on $M$ in general and in the case of 4-torus $C^2/Z^4$?
For the case of surfaces ($dim_C=1$), ...
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The type of a Riemann surface arising from a polynomial vector field
Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$
It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...
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Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve
Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us ...
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Deformation of Canonical singularities
Let $p: X \rightarrow T$ be a flat family of normal projective varieties over a variety $T$. Assume that $X_{t_{0}}=p^{-1}(t_{0})$, for a $t_{0}\in T$, has only canonical singularities of index $1$. ...
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Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms
Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ (in the sense that $Y$ is hyperelliptic and the morphism $X\...
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extending to bimeromorphic maps
A meromorphic map of complex spaces (in the sense of Remmert) $f : X \to Y$ is a multivalued map such that its graph $\Gamma$ is an analytic subset of $X\times Y$ and off some analytic subset $Z \...
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Are open immersions in analytic geometry transverse?
lately I have been interested in functional analysis, especially with a view towards its applications in the world of (complex) analytic geometry. I have been using R. Taylor's book Several complex ...
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Reference request for instantons
I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...
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Extending vector bundles from subvarieties
Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there ...
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Where can I find a translation of Caspar Wessel's "Om directionens analytiske betegning?"
I found a listing on Google books for a book containing the desired English translation, together with some biographical information on Wessel, and entitled On the Analytical Representation of ...
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Complex structure on the set of prequantization line bundles
For geometric quantization, the set of equivalence classes of prequantization line bundles of a quantizable symplectic manifold $(M, ω)$ is parametrized by $H^1(M, S^1)$ which represents the ...
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Is there some lattice not rigid
I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain ...
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If a compact real submanifold of $\mathbb{CP}^n$ is approximable by complex algebraic curves, is it algebraic?
To make this into a separate question:
If the supports of a sequence of complex algebraic curves in $\mathbb{CP}^n$ (images of non-constant holomorphic maps from compact Riemann surfaces) converge to ...
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Complex structures on Riemann surfaces
This is cross posted from math.SE: https://math.stackexchange.com/q/876432/9
Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a ...
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A topological criterion for connectedness of a semi-ample divisor
I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof?
Statement. ...
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Embedding dimension: local finiteness & intuition for more general spaces
Can every complex analytic space be covered by Stein spaces of finite embedding dimension?
I am almost sure that ought to be true. Here the definition of embedding dimension I have in mind is
$$
\...
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History of the connection between Riemann surfaces and complex algebraic curves
As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...
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Universal cover of elliptic curve without a point
This is something that I perhaps would be expected to know, but don't. Let $E_\tau$ be the elliptic curve ${\mathbb C}/({\mathbb Z +\mathbb Z} \tau)$. Consider the complement of a point in $E_\tau$,
...
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function field analogy and global/absolute geometry
The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...
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Basic questions on the Hilbert scheme/ Douady space
Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More precisely,...
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Complex geometry text/research introduction for the analyst
To give some background, I am mainly an analyst trained in harmonic/functional and do work on geometric pde's and spectral multipliers. Of late, I am trying to learn more about (research level) ...
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517
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Second betti number of compact analytic spaces
Let $V$ be a proper singular complex algebraic variety, possibly nonprojective ($dim(V)=n>0$). I would like to know:
1) if its second Betti number is non zero,
2) same question but now $V$ is a ...
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What is the relation between Beilinson's conjectures and Standard conjectures of algebraic cycles?
Do Standard conjectures on the K-theory of varieties over finite field have implications in the motivic cohomology of Z where exist the correct formalism of Beilinson's conjectures?
What is the ...
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How does one show that slope stability of a vector bundle is an open condition with respect to the polarisation?
I would like a source for the following result, which I expect to be true (probably well known):
Let $X$ be a complex projective variety, $L$ an ample line bundle and $E$ a slope stable vector bundle ...
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A semi-ampleness criterion for homogeneous bundles on homogeneous spaces?
Let $X$ be a (compact) homogeneous space and $V$ be a homogeneous vector bundle on $X$ of rank $n$, and such that $\operatorname{dim}X\ge n$. Suppose $V$ has a section $s$, whose zeros $s=0$ form a ...
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Flat family with special fiber $\mathbb{C}\mathbb{P}^1$
Let $C=Spec \mathbb{C}[t]/(t^{n+1})$. Let $X$ be an algebraic (or complex analytic) scheme, flat over $C$ with the structure morphism $f\colon X\to C$. Assume that the special fiber is isomorphic to $\...
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Holomorphic Foliations having transverse sections
In the introduction to the paper "On the Geometry of Holomorphic Flows and Foliations Having Transverse Sections" by Ito and Scardua, one reads the following "a holomorphic codimension one foliation ...
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When flatness of a morphism implies smoothness?
EDIT: Let $f\colon X\to C$ be a flat proper morphism of complex algebraic (or analytic) varieties. Assume the special fiber over a point $p\in C$ is smooth.
Is it true that there exists a ...
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Density of rational functions in open Stein
I repost here, after I tried here.
Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb ...
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is grassmannian rational connected or not [closed]
I wan to know if Grassmannians are rational connected? Any reference describe how to tell if a variety is rational connected or not?
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Can someone tell me properties of Douady space?
I want to know the parallel properties of Douady space with respect to Hilbert scheme. For example I want to know what is the irreducible component of Douady space, what if I consider a family of ...
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A question related to the Grauert semi-continuity theorem
Let $f\colon X\to Y$ be a proper holomorphic map of complex analytic manifolds. Assume $f$ to be submersive for simplicity, but probably it is not important. Let $\mathcal{F}$ be a coherent sheaf on $...
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Generalizations of de Franchis and function field Mordell
The classical de Franchis theorem, as generalized by S. Kobayashi and T. Ochiai ("Meromorphic mappings onto compact complex spaces of general type," Inventiones, 1975), states that if $X$ is a complex ...
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Non projective hyperbolic compact complex space
A famous conjecture by Kobayashi (perhaps slightly revisited subsequently) states that every compact hyperbolic Kähler manifold $X$ has ample canonical bundle.
This implies in particular that $X$ is ...
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A generalization of the Grauert direct image theorem
EDIT: Let $f\colon X\to Y$ be proper holomorphic submersive map of complex analytic manifolds. Let $\mathcal{F}$ be the sheaf of holomorphic sections of a holomorphic vector bundle over $X$. Assume ...
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The stability of vector bundle with trivial Chern classes is independent of ample divisor, a direct proof?
Let $X$ be a smooth projective variety over $\mathbb{C}$. For an ample divisor H, we can define the slop of vector bundle with respect to $H$, then we can define stablilty of vector bundle with ...
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Analytic representatives for Kahler classes
If we are given compact complex manifold $X$ and a Kahler class $[\omega]$,
can we always find a positive definite representative $\omega \in [\omega]$ that is
real analytic?
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639
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what is the first cohomology group of structure sheaf of grassmannian [closed]
I want to know if the first cohomology group of structure sheaf of grassmannian vanishes.
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Hodge numbers of diffeomorphic complete intersections
Is there an example of two complex projective complete intersections that are diffeomorphic but have different Hodge numbers?
Edit: as written by Daniel Loughran in the comments below, complete ...
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What are the automorphisms of a Grassmannian?
I want to know what are the holomorphic automorphisms of a Grassmannian. Can someone tell me this?
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Projectively flat Hermitian curvature proportional to Kähler form?
Is there a classification of the holomorphic Hermitian vector bundles $\pi:E\rightarrow M$, over a given complex Hermitian manifold, which are projectively flat and the curvature is proportional to ...
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A geometric characterization of smooth points of a complex algebraic variety
Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric.
Fix an arbitrary point $p\in X$. Let $...
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128
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complementary bundle for a divisor
For a divisor in a complex manifold, what is known about a complementary bundle to the divisor in the manifold (either for the tangent or the cotangent bundle). Is there a description in terms of ...
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Show properness of Ahlfors map
If have got a $k$-fold connected surface $G$, which is bounded by n distinct, non-intersecting Jordan-curves. By Ahlfors it is known that there exists a unique function $\phi$ which maps G to the unit ...