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.
149 questions
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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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125
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Calibrated submanifolds in Spin(7) and Calabi-Yau threefold
Suppose I have a Cayley cycle in a $Spin(7)$ holonomy manifold $M$, i.e. a calibrated submanifold. In the special case that $M=CY_3\times T^2$, is it possible that the Cayley cycle reduces to a Kahler ...
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What is the meaning of rationality for these series?
Let me start with a couple of examples of rationality.
Let $X$ be a nonsingular, projective Calabi-Yau threefold. Let $\beta\in H_2(X)$ be a homology class. The rationality of the reduced Donaldson-...
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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
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Calabi-Yau manifolds and knot theory
In the paper "The Volume Conjecture and Topological Strings" it is said that the mirror Calabi-Yau threefold is given by
$X := \{ (x,y,u,v) \in \mathbb{C^* \times\mathbb{C^*} \times \mathbb{C} \...
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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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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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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, ...
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Constraints on the base of an elliptically fibered Calabi-Yau threefold
Let $X\to B$ be an elliptic fibration over a base $B$. I assume that both $X$ and $B$ are smooth projective varieties. The elliptic fibration has a rational section.
If $X$ is a Calabi-Yau variety (...
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How do you get the spectral curve from a Calabi-Yau?
In N=2 Quantum Field Theories and Their BPS Quivers by Alim, Cecotti, Córdova, Espahbodi, Rastogi and Vafa the authors give a recipe which constructs from a pair $(C,\phi)$ consisting of a Riemann ...
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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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Fibration when central fibre is a Calabi-Yau variety with canonical singularities
Let $f\colon X\to Y$ be a surjective proper holomorphic fibre space such that $X$ and $Y$ are projective varieties and central fibre $X_0$ is Calabi-Yau variety with canonical singularities, then can ...
user21574
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Find the Picard Fuchs operator of a four parameter fundamental period
In my research, we have constructed a Calabi-Yau as a hypersurface of a toric variety and we could compute the fundamental period of the complex moduli, which is a series in four parameters. Say
\...
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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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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
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Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)
Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$.
I have seen another post on ...
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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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Holomorphic vector fields on compact complex manifolds with trivial canonical bundle
Let $M$ be a compact complex manifold whose canonical bundle $K_M$ is holomorphically trivial. Is it possible for $M$ to admit a non-zero holomorphic vector field with zeroes? Equivalently, using a ...
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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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Weil-Petersson metric is quasi isometric with which model?
Let $\mathcal M_g$ be the moduli space of curves of genus $g$. If we take $X^{reg}=X\setminus D$, where D is a divisor with normal crossings. Endow
$X^{reg}$ with a complete Kahler metric which has a ...
user21574
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Ricci flat metric on pair (X,D)
Let $(X,\omega)$ be a Calabi-Yau variety and $D$ be a simple normal crossing divisor on $X$ with conic singularities with cone angle $2\pi\theta$, $0<\theta<1$ such that $K_X+D>0$, then is ...
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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 ...
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Line bundles with vanishing cohomology on Calabi-Yau manifold
Suppose we have some line bundle $L(D)$ on Calabi-Yau threefold. Let's call this line bundle "rigid" if $H^0(X,L(D)) \simeq \mathbb{C}$ and $H^i(X,L(D))=0$ for $i=1,2,3$.
Is anything known about such ...
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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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Weil Petersson metric on moduli space of Calabi Yau manifolds
Let $f:(X,D)\to Y$ be a holomorphic fibre space where $D$ is divisor with conic singularities and let fibres $(X_s,D_s)$ are log Calabi-Yau pair .i.e $K_X+D$ is nummerically trivial, then we have ...
user21574
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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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central charge and Calabi-Yau dimension
I would like to know if there is any setting where the two notions of
central charge of 2D conformal field theories,
Calabi-Yau dimension of fractionally Calabi-Yau categories
can be understood as "...
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Flat cohomology of an ordinary liftable Calabi-Yau threefold
Let $k$ be a perfect field of characteristic $p>0$ and consider an ordinary liftable Calabi-Yau threefold $X_{0}/k$. By this I mean that $H^{i}(X_{0},B_{X_{0}/k}^{j})=0$ for all $i\geq 0$ and $j\...
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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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Can a rigid CY threefold have infinitely many automorphisms
Let $X$ be a rigid Calabi-Yau threefold. Does $X$ have only finitely many automorphisms?
N.B. A smooth projective threefold $X$ over $\mathbb C$ is a rigid Calabi-Yau variety if $h^i(X,\mathcal O_X) =...
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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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Singularities of the moduli stack of Calabi-Yau threefolds
Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack?
In many cases I ...
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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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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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Resolving nodes of a quintic CY 3-fold
Let's consider the following quintic 3-fold $X$:
\begin{equation}
\{(x_i) \in \mathbb{P}^4 \ | \ x_1f(x)-x_2g(x)=0\}
\end{equation}
for generic homogeneous polynomials $f(x),g(x)$ of degree four. It ...
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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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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 ...
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Contracting a rational curve in a Calabi-Yau threefold
Let $X$ be a Calabi-Yau threefold and $C \subset X$ be a rational curve with $N_{C/X}\cong \mathcal{O}\oplus \mathcal{O}(-2)$. Can one contract the curve $C$? Assuming the answer is yes, what kind of ...
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State of the art of BPS and Donaldson-Thomas invariants for toric Calabi-Yau threefolds
I am trying to understand what has been done with regards to computing BPS invariants and Donaldson-Thomas type invariants of Calabi-Yau threefolds. To make the question more focused, let's say that I ...
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2
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L-functions of Calabi-Yau varieties
This question might not be suitable for MO since i know nothing about Calabi-yau varieties aside the fact that they are used in string theory to compactify additional dimensions, but still, it makes ...
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The moduli space of special Lagrangian submanifolds
Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures (symplectic and complex) on the base space $B$. A theorem of ...
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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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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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1
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Properties of extreme rays
Let $X$ be a projective variety over $\mathbb{C}$, then the effective curves module the numerical equivalence form a cone. For my understanding, if the extreme ray $[C]$ has the property that $[C]\...
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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
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2
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467
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Necessary condition on Calabi-Yau manfiold to be a hypersurface in a Fano manifold
Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it
being a member of a base-point-free linear system in a nef-Fano fourfold?
What, in anything, is known ...
- 5,186
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0
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324
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2 - Calabi Yau algebras and bimodule coherence
Let $\Pi:=\Pi(Q)$ be the preprojective algebra of a connected non-Dynkin quiver over an algebraically closed field of characteristic zero.
In H. Minamoto "Ampleness of two-sided tilting complexes", ...
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2
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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 Yau's theorem implies the existence of hyperkähler metric on complex symplectic manifolds?
Every expository article on hyperkähler manifolds that I have read states without detailed proof the following fact:
It follows from Yau's theorem (i.e. a compact Kähler manifold $M$ with $c_1(M)=0$ ...
- 1,804
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1
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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 (...