All Questions
Tagged with complex-geometry dg.differential-geometry
869
questions
4
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1
answer
129
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Lie algebra cohomology and Lie groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}\DeclareMathOperator\Sp{Sp}$Let $G$ be a complex Lie group and $\mathfrak{g}$ its Lie algebra (which is over $\mathbb{C}$). My first question is:
(...
- 627
0
votes
0
answers
103
views
Analogue of Bochner's formula for compact Kähler manifolds
Let $X$ be a compact Kähler manifold and $(E,h)$ a Hermitian vector bundle over $X$. Suppose that $\nabla$ is a Hermitian-Einstein connection on $E$, that is $$i\Lambda F_\nabla = \lambda\text{id}_E.$$...
- 103
1
vote
0
answers
164
views
How to "eliminate" the log pole of a logarithmic $(p,q)$-form？
Let $X$ be a compact complex manifold, and $D=\sum_{i=1}^{r} D_i$ be a simple normal crossing divisor on $X$. Let $\alpha$ be a logarithmic $(p,q)$-form, namely, on an open subset $U$, we can write
$$\...
- 295
0
votes
0
answers
66
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Details of the proof of the inequality $ \int_{X}\left(2 r \mathrm{c}_{2}(E)-(r-1) \mathrm{c}_{1}^{2}(E)\right) \wedge \omega^{n-2} \geq 0.$
I'm trying to make sense of the following proof.
Let $E$ be a holomorphic vector bundle of rank $r$ on a compact hermitian manifold $(X, g)$. If $E$ admits an Hermite-Einstein structure then $$
\int_{...
- 103
3
votes
1
answer
194
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Definition for the Chern–Weil formula?
I'm reading Yang–Mills connections and Einstein–Hermitian metrics by Itoh and Nakajima. On definition 1.8 they define a notion for an Einstein–Hermitian connection $A$ by $$K_A = \lambda(E)\mathrm{id}...
- 103
3
votes
0
answers
111
views
Relative $dd^c$-lemma
Let $f\colon X\to Y$ be a surjective map of compact Kähler varieties. Pick an open subset $U\subset Y$ and let $X_U$ be the preimage of $U$. Does the $dd^c$-lemma hold on $X_U$? Namely, let $\alpha$ ...
- 153
3
votes
1
answer
210
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A formula resembling the integral mean value on Kähler manifolds
I am reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I have a problem when reading the proof of the following theorem:
Theorem. ...
- 173
2
votes
0
answers
102
views
Inequality of Lübke
The following famous inequality by Lübke is given in Differential Geometry of Complex Vector Bundles by S. Kobayashi
Let $(E, h)$ be an Hermitian vector bundle of rank $r$ over a compact Hermitian ...
- 103
1
vote
0
answers
91
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Curvature and Hermitian-Einstein conditions
The following is from a set of lecture notes I'm following and I have had some difficulties understanding it.
Let us discuss a few equivalent formulations of the Hermite-Einstein condition ($\Lambda_\...
- 111
4
votes
1
answer
254
views
A closed $(1,1)$-form $\eta$ is harmonic if and only if $\Lambda\eta = \text{constant}$
Let $X$ be a Kähler manifold and consider the dual operator $\Lambda$ of the Lefschetz operator $L$. Is the following a true statement?
A closed $(1,1)$-form $\eta$ is harmonic if and only if $\...
- 103
1
vote
0
answers
106
views
Mean curvature as a contraction
I'm going over some of Kobayashi's work on complex vector bundles and trying to state some of the notions in a more familiar language to me.
The set up is the following. We have a hermitian vector ...
- 103
1
vote
0
answers
58
views
Expression of the Riemannian metric on the Siegel domain?
I'm looking for proof that, for the complex Siegel domain in $\mathbb C^{n}$ defined by:
$$\mathcal H_{n} = \{ z=(z_1,\dots,z_n) \in \mathbb C^{n} \mid \operatorname{Im}(z_{n}) > \sum_{j=1}^{n-1} |...
- 640
1
vote
0
answers
159
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Conceptual understanding of the definition for Hermite-Einstein metrics
I'm studying holomorphic vector bundles $(E,h)$ on Kähler manifolds that admit a Hermite-Einstein metrics. Particularly, I'm trying to find the motivation for the definition.
An hermitian structure $...
- 63
3
votes
2
answers
318
views
A paper of Borel (in German) on compact homogeneous Kähler manifolds
I am trying to understand the statement of Satz 1 in Über kompakte homogene Kählersche Mannigfaltigkeiten by Borel. Here is the statement in German
Satz I: Jede zusammenhängende kompakte homogene ...
4
votes
0
answers
106
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A compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group
Is it possible to have a compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group? It seems not to be the case, but a precise argument of reference would be great!
Edit: ...