All Questions
Tagged with complex-geometry kahler-manifolds
353
questions
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
4
votes
2
answers
215
views
Locality of Kähler-Ricci flow
Let $(M,I, \omega)$ be a compact Kähler manifold with $c_1(M)=0$. Denote by $\operatorname{Ric}^{1,1}(\omega)$ the Ricci (1,1)-form, that is, the curvature of the canonical bundle. It is known ("...
- 9,157
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
views
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
0
votes
0
answers
108
views
Reference request. Looking for a specific compact complex manifold
For my research I need to construct a compact complex manifold with quartic ramification loci. By quartic ramification loci I mean that $L_1,L_2,L_3$ are complex algebraic varieties of degree four and ...
- 189
1
vote
0
answers
25
views
Variation of the metric on Kähler quotient
We can use Kähler quotient to produce a family of Kähler metrics on quotient space.
My question is: how do we calculate the variation of these metric?
This seems to be a natural question but I can't ...
- 21
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
views
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: ...
1
vote
0
answers
124
views
Compact complex manifolds with nef canonical bundle have nonnegative Kodaira dimension
Let $X$ be a compact Kähler manifold with nef canonical bundle. The (Kähler extension of the) abundance conjecture asserts that $K_X$ is semi-ample, and thus $K_X^{\otimes m}$ admits a section for ...
- 265
3
votes
1
answer
172
views
Request for non-Einstein positive constant scalar curvature Kähler surfaces
I am curious about concrete examples of compact cscK manifolds in complex dimension two, in particular cscK surfaces with positive scalar curvature.
There are of course the Fano (del Pezzo) Kähler-...
0
votes
1
answer
182
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?
2
votes
0
answers
59
views
Lefschetz operator on bundle-valued forms
For a holomorphic vector bundle $V \rightarrow X$ endowed with a Hermitian structure, one may define the corresponding Dolbeault-like operators $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,...
- 121
5
votes
1
answer
237
views
Rozansky-Witten invariants of hyperkahler manifolds and independence of complex structure
Recently I have been learning about Rozansky-Witten invariants, mainly through Hitchin-Sawon's paper "curvature and characteristic numbers of hyperkahler manifolds" and through Justin Sawon'...
- 91
2
votes
0
answers
137
views
Hypercomplex structures and tangent space decompositions
For any almost complex manifold we have a decomposition of its tangent space into two subspaces $T = T^{(1,0)} \oplus T^{(0,1)}$. For an almost hypercomplex manifold we have three almost-complex ...
1
vote
0
answers
137
views
Non-compact extremal Kähler spaces
I want to ask about a generalisation of the Calabi functional to non-compact Kähler spaces. My interest is mostly in Kähler surfaces, so I will assume real dimension $4$. In my work, I have found an ...
