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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( …
Christian Fischmann's user avatar
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 …
Christian Fischmann's user avatar
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 …
Christian Fischmann's user avatar
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 …
Christian Fischmann's user avatar