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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Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?

In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the ...
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6 votes
1 answer
184 views

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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4 votes
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198 views

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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6 votes
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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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2 votes
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190 views

What is a moduli space of Calabi-Yau threefolds?

A Calabi-Yau threefold is a compact Kahler threefold which is simply connected and has trivial canonical bundle. So my question is as in the title. What is the moduli space of such objects? I'm ...
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3 votes
0 answers
127 views

Does existance of crepant resolution of tangent space imply existance of crepant resolution globally in the algebraic setting?

Suppose $X$ is smooth proper algebraic $\mathbb C$-variety with algebraic action of a finite abelian group $G$. Suppose I know that $X/G$ (good geometric quotient) exists and it is normal Gorenstein ...
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3 votes
1 answer
158 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 $\neq 2, 3$. ...
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3 votes
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When do crepant resolutions of quotients of Calabi-Yau varieties exist?

Suppose I have a Gorenstein variety $X$ over $\mathbb{C}$ with trivial canonical bundle, and the action of a finite group $G$ on $X$, which acts trivially on the canonical bundle. Question. When does ...
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3 votes
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134 views

Why does the bisectional curvature blow up?

Suppose we have $X = X_1 \times X_2$, where $X_1$ is a Calabi-Yau manifold (i.e. $c_1(X_1) = 0$) and $X_2$ is a compact Kähler manifold of $c_1(X_2) < 0$. We can consider the metric $\omega(t, x_1, ...
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1 answer
291 views

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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5 votes
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138 views

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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2 votes
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216 views

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 ...
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3 votes
1 answer
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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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4 votes
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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 ...
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9 votes
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Is there a Calabi-Yau threefold with $h^{1,1}=1$ and $h^{1,2}=0$?

Is there a Calabi-Yau threefold with $h^{1,1}=1$ and $h^{1,2}=0$?
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4 votes
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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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2 votes
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geometrical or physical interpretation of second Chern classes of Calabi-Yau threefold

It's my first post. Consider Calabi-Yau threefold $M$ and its tangent bundle $TM$. I know $c_1(TM)=0$ means metric on $M$ is a solution of vacuum Einstein equation. Then my question is "are there any ...
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5 votes
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229 views

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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7 votes
1 answer
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What is the geometrical meaning of higher Chern forms and classes?

Let $M$ be a complex manifold, $R^{\nabla}$ be the curvature operator for connections $\nabla$. Consider a polynomial function $f:\operatorname M_n(\mathbb{C})\to\mathbb{C}$. For the gauge group $\...
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7 votes
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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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2 votes
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Calabi $C^3$ estimate

I have a question regarding a computation analogous to the Calabi $C^3$ estimate which is used in the proof of the Calabi--Yau theorem. Motivation: Establishing Liouville type theorems for complex ...
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9 votes
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320 views

Higher homotopy groups of Calabi-Yaus

Is something known about the higher homotopy groups of Calabi-Yau threefolds? For example, one of the easiest CYs is the quintic, defined as the anticanonical divisor in $\mathbb{CP}_4$. What are its ...
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7 votes
1 answer
316 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 ...
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5 votes
1 answer
257 views

$K_X+B \equiv 0$ 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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6 votes
0 answers
177 views

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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4 votes
0 answers
183 views

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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3 votes
0 answers
113 views

Open Gromov-Witten invariants via lozalization with $\mathbb{C}^{*}$ (not $S^1$) action

Amplitudes of open A-model on a Calabi-Yau 3-fold $X$ with branes are given by the open Gromov-Witten invariants of $X$. It is known how to compute them if there is a toric action on a manifold, which ...
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14 votes
2 answers
2k views

Deformations of Calabi-Yau manifolds

Let $X$ be a compact complex smooth manifold with holomorphically trivial canonical class. It is true that any (sufficiently small?) deformation of the complex structure of $X$ also has ...
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  • 19k
7 votes
1 answer
559 views

Hodge Numbers and Leray Spectral Sequence

Mark Gross' notes survey of SYZ fibrations and toric degenerations begin by explaining why dual torus fibrations interchange Hodge numbers. But he defined the Hodge numbers in an unusual way $$h^{p,q}(...
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1 vote
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A question about the proof of McLean's theorem: why is the space of $C^{k,\alpha}$ exact $p$-forms a Banach space?

In the famous paper "Deformations of Calibrated Submanifolds" by Robert McLean, he showed that given a smooth compact special Lagrangian $L$ in a Calabi-Yau manifold $(X^{2n},\omega,\Omega)$, there is ...
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4 votes
0 answers
158 views

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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13 votes
0 answers
647 views

Kähler-Ricci flow approach for Beauville-Bogomolov type decomposition?

Is there any Kähler Ricci flow method for solving structure theorems in Algebraic geometry In fact If $X$ be a Calabi-Yau manifold then we can descend the Kähler Ricci flow to its finite etale ...
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6 votes
1 answer
173 views

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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2 votes
0 answers
299 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 ...
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11 votes
1 answer
750 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 ...
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6 votes
0 answers
252 views

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 ...
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2 votes
0 answers
111 views

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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3 votes
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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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10 votes
0 answers
619 views

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 ...
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10 votes
1 answer
593 views

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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5 votes
0 answers
435 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 ...
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7 votes
0 answers
634 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 ...
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2 votes
1 answer
387 views

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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1 vote
1 answer
427 views

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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8 votes
1 answer
343 views

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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  • 2,113
5 votes
1 answer
319 views

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 ...
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1 vote
1 answer
428 views

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 ...
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1 vote
1 answer
136 views

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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  • 2,883
4 votes
0 answers
247 views

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\...
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0 votes
1 answer
486 views

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 ...
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