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Results tagged with complex-geometry
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user 21564
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.
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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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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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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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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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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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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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$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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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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Hermitian Christoffel Symbols
Does anyone know of some good references for computing Christoffel symbols for Hermitian metrics?
A quick Google search turns up this. The following formula appears on page 4:
$$\Gamma_{AB}^C = …
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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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Does equality of Laplacians imply Kähler?
This question follows on from this one.
Let $(X, \omega)$ be a Hermitian manifold and define the Laplacians $\Delta_{\partial} = \partial\partial^* + \partial^*\partial$ and $\Delta_{\bar{\partial}} …
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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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Riemannian metrics as sections of a vector bundle
Let $\pi \colon E \to M$ be a smooth vector bundle. A Riemannian metric on $E$ can be regarded as a global section of the vector bundle $(E\otimes E)^{\ast}$, or more specifically, the subbundle $(S^2 …
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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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