All Questions
13 questions
35
votes
1
answer
1k
views
What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?
By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ (h^{...
- 5,186
11
votes
1
answer
1k
views
Is $Sym^g$ of a Riemann Surface of genus $g$ Calabi-Yau?
The $g$-fold symmetric product of a Riemann surface of genus $g$ naturally has both a symplectic structure as well as a complex structure. Is it in fact Calabi-Yau? If so, is anything known about a ...
- 2,283
11
votes
1
answer
822
views
Large Complex Structure Limit of Calabi-Yau family and uniqueness of limit
Let $\mathcal X$ be a smooth complex manifold of dimension $n+1$. We say $\mathcal X \to ∆$ is a large complex structure limit if and only if it’s maximal unipotent degeneration .
$T: H^n(\mathcal ...
user21574
7
votes
2
answers
521
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 ...
- 375
7
votes
0
answers
760
views
Examples of Maximal degeneration of Deligne on Calabi-Yau degeneration
Pierre Deligne in his celebrated paper entitling "Local behavior of Hodge structures at infinity" introduced Maximal degenerations of Calabi-Yau manifolds.
Let $\pi:X\to \mathbb C^*$ be a family of ...
user21574
5
votes
0
answers
629
views
connectedness of moduli space of Calabi-Yau 3-folds by symplectic surgery theory
"Motto" Moduli space of Calabi-Yau varieties can be connected by using Symplectic surgery theory.
Miles Reid’s Fantasy:“There is only one Calabi-Yau space”
i.e "All CY connected through conifold ...
user21574
4
votes
2
answers
460
views
A question on the topological change of dualizing a SLAG fibration.
Let $S$ be a K3 surface and $\pi:S\rightarrow B$ be a SLAG $T^2$-fibration. I am struggling with a statement that
Fiberwise dualization does not change the topology of $S$.
Here by fiberwise ...
- 41
4
votes
1
answer
237
views
Mirror partners of some Calabi-Yau threefolds
I don't have experience in mirror symmetry, hence I am not sure that my question is of research level. Sorry in advance.
Let $k$ be an algebraically closed field of characteristic $\neq 2, 3$. ...
- 1,833
3
votes
1
answer
479
views
How to construct (another) Landau-Ginzburg model for a compete intersection Calabi-Yau?
For Calabi-Yau variety $X$ which is a complete intersection
$$
f_1=f_2=\ldots=f_r=0
$$
in ${\mathbb P }^n$ (hence $\mathrm{dim}\,X=n-r$) it is possible to construct a Landau-Ginsburg model in the ...
- 600
3
votes
0
answers
209
views
Proof of the existence of a mirror Calabi–Yau manifold
Let $X$ be a Calabi–Yau threefold. Here, Calabi–Yau is understood to a mean a smooth simply connected projective threefold with $h^1(\mathcal{O}_X) = h^2(\mathcal{O}_X)=0$ and holomorphically trivial ...
- 1,379
3
votes
0
answers
120
views
What sort of spaces show up as intersection complexes of toric degenerations of Calabi-Yau Varieties?
Roughly, a toric degeneration is a proper flat family $f:\mathcal{X}\to D$ of relative dimension $n$ with the properties that $\mathcal{X}_t$ is an irreducible normal Calabi-Yau and $\mathcal{X}_0$ is ...
- 191
2
votes
0
answers
349
views
SYZ conjecture for varieties of general type or Fano
Let $X$ and $Y$ are Calabi-Yau varieties and mirror to each other. Then from HMS the Fukaya Floer category of Lagrangian intersections in $X$, is equivalent to bounded derived category of coherent ...
user21574
1
vote
0
answers
271
views
Are there connections between Calabi-Yau manifolds and number theory?
I am interested in understanding whether there are any significant connections between Calabi-Yau manifolds and number theory. Calabi-Yau manifolds are central objects in algebraic geometry and string ...
