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Questions about Kähler manifolds and Kähler metrics.
2
votes
1
answer
1k
views
Ricci Curvature and the Chern Class of the Levi-Civita
For a (compact) Kahler manifold $M$, the Ricci tensor is the symmetric $2$-form
$$
r(u,v) = \text{tr}\big( w \mapsto (D_wD_u - D_uD_w - D_{[u,w]})v\big).
$$
The Ricci curvature is the $2$-form
$$
r( …
4
votes
1
answer
346
views
The de Rham complex of a quaternion-Kahler manifold
As we all know, for a complex manifold $M$, its de Rham complex admits a decomposition into a double complex called the Dolbeault complex. If $M$ also admits a Kahler metric, then we get the wonderful …
1
vote
1
answer
180
views
Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics
Recal that $\frak{sl}_2$ is the Lie algebra with basis elements $e,f,h$, and bracket
$$
[e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f.
$$
For $M$ a $2n$-complex manifold, the Lefschetz identities tell us …
6
votes
1
answer
272
views
Equivariant Almost Complex Structures on the Full Flag Manifolds
On complex projective space ${\bf CP}^m$, there exists a unique $SU(m+1)$-equivariant almost-complex structure. What happens for the case of the full flag manifold of $SU(m+1)$, which is to say the sp …