All Questions
10 questions
5
votes
1
answer
820
views
Coincide between Chern-connection and Levi-Civita connection
I am a beginner in complex geometry and I am going to show Levi-Civita connection $\nabla$ and the Chern connection $D$ are the same on the holomorphic tangent bundle $T^{1,0}M$ on Kahler manifold. By ...
- 273
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
3
votes
1
answer
297
views
Torsion free (1,0)-connections on the holomorphic tangent bundle?
Let $M$ be a complex manifold. Consider a connection $\nabla$ on the holomorphic tangent bundle $T^{1,0}M$. The torsion of $\nabla$ is defined as the torsion of the induced connection $D$ on the real ...
- 1,457
2
votes
2
answers
277
views
Do "associative" connections exist / arise naturally in some context?
Here is a little bit of curiousity that's been itching me, let's hope it doesn't get me killed, meow.
Definition: Let $M$ be a smooth manifold. A connection $\nabla$ on $TM$ is called associative ...
- 7,127
2
votes
0
answers
56
views
Finding the (1,1) component of $e^{-\mathbb{A}^2}$ for $\mathbb{A}$ a superconnection
Let $E=E^+\oplus E^-$be a holomorphic superbundle over a compact Kahler manifold, and $v:E^+\oplus E^-$ an odd bundle map. Assume that both $E^+$ and $E^-$ are endowed with Hermitian metrics, and ...
- 789
2
votes
0
answers
224
views
Hermitian connections on real hypersurfaces of $\mathbb C^{n+1}$
I would like to find some non-trivial and "geometric" examples of hermitian connections (that is compatible with a give hermitian metric) on complex hermitian vector bundles over a smooth real ...
- 7,902
1
vote
1
answer
260
views
Flat connection, finite-dimensional space of covariant constant one forms
hallo,
I have the following question: Let $U\subset \mathbb{R}^{n}$ be an open subset. Furthermore, let $\nabla$ be a flat connection on $U$ (not necessary Levi-Civita). How can one show that the ...
- 339
1
vote
3
answers
572
views
Special connection of vector bundle over real manifold
Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...
- 131
1
vote
0
answers
195
views
Can logarithmic connection on holomorphic vector bundle induce logarithmic connection on dual bundle?
Let $(X,\omega)$ be a compact K"ahler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $D_i$ intersecting ...
- 295
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