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Results tagged with complex-geometry
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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.
21
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Does every group arise as the fundamental group of a complete Kähler manifold?
The fundamental group of a manifold is countable, and every countable group $G$ arises as the fundamental group of a (smooth) manifold; see this comment or this answer for a construction of an open su …
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Complex manifold with subvarieties but no submanifolds
I previously asked this question on MSE and offered a bounty but received no responses.
There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. Fo …
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84
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1
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Is there a complex surface into which every Riemann surface embeds?
This question was previously asked on Math SE.
Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma …
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14
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Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central...
A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian Compl …
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1
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For compact complex surfaces $h^{1,0}$ is either $h^{0,1}$ or $h^{0,1} - 1$. Do we need to u...
In the Wikipedia article on the Enriques-Kodaira classification, before the classification itself, the following sentence appears:
For compact complex surfaces $h^{1,0}$ is either $h^{0,1}$ or $h^ …
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12
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How much algebraic geometry do I need to study complex geometry?
As one can deduce from the questions I have asked on MO, I'm interested in complex geometry. I am aware that there are many facets to the field, some of which I am more comfortable with than others. T …
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Examples of compact complex non-Kähler manifolds which satisfy $h^{p,q} = h^{q,p}$
The existence of a Kähler metric on a compact complex manifold $X$ imposes restrictions on it's Dolbeault cohomology; namely, $h^{p,q}(X) = h^{q,p}(X)$ for every $p$ and $q$. I am looking for some exp …
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If $L$ is positive, $E\otimes L^k$ is Nakano positive for some $k$
I'm trying to prove the following:
Let $L$ be a positive holomorphic line bundle on a compact complex manifold $X$. For any hermitian holomorphic vector bundle $E$ on $X$, there is $k \in \mathbb{ …
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Weitzenböck Identity for $\Delta_{\bar{\partial}_E}$
This question is related to this MO question and this MSE question.
Let $E$ be a hermitian holomorphic vector bundle over a hermitian manifold $X$. The bundle $\bigwedge^{\bullet,\bullet}X\otimes E$ …
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$E$ is a holomorphic vector bundle if and only if there is a Dolbeault operator $\bar{\parti...
I am looking for a reference which shows that the following statements are equivalent for a complex vector bundle $E$:
$E$ is a holomorphic vector bundle.
There is a Dolbeault operator $\bar{\partia …
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Three-dimensional compact Kähler manifolds
Consider the problem of trying to identify which $n$-dimensional compact complex manifolds can be endowed with a Kähler metric.
$\underline{n = 1}:$ Any hermitian metric on a Riemann surface is a Käh …
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2
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1
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Local expression involved in the definition of positivity of vector bundles
This is question follows on from this one.
In the linked question, the hermitian form $\theta_E$ on $T^{1,0}X\otimes E$ is defined globally as $$\theta_E(v\otimes\sigma,v\otimes\sigma):=h(i\Theta_E(v …
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Decomposition of hermitian form used in the definition of Griffiths/Nakano positivity
Let $E$ be a hermitian holomorphic vector bundle over a complex manifold $X$. Then $\Theta(E)$, the curvature of $E$, is a section of $\bigwedge^{1,1}X\otimes\operatorname{End}(E)$. However, we have t …
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Alternative Almost Complex Structures
Originally posted on Maths Stack Exchange.
Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure give …
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answer
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Where do the Kähler Identities first appear?
The Kähler identities (sometimes known as the Hodge identities) are an important collection of relationships between operators on the exterior algebra of a Kähler manifold. These relationships general …
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