All Questions
Tagged with complex-geometry hermitian-manifolds
12 questions
7
votes
2
answers
497
views
Geometrical meaning of admissible hermitian metric on a line bundle
Let $(X,\Omega)$ be a complex compact Kahler manifold, where $\Omega$ is the fundamental $(1,1)$-form. Moreover let $L$ be a holomorphic line bundle on $X$.
A (smooth) hermitian metric $h$ on $X$ ...
- 1,237
6
votes
1
answer
518
views
Where do the (Akizuki)-Nakano Identities First Appear
The answers to this M.O. question give a history of the Kaehler identities. The identities can be extended to the vector bundle-valued setting, and play a central role in the proof of the Kodaira ...
4
votes
2
answers
399
views
Lie super algebra presentation of the Kähler identities
For any Kähler manifold $(M,h)$, with Lefschetz operators $L$ and $\Lambda$, and counting operator $H$, we have the following the well-known Kähler-Hodge identities:
\begin{align*}
[\partial,L] = 0, ...
- 643
4
votes
1
answer
1k
views
Confusion about complex differential forms
I follow Kobayashi "Differential Geometry of Complex Vector Bundles", pages 11-12, prop. 4.9. Given a rank-$r$ Hermitian holomorphic vector bundle $(E,h)$ over a complex manifold $M$, there exists a ...
- 143
2
votes
1
answer
483
views
Curvature forms of holomorphic line bundles
Let $M$ be a compact complex manifold, $L$ a holomorphic line bundle over $M$, and $\nabla$ a connection extending the holomorphic structure map $\overline{\partial}$ of $L$. In general can it happen ...
- 969
2
votes
0
answers
81
views
Restriction of an almost-complex structure to a complex structure on a sub-manifold?
I have been thinking about this recent question of mine a bit more and came to the following question: Consider a manifold $M$ endowed with a non-integrable almost complex structure $J$. Can it happen ...
1
vote
1
answer
707
views
de Rham closed harmonic form on a Kähler manifold
For a compact Kähler manifold, we say that a form is primitive if it is contaned in the kernel of the dual Lefschetz operator, or the co-Lefschetz operator. For all examples I know, a primitive form $\...
- 449
1
vote
0
answers
112
views
Mean curvature as a contraction
I'm going over some of Kobayashi's work on complex vector bundles and trying to state some of the notions in a more familiar language to me.
The set up is the following. We have a hermitian vector ...
- 103
1
vote
0
answers
86
views
Representatives of line bundle cohomology over tori
Let $V^n$ a be a $\mathbb{C}$-vector space. For $U\subset V$ a complete lattice, the holomorphic line bundles over $V/U$ are classified (see e.g. `Abelian varieties', D. Mumford) by data $(H,\alpha)$ ...
1
vote
0
answers
162
views
Warped product manifold with real and complex parts
Is possible to define a warped product manifold $M=(N,g_N) \times f(F, g_F)$ where $(N, g_N)$ is a Riemannian manifold with Riemannian metric (i.e., real manifold with real structure) and $(F, g_F)$ ...
- 272
0
votes
1
answer
255
views
Torsion free Chern connections and Kähler manifolds
Let $(M,h)$ be an Hermitian manifold and let $\nabla$ be the associated Chern connection. Is it true that $(M,h)$ is Kähler if and only if $\nabla$ is torsion free?
- 417
0
votes
0
answers
73
views
Some calculation about Chern connection
The Chern connection is the unique connection satisfying $\nabla^{0,1}=\bar{\partial}$ and
$$
\partial_k\langle u, v\rangle=\left\langle\nabla_k u, v\right\rangle+\left\langle u, \nabla_{\bar{k}} v\...
- 809