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. ...

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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} = ...
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151 views

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 ...
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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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147 views

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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264 views

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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60 views

How exactly do we construct the $T^2\times \mathbb{R}$ toric Calabi-Yau three-fold?

I am trying to understand why and how the functions $r_{a}(z) = |z_1|^2 - |z_3|^2$, $r_{b}(z)=|z_2|^2 - |z_3|^2$ and $r_{c}(z)=\Im(z_1z_2z_3)$ "generate" the toric CY threefold $T^2 \times \mathbb{R}$ ...
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255 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 ...
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164 views

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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60 views

Conifold singularities on the mirror quintic

It is well-known that conifold transitions go in the reversed direction under mirror symmetry. My question is, what is the mirror picture of a degeneration of the generic quintic CY3 to the one ...
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1answer
120 views

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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90 views

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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2answers
518 views

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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314 views

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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291 views

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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2answers
247 views

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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515 views

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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646 views

Why quintics are Calabi-Yau?

Why quintics are Calabi-Yau? Is there a explicit formula of the holomorphic volume form?
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235 views

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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196 views

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 ...
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868 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 $ ...
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2answers
315 views

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 ...
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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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2answers
188 views

$Aut(X)$ 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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665 views

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 $c_1(M)=0$ ...
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1answer
187 views

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 ...
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307 views

Cohomology and conifold transition for the quintic

Let $Y\subset \mathbb{C}P^4$ be the quintic threefold given by the equation $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4+5X_0X_1X_2X_3X_4=0$$ it has 125 singular points whose links are homeomorphic to $S^2\times ...
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204 views

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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Moduli space of K3 surfaces and Bogomolov-Tian-Todorov theorem

The famous Bogomolov-Tian-Todorov theorem says that the moduli space of Calabi-Yau manifold is smooth, that is locally a complex manifold. Doesn't this contradicts to the fact that the moduli space ...
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3answers
337 views

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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Why is the mirror of resolved conifold the deformed conifold?

I often see a sentence like "the mirror of resolved conifold is the deformed conifold" in physics literature. I would like to ask, in what sense is this true? What is known for mirror symmetry for ...
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233 views

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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256 views

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 ...
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145 views

Physical invariants of Calabi-Yau manifolds and G2 manifolds

Physicists said that for a given Calabi-Yau 3-fold with the topological Euler number $e$, $|e|/2$ corresponds to the number of generations of the elementary particles. My question is: what ...
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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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2answers
203 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 ...
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Is Gromov-Witten theory of Calabi-Yau threefolds of Type A trivial?

There are some Calabi-Yau threefolds that do not contain any rational curves, e.g. Calabi-Yau threefold of type A in the paper "Calabi-Yau threefolds of quotient type" by Oguiso and Sakurai. My ...
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2answers
390 views

What can one say about (differentiable) topological structure of CY3s?

It is known that there is unique differantial topological structure on the elliptic curves or K3 surfaces over $\mathbb{C}$. Since we know tons of Hodge diamonds for Calabi-Yau threefolds, we cannot ...
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339 views

Is a flop on Calabi-Yau threefolds always Atiyah flop?

Is it true that any flop on a Calabi-Yau threefold is given by the Atiyah flop? That is, there always exists a rigid rational curve $\mathbb{P}^1$ with normal bundle ...
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441 views

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 ...
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250 views

A question on fibered Calabi-Yau threefolds

Let $\phi:X\rightarrow \mathbb{P}^1$ be a fibered Calabi-Yau threefold with a general fiber $F$. The following are known $\phi=\Phi_{mF}$ for some $m\in \mathbb{N}$, where $\Phi_D$ stands for the ...
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412 views

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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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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742 views

Derived category of varieties and derived category of quiver algebras

I have heard that derived category of coherent sheaves $\mathrm{Coh}(X)$ on any Fano varieties $X$ may be realized as derived category $\mathrm{Coh}(\mathrm{Rep}(Q,W))$ of representation of quiver $Q$ ...
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1answer
506 views

For which Calabi-Yau threefolds is SYZ conjecture known to hold?

I would like to know for which Calabi-Yau threefolds SYZ conjecture is known to hold. I am aware of works by Gross-Wilson (Borcea-Voisin CY3s) and Ruan (quintic CY3), but they are quite classical ...
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677 views

Today's world record on the Betti numbers of Calabi-Yau three-folds.

What are largest betti numbers $b_2$ and $b_3$ of three-dimensional Calabi-Yau manifolds that are discovered for today? Is there some nice reference?
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383 views

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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290 views

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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612 views

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 ...