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Results tagged with complex-geometry
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user 1648
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.
35
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5
answers
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The Relationship between Complex and Algebraic Geomety
I have recently begun to study algebraic geometry, coming from a differential geometry background. It seems that there is a deep link between complex manifolds and complex varieties. For example, one …
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31
votes
3
answers
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Rep Theory Consequences of Bott--Weil--Borel
I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory o …
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15
votes
2
answers
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When is a Form a Kähler Form?
Let $M$ be a complex manifold, and $\omega$ a closed $2$-form. When is $\omega$ a Kähler form? I mean, when does there exist a Kähler metric for which $\omega$ is the corresponding form.
I would (wil …
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13
votes
3
answers
2k
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Global Algebraic Proof of the Kahler Identities?
I'm looking at Kahler geometry at the moment and admiring how it manages to do so much with clean global algebraic arguments. One of the big exceptions to all this, however, is the proof of the Kahler …
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8
votes
1
answer
630
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Kähler Structure for Projective Varieties over a Finite Field
(i) In 1960 Serre proved a famous analogue of the Weil conjectures for Kähler manifolds. This poses an obvious question: Does there exist an analogue of a Kähler structure for (non-singular) projectiv …
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8
votes
2
answers
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Hyper-complex and quaternionic Kähler Geometry
What is the exact relationship hyper-complex and quaternionic Kahler manifolds? From Wikipedia we get that hyper-Kahler manifolds are both hyper-complex and quaternionic Kahler. Thus, the two families …
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7
votes
3
answers
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Did Kahler say "a long list of miracles occur"?
I've been reading Moroianu's Kahler geometry notes and found a unattributed quote that says that if the Kahler properties hold, then
"a long list of miracles occur"
I am guessing that this quote belo …
- 3,409
6
votes
0
answers
304
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Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Bor...
For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than t …
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6
votes
2
answers
874
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Why can the Dolbeault Operators be Realised as Lie Algebra Actions
I've been looking at an example in the non-commutative geometry literature and I'm having trouble figuring out what the classical motivation is. I'll just describe the classical case here: Recall that …
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6
votes
1
answer
681
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Classifying Globally Generated Holomorphic Line Bundles over a Flag Manifold
I was recently looking back at an old question of mine, where I asked about the classification of the line bundles over a general complex flag manifold. Pavel Etingof gave the following excellent answ …
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5
votes
2
answers
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Global Definition of the Dolbeault Complex of a Vector Bundle
For an $2n$-dimensional complex manifold $M$, and a smooth vector bundle $E$ over $M$, it is well-known (see Voisin, Huybrechts) that there exists an operator $\overline{\partial}$, built locally from …
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5
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0
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355
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Is the SUSY Algebra isomorphic for all Kähler Manifolds?
For a Kähler manifold, the graded algebra generated by $\partial,\overline{\partial},\partial^*,\overline{\partial}^\ast$, the Lefschetz operator, and the dual Lefschetz operator, is called the supers …
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5
votes
1
answer
666
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When are the Smooth Sections of a Bundle Generated as a Module (over Smooth Functions) by th...
For a holomorphic vector bundle $E$ over a complex manifold $M$, we denote its space of smooth sections by $\Gamma^{\infty}(E)$, and its space of holomorphic sections by $\Gamma^{hol}(E)$. Now I've be …
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5
votes
4
answers
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Questions Suggested by the Parabolic Subgroup Definition
Take the following definition:
"A parabolic subgroup of a linear algebraic group defined over $\mathbb{C}$ is a subgroup, closed in the Zariski topology, for which the quotient space is a projective a …
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4
votes
1
answer
443
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Dolbeault Operators for $CP^1$ as $\mathfrak{su}(2)$ Actions.
This question is related to a previous question of mine. More specifically, it results from my attempts to understand the simplest incarnation of a phenomenon mentioned therein.
Put a grading on the …
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