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Results tagged with complex-geometry
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user 167165
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.
1
vote
0
answers
153
views
Fundamental $1$-form for a Riemannian manifold?
Take a Hermitian manifold $(M,I,g)$ where $I$ is the complex structure and $g$ is the Hermitian metric. The associated fundamental $2$-form
$
g(\cdot,I(\cdot))
$
captures a lot of the information abou …
- 203
6
votes
1
answer
336
views
Atiyah-Singer for Riemannian and Kaehler manifolds
I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special case of
A Ri …
- 203
2
votes
0
answers
108
views
A tri-grading on the de Rham complex of a Lie group?
The tangent space of a (compact) Lie group $G$ is given by its Lie algebra. Assuming for convenience that $G$ is connected, its Lie algebra $\frak{g}$ decomposes into a Cartan part, as well as positiv …
- 203
1
vote
1
answer
138
views
Quaternion-Sasakian manifolds and special holonomy Sasakian manifolds
Two well-known slogans are
A Sasakian manifold is the odd dimensional analogue of a Kähler manifold
and
A $3$-Sasakian manifold is the odd dimensional analogue of a hyper-Kähler manifold
Does this ana …
- 203