All Questions
Tagged with complex-geometry calabi-yau
45
questions
2
votes
0
answers
124
views
Hodge coniveaux of Calabi-Yau manifolds
Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
- 279
1
vote
0
answers
110
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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
1
vote
0
answers
180
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Betti numbers of threefolds with trivial canonical class
I am interested in a simply-connected compact complex manifold $M$ of dimension three with trivial canonical class.
Note that if it is K"ahler, then it is a Calabi-Yau threefold.
Its independent ...
- 1,831
2
votes
0
answers
144
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 ...
- 1,831
3
votes
0
answers
182
views
Smallest Hodge numbers of Calabi-Yau threefolds ever found
By a Calabi-Yau threefold, I mean a simply-connected smooth compact K"ahler threefold with trivial canonical class.
It has two independent Hodge numbers $h^{1,1}$ and $h^{1,2}$.
What is the ...
- 1,831
4
votes
0
answers
147
views
Calabi–Yau theorem and complex Monge–Ampère equation for transversally Kähler manifolds
Let $M$ be a compact smooth
manifold, and $F\subset TM$ a smooth
foliation. It is called transversally Kähler
if the normal bundle $TM/F$ is equipped with
a Hermitian structure (that is, a complex ...
- 9,157
5
votes
1
answer
283
views
Mirror symmetry for K3 fibered Calabi-Yau threefolds
By a K3 fibered Calabi-Yau threefold, I mean a smooth projective threefold $X$ with trivial canonical class and
$h^{1,0}(X) =h^{2,0}(X) = 0$ that has a fibration $X \rightarrow \mathbb P^1$ whose ...
- 1,831
5
votes
0
answers
286
views
Examples or references for this claim about elliptic Calabi-Yau threefolds
In this article (page 2) , the authors say:
"it is expected, based on known examples, that Calabi–Yau threefolds of large Picard
rank are always elliptically fibered, perhaps after flopping a ...
- 1,831
2
votes
0
answers
423
views
What is a moduli space of Calabi-Yau threefolds?
A Calabi-Yau threefold is a compact Kahler threefold which is simply connected and has trivial canonical bundle.
So my question is as in the title. What is the moduli space of such objects? I'm ...
- 896
3
votes
0
answers
146
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
1
answer
369
views
(1/2) K3 surface or half-K3 surface: Ways to think about it?
I heard from string theorists thinking of the so-called "(1/2) K3 surface" or "half-K3 surface" as follows:
Let $T^2 \times S^1$ be a 3-torus with spin structure periodic in all directions. $T^2 \...
- 10.4k
9
votes
0
answers
831
views
Is there a Calabi-Yau threefold with $h^{1,1}=1$ and $h^{1,2}=0$?
Is there a Calabi-Yau threefold with $h^{1,1}=1$ and $h^{1,2}=0$?
- 726
2
votes
0
answers
161
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 ...
- 21
7
votes
1
answer
741
views
What is the geometrical meaning of higher Chern forms and classes?
Let $M$ be a complex manifold, $R^{\nabla}$ be the curvature operator for connections $\nabla$.
Consider a polynomial function $f:\operatorname M_n(\mathbb{C})\to\mathbb{C}$. For the gauge group $\...
- 321
7
votes
2
answers
488
views
Multiple mirrors phenomenon from SYZ and HMS perspective
There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties ...
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