All Questions
Tagged with calabi-yau mirror-symmetry
22 questions
1
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0
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270
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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 ...
3
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0
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209
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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
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120
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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
4
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237
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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
5
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247
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Calabi-Yau structures on dg-categories
A (smooth) dg algebra is called (left) Calabi-Yau if (see for example here)
$$ A^! = A[-n]$$
Here we use the inverse dualizing complex $A^!=\mathbf{R}\operatorname{Hom}_{(A^e)^{op}}(A,A^e)$. In ...
- 898
2
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507
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Embedding Calabi-Yau manifolds in projective space
When studying homological mirror symmetry, a lot of work is done not in the setting of complex manifolds, but of smooth (quasi-)projective varieties, see e.g. a paper from Orlov. However, the actual ...
- 898
7
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2
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520
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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
2
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349
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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
11
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1
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822
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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
12
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0
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731
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Any progress on Strominger-Yau and Zaslow conjecture?
In 2002 Hausel - Thaddeus interpreted SYZ conjecture in the context of Hitchin system and Langlands duality. Let briefly explain it
Let $\pi : E \to Σ$ a complex vector bundle of rank $r$ and ...
user21574
5
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0
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628
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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
7
votes
0
answers
760
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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
3
votes
1
answer
479
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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
7
votes
2
answers
1k
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Mirror Symmetry for Quaternionic-Kähler Manifolds
I take the following quote from Huybrecht's notes on hyperkähler manifolds and mirror symmetry:
Mirror symmetry in a first approximation predicts for any Calabi-Yau manifold (M,g) the existence of ...
- 465
35
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1
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1k
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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
2
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1
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323
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Special Lagrangians and fat
I am unable to find the MO comments about the first use of the phrase "fat slags" in an article. On page 26 of this we find "these correspond to thickenings of the
corresponding special Lagrangian (...
2
votes
0
answers
375
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fake Calabi-Yau threefold
(1) What is a "fake Calabi-Yau threefold"? Can I describe as a complex or symplectic manifold with trivial canonical bundle, but no compatible Kahler structure? Some mathematicians actually seem to ...
- 119
4
votes
2
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460
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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
9
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3
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915
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Why is the mirror of rigid Calabi-Yau threefold singularity theory?
Mirror symmetry relates two Calabi-Yau threefolds with mirrored Hodge diamonds. Since Calabi-Yau threefold is Kahler, this naive correspondence does not hold for rigid Calabi-Yau threefolds. Here ...
- 583
11
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1
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1k
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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
7
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3
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2k
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what is large compex structure limit of CY moduli space
What is the Large Complex Structure limit(LCL) of complex moduli space of a Calabi-Yau 3-fold and why do we need to consider LCL in Mirror symmetry.
- 3,218
14
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1
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3k
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Witten's topological twisting
I am reading the Witten's topological twisting for $N = 2$ Superconformal Field Theory(SCFT) http://arxiv.org/abs/hep-th/9112056
In this paper Witten constructed 2 TQFTs i.e. A-model and B-model from ...
- 3,218