All Questions
15 questions
1
vote
0
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130
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Is Stenzel's Ricci-flat metric on $T^*\mathbb{CP}^n$ hyperkahler?
In a well-known paper, Stenzel constructed complete Ricci-flat Kahler metrics on the total spaces of cotangent bundles of $S^n$, $\mathbb{RP}^n,$ $\mathbb{CP}^n$, $\mathbb{HP}^n$, and $\mathbb{OP}^2$.
...
- 871
2
votes
0
answers
151
views
Minimal Betti numbers of simply-connected threefolds with trivial canonical class
By a threefold, I mean a compact complex manifold of dimension three.
For a simply-connected threefold with trivial canonical class, its Betti numbers satisfy:
$$b_2 \ge 0, b_3 \ge 2.$$
I am wondering ...
- 1,841
3
votes
0
answers
150
views
Why does the bisectional curvature blow up?
Suppose we have $X = X_1 \times X_2$, where $X_1$ is a Calabi-Yau manifold (i.e. $c_1(X_1) = 0$) and $X_2$ is a compact Kähler manifold of $c_1(X_2) < 0$. We can consider the metric $\omega(t, x_1, ...
- 59
3
votes
0
answers
166
views
geometrical or physical interpretation of second Chern classes of Calabi-Yau threefold
It's my first post.
Consider Calabi-Yau threefold $M$ and its tangent bundle $TM$. I know $c_1(TM)=0$ means metric on $M$ is a solution of vacuum Einstein equation. Then my question is "are there any ...
- 31
13
votes
0
answers
743
views
Kähler-Ricci flow approach for Beauville-Bogomolov type decomposition?
Is there any Kähler Ricci flow method for solving structure theorems in Algebraic geometry
In fact If $X$ be a Calabi-Yau manifold then we can descend the Kähler Ricci flow to its finite etale ...
user21574
2
votes
1
answer
408
views
Asymptotic formula for Ricci flat metric
Let $(\mathcal X,\mathcal D)\to T$ be a surjective holomorphic fibre space of K\"ahler manifolds of pairs such that fibers $(X_s,D_s)$ admit Ricci flat metric in bounded geometric sense (conic, ...
- 63
0
votes
1
answer
522
views
Canonical metric on moduli space of singular Calabi-Yau varieties
Let $\pi:X\to Y$ be a surjective holomorphic map with connected fibers
and let fibers are singular Calabi-Yau varieties (i.e. numerical dimension is zero) then is it possible to construct canonical ...
user21574
15
votes
3
answers
1k
views
Can Calabi-Yau manifolds have nonabelian discrete symmetry groups?
A particle physicist asked me the above question. Let me try to make it more precise. Suppose $M$ is a 3-dimensional Calabi-Yau manifold: that is, a compact Kähler manifold of complex dimension ...
- 22.3k
5
votes
1
answer
430
views
Explicit metrics on non-compact Calabi-Yau threefolds
I would like to know which explicit metrics on non-compact Calabi-Yau (CY) threefolds are known.
For instance, an important class of such spaces can be constructed algebraically, including local $\...
- 333
6
votes
1
answer
413
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A technical question in Feix's construction of hyperkahler metric on cotangent bundles
I am now reading Feix's paper Hyperkahler metrics on cotangent bundles
and I have a technical question to ask.
In his paper, for an analytic Kähler manifold $(X,J,\omega)$, Feix considered its ...
- 783
3
votes
1
answer
818
views
Questions on the Hodge Dual of the Kähler Class
Let $M$ be a compact complex surface, i.e., a complex manifold with complex dimension two. Also, let us denote its Kähler metric by $g$ and its Kähler form by $K$. Let us denote the Kähler class by $[...
- 424
18
votes
3
answers
1k
views
Moishezon manifolds with vanishing first Chern class
Suppose $M$ is a Moishezon manifold with $c_1(M)=0$ in $H^2(M,\mathbb{R})$. Does it follow that $K_M$ is torsion in $\mathrm{Pic}(M)$?
This is true whenever $M$ is Kähler (and therefore projective) ...
- 6,871
2
votes
2
answers
596
views
calabi conjecture on compact manifolds
hi,
is the calabi conjencture formulated for compact manifolds with boundary ? or only for those without boundary ? excuse me if the question is too trivial but in my literature it isn't mentioned ...
- 213
18
votes
1
answer
3k
views
Theorem of Bryant in higher dimensions
I have the following question. I read about Bryant's theorem which says that: any real-analytic 3-dimensional Riemannian manifold $(Y,g)$ with real-analytic metric $g$ can be isometrically embedded as ...
- 221
13
votes
4
answers
3k
views
Calabi - Yau Manifolds
I just started reading about Calabi-Yau manifolds and most of the sources I came across defined Calabi-Yau manifold in a different way. I can see that some of them are just same and I can derive one ...
- 3,218