Questions tagged [calabi-yau]
Calabi-Yau manifolds are higher dimensional generalizations of elliptic curves and K3 surfaces. They can be defined as the compact complex Kähler manifolds with trivial canonical bundle, and play a central role in mirror symmetry. This tag can also be used for Calabi-Yau algebras and categories. These algebraic notions are inspired by the properties of the derived categories of coherent sheaves on Calabi-Yau manifolds.
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Can we define topological quantum field theories on Calabi-Yau manifolds?
Calabi Yau manifolds are Kähler manifolds with vanishing first Chern class. According to the conjecture of E. Calabi , for a Kähler manifold M , if
$c_1 (M) = 0 $ , then M would admit a Ricci-flat ...
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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
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Positivity properties of virtual Hodge numbers of Calabi-Yaus
Let $X$ be a normal, projective complex variety with an anticanonical divisor $D$. Do the virtual Hodge numbers of the noncompact Calabi-Yau variety $X$ \ $D$ enjoy some sort of positivity property?
...
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$Aut(X)$ and $Bir(X)$ for Calabi-Yau 3-folds with $\rho(X)=1$
Let $X$ be a Calabi-Yau 3-fold with Picard number one. How can one show that the automorphism group $Aut(X)$ is finite and moreover coincides with the birational automorphism group $Bir(X)$?
It seems ...
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Why is the base of SLAG fibration of CY3 expected to be $S^3$?
The SYZ conjecture roughly says that any Calabi-Yau threefold $X$ has a special Lagrnagian fibration $\pi:X\rightarrow B$ by 3-tous with section and one of its mirror partners $\check{X}$ is obtained ...
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Questions on how SYZ conjectures is deduced from HMS conjeture.
The Strominge-Yau-Zaslow conjecture is roughly the following. Any Calabi-Yau $m$-manifold $X$ admits a special Lagrangian $T^m$ fibration (maybe at around a special point in its complex moduli space) ...
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Is there any relativistic interpretation on considering Kaehler-Einstein metrics and Calabi-Yau manifolds?
Perhaps I sould ask this question on a physics forum, but I am curious about answers coming from mathematicians.
Calabi-Yau manifolds are examples of Ricci-flat Kaehler manifolds. As we know, in the ...
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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 ...
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How many Coulomb branches do we (conjecturally) know?
Physics preamble: Attached to any $3$ or $4$ dimensional SCFT it is expected to be a Poisson variety $\mathcal{M}_C$ called the Coulomb branch. It should admit a symplectic resolution.
Moreover, ...
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Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?
For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$.
If ...
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Analytical point of view of Kawamata's Unipotent reduction condition for Calabi-Yau family
Motivation: Unipotent reduction condition is very important for study of family of algebraic varieties. For example for algebraic fiber space if we have such condition then the direct image of ...
user21574
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Moduli space of log Calabi-Yau varieties exists?
Let $\mathcal M^{(X,D)}$ be a moduli space of pair varieties $(X,D)$ which $K_X+D$ is trivial and $D$ is a divisor with conic singularities on Kaehler variety $X$. I am looking for a proof that such ...
user21574
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Why quintics are Calabi-Yau?
Why quintics are Calabi-Yau? Is there a explicit formula of the holomorphic volume form?
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What information is required for SYZ mirror symmetry?
Let $X$ be a Calabi-Yau threefold. The Strominger-Yau-Zaslow conjecture suggests that $X$ should have a special Lagrangian $T^3$-fibration (when $X$ lies near a large complex structure limit) and a ...
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$K_X+B \equiv 0$ implies $K_X + B \sim_\mathbb{Q} 0$?
Let $(X,B)$ be a projective log canonical pair (here I mean $B \geq 0$). Assume that the coefficients of $B$ are rational, and that $K_X+B \equiv 0$. Is it true that $K_X + B \sim_\mathbb{Q} 0$? I ...
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Lelong number of Ricci flat metric
Let $M$ be a compact Kahler Calabi-Yau variety which admit Ricci flat metric $\tilde\omega$, $Ric(\tilde \omega)=0$, then the Lelong number $\tilde \omega$ is zero?
In general if $\omega$ satisfies ...
user21574
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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 ...
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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 $\...
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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 ...
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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 ...
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A Calabi-Yau manifold with finite simple fundamental group?
Is there a known example of a Calabi-Yau manifold (say, a Kähler compact manifold with $c_1$ torsion) with finite simple (non cyclic) fundamental group, for instance $\mathfrak{A}_5$? I am pretty sure ...
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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
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Injective homomorphism induced by cup product in cohomology
Let $M$ be an irreducible holomorphic symplectic manifold of dimension $\geq 4$. In his paper 'A survey of Torelli and Monodromy results', Markman claims (discussion after Theorem 9.7) that the cup ...
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Crepant resolution of isolated fourfold singularity
I stumbled upon this isolated singularity of a Calabi-Yau fourfold:
\begin{equation}
x_1x_2+x_3x_4+x_5^2=0
\end{equation}
as a hypersurface in $\mathbb{C}^5$.
Clearly, I can resolve this by a simple ...
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vector multiplet/hypermultiplet moduli space of String Theory
What is vector multiplet and hypermultiplet moduli space associated to IIA/B string theory (or in general to a N = 2 Supersymmetric theory) ?
The vector multiplet moduli space is special Kahler while ...
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Are there any symplectic but not holomorphic Calabi-Yau manifolds in real dimensions 4 and 6?
Are there any symplectic but not complex Calabi-
Yau manifolds in real dimensions 4 and 6?
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Factoriality of one-nodal Calabi-Yau threefolds
Let $X$ be a projective Calabi-Yau threefold with a single ordinary double point at $x \in X$, and smooth elsewhere. Is $X$ necessarily factorial?
I suspect that the answer is "yes", for the ...
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Examples of Calabi-Yau that are birational to each other?
I was told that Calabi-Yau's can be birational to each other but not isomorphic (biholomorphic).
But I've never seen explicit examples. Can anybody here show me one?
(E.g. maybe an explicit ...
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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 ...
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Why should noncommutative CYs be dgas?
For a (commutative) CY manifold of dimension $n$, Serre duality implies that there are ifunctorial isomorphisms
$$\operatorname{Hom}_{D^b(X)}(E,F)\cong\operatorname{Hom}_{D^b(X)}(F,E[n])^*$$
in the ...
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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 $\neq 2, 3$. ...
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Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$?
On a compact Kahler manifold, let $g$ be the unique Kahler-Einstein metric with $\mathrm{Ricc}(g)=-g$, proved to exist by Yau and Aubin when the first Chern class $C_1(M)<0$.
Question: Does $g$ ...
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Explicitly computing Donaldson-Thomas invariants (of CY 3-folds)
I am interested in the explicit computation of generating functions of rank 1 and higher rank Donaldson-Thomas (DT) invariants. In particular, I am interested in DT invariants of K3 fibered Calabi-Yau ...
user35360
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Examples of Calabi-Yau manifolds with $\mathbb{T}^2$ symmetry
I want to know if there exists examples of Calabi-Yau manifolds with $\mathbb{T}^2$-invariant $SU(n)$-structure. In particular these actions are both Killing and holomorphic. I am especially ...
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When is the birational Torelli problem for CY threefolds true?
I am aware from Borisov, Căldăraru, Perry and Ottem, Rennemo that what is known as the birational Torelli problem is false in general for Calabi-Yau threefolds, but I would like to know if there are ...
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BCOV's holomorphic anomaly equation at genus one
BCOV in their famous paper (http://arxiv.org/abs/hep-th/9309140) state the genus one holomorphic anomaly equation (on page 53) to be $$\partial_i \partial_{\bar{j}} F_{1} = \frac{1}{2}C_{ikl}\bar{C}^{...
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Singularities induced by the toric ambient spaces
Let $\Delta \subset \mathbb{R}^4$ be a (reflexive) polytope and $X$ be the hypersurfacedefined by a generic section of the any-canonical bundle of the toric variety $\mathbb{P}_{\Delta}$. Are there ...
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The link of a singular quintic hypersurface in CP^4
Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by
$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$
we get a singular variety for $\epsilon=0$ with 125 singular points.
I know ...
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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 ...
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Canonical differential on K3 surface
On an elliptic curve over $\mathbb{Q}$, we can associate a canonical Neron model and with it a Neron differential, whose coefficients in some natural coordinates yield the Dirichlet coefficients of ...
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Kuranishi family and smoothing of Calabi-Yau n-fold
Consider $X$ be a Calabi-Yau n-fold with at
most one ordinary double point singularity. Suppose $X$ is smoothable. Then it is known that the Kuranishi family of $X$ is a smoothing of $X$.
Now, ...
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Matsushita theorem on framed variety (X,D)
I have a question about fibrations on Irreducible log holomorphic symplectic manifolds. Lets give some introduction
Motivation; A holomorphic symplectic manifold (HSM) is a $2n$-dimensional compact K\...
user21574
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Calabi-Yau theorem on arithmetic variety
Let $\mathcal X\to \mathrm{Spec}(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$. Let $\omega$ be a Kähler current of $\mathcal X(\...
user21574
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Moduli space of CY3 with $h^{2,1}=1$ is $\mathbb{P}^1\setminus \{p_i\}_i$?
It seems to me that all known CY3 with $h^{2,1}=1$ has the complex moduli space of the form $\mathbb{P}^1\setminus \{p_i\}_i$ for some $i \in \mathbb{N}$.
Is this true for all CY3 with $h^{2,1}=1$? ...
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Ricci flat metrics on holomorphic vector bundles over Riemann surfaces
I am interested in the local geometry of holomorphic curves in Calabi-Yau threefolds. The setup and question are then the following:
Consider a $\mathbb{C}^2$ bundle over a compact Riemann surface $\...
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Example of Calabi-Yau 3-fold fibered by both K3 surface and abelian surface?
I am looking for a compact Calabi-Yau 3-fold which is fibered by both K3 surface and abelian surface (with possibly singular fibers). Are there any examples?
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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 $[...
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(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 \...
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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 ...
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How to find a rational $\mathbb{F}_{\!q}$-curve on a quite classical Calabi–Yau threefold?
Take a finite field $\mathbb{F}_{\!q}$ such that $q \equiv 1 \pmod 3$, i.e., $\omega \mathrel{:=} \sqrt[3]{1} \in \mathbb{F}_{\!q}$, $\omega \neq 1$. Also, for $i \in \{0,1,2\}$ consider the elliptic ...
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