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Results tagged with complex-geometry
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user 6871
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.
5
votes
Can a metric conformal to a Kahler metric be Kahler?
The paper by Apostolov, Calderbank, and Gauduchon that Francesco mentions find different Kaehler structures whose associated Riemannian metrics are conformal to each other. But they correspond to diff …
- 3,540
6
votes
book on calabi yau manifolds
I would also add the following book:
Dominic Joyce, Compact Manifolds with Special Holonomy
The early parts of the book include an introduction to the Riemannian geometry of Calabi-Yau manifolds. It …
- 3,540
1
vote
local kählerforms on complex manifold
(I decided to repost multiple comments as an answer. It's partly an answer if I understand the OP correctly.)
I think the OP does not necessarily want to conclude that these "local" forms are the res …
- 3,540
5
votes
Hyper-complex and quaternionic Kähler Geometry
[First paragraph has been edited, after Vitali's comments below.] According to one convention, hyperKahler manifolds are not actually quaternionic-Kahler. This is the case if you define hyperKahler as …
- 3,540
5
votes
Is a simply connected Ricci-flat Kaehler manifold a Calabi-Yau manifold?
José is correct, with the caveat that Gunnar mentioned - you need simple-connectedness to know that reduced holonomy = holonomy. Below I expand a bit more on the details. [Thanks to Tim Perutz for cat …
- 3,540
27
votes
5
answers
7k
views
References for "modern" proof of Newlander-Nirenberg Theorem
Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In particu …
7
votes
References for "modern" proof of Newlander-Nirenberg Theorem
In my continued searches for modern proofs of Newlander-Nirenberg, I found this great source: it's "Applications of Partial Differential Equations to Some Problems in Geometry", a set of lecture notes …
26
votes
Accepted
When is a Form a Kähler Form?
I decided to make my comment into a more detailed answer. When $M$ has an almost complex structure $J$, then one can talk about smooth complex-valued differential forms of type $(p,q)$ in the usual wa …
- 3,540
10
votes
Accepted
Calabi - Yau Manifolds
There are several different "definitions" of Calabi-Yau manifolds, not all equivalent, and not all contained in one general definition. A good discussion of some of these inequivalent definitions can …
- 3,540