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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.
6
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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
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 …
- 3,409
31
votes
3
answers
3k
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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 …
- 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 …
- 3,409
5
votes
2
answers
2k
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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 …
- 3,409
5
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0
answers
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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3
votes
0
answers
280
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Equivariant Tangent Bundle Decomposition
Given a $G$-homogeneous space $M$, for $G$ a (Lie) group, we have a canonical $G$-action on the tangent bundle $T(M)$ of $M$. If $M$ is a complex manifold, then we have a decomposition of $T(M) \otime …
- 3,409
2
votes
1
answer
1k
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Global Lichnerowicz Formula Proof (in the Kahler case)?
For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection w …
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8
votes
2
answers
1k
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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 …
- 3,409
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 …
- 3,409
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 …
- 3,409
7
votes
3
answers
1k
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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
4
votes
2
answers
654
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Semi-Simple Kahler Groups?
We say that a Kahler manifold is a Kahler group if it is also a Lie group. I would like to know which semi-simple Lie groups are also Kahler groups?
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4
votes
0
answers
382
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Atiyah--Singer for the Complex Projective Line
I'm trying to understand Atiyah--Singer by looking at the usual starting point of $CP^1$ and the Dirac--Dolbeault operator. If I've reduced everything down correctly, then in this case the theorem giv …
- 3,409
13
votes
3
answers
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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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