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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Hodge coniveaux of Calabi-Yau manifolds

Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
Pène Papin's user avatar
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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 ...
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Do there exist Calabi-Yau 3-folds that contain a finite number of elliptic curves?

The moduli space $M_1(X, e)$ of degree $e$ elliptic curves on $X$ has virtual dimension zero if $X$ is a Calabi-Yau 3-fold. I am wondering if there is an example of such an $X$ so that each $M_1(X, e)$...
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Contracting a family of rational curves in a Calabi Yau threefold

Suppose we have a Calabi-Yau 3-fold $X$ (not necessarily compact, over $\mathbb{C}$) that contains a ruled surface over a smooth curve $C$ of genus $g$. I am using a strong definition of a ruled ...
Sungwoo's user avatar
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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, ...
Pulcinella's user avatar
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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 ...
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Is Stenzel's Ricci-flat metric on $T^*\mathbb{CP}^n$ hyperkahler?

In a well-known paper, Stenzel constructed complete Ricci-flat Kahler metrics on the total spaces of cotangent bundles of $S^n$, $\mathbb{RP}^n,$ $\mathbb{CP}^n$, $\mathbb{HP}^n$, and $\mathbb{OP}^2$. ...
Jesse Madnick's user avatar
1 vote
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Betti numbers of threefolds with trivial canonical class

I am interested in a simply-connected compact complex manifold $M$ of dimension three with trivial canonical class. Note that if it is K"ahler, then it is a Calabi-Yau threefold. Its independent ...
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Minimal Betti numbers of simply-connected threefolds with trivial canonical class

By a threefold, I mean a compact complex manifold of dimension three. For a simply-connected threefold with trivial canonical class, its Betti numbers satisfy: $$b_2 \ge 0, b_3 \ge 2.$$ I am wondering ...
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Iterated integrals on higher dimensional Calabi-Yau manifolds?

I recently read about the construction of closed quasi-periodic differential forms on elliptic curves (1-dim Calabi-Yaus) via the Kronecker-Eisenstein series. I now wonder if similar constructions are ...
DaveWasHere's user avatar
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181 views

Smallest Hodge numbers of Calabi-Yau threefolds ever found

By a Calabi-Yau threefold, I mean a simply-connected smooth compact K"ahler threefold with trivial canonical class. It has two independent Hodge numbers $h^{1,1}$ and $h^{1,2}$. What is the ...
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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 ...
Misha Verbitsky's user avatar
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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 ...
Zhenhua Liu's user avatar
5 votes
1 answer
266 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 ...
Basics's user avatar
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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 ...
AT0's user avatar
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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 ...
Basics's user avatar
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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 ...
Cranium Clamp's user avatar
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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 ...
Mykola Pochekai's user avatar
4 votes
1 answer
215 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 $\neq 2, 3$. ...
Dimitri Koshelev's user avatar
3 votes
0 answers
190 views

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 ...
Josh Lam's user avatar
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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, ...
Shiyu's user avatar
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1 answer
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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 ...
Dimitri Koshelev's user avatar
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0 answers
216 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 ...
Markus Zetto's user avatar
2 votes
0 answers
426 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 ...
Markus Zetto's user avatar
3 votes
1 answer
362 views

(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 \...
wonderich's user avatar
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2 answers
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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
0 answers
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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 $h^{1,2}=0$?
rj7k8's user avatar
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1 answer
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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 ...
u184's user avatar
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0 answers
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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 ...
anonymous's user avatar
5 votes
0 answers
294 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 ...
abx's user avatar
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7 votes
1 answer
716 views

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 $\...
Mishkaat's user avatar
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0 answers
192 views

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 ...
Mishkaat's user avatar
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2 votes
0 answers
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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 ...
AmorFati's user avatar
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9 votes
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380 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 ...
Federico Carta's user avatar
7 votes
2 answers
475 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 ...
paul's user avatar
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5 votes
1 answer
297 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 ...
Stefano's user avatar
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6 votes
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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 ...
doetoe's user avatar
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4 votes
0 answers
193 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 ...
xir's user avatar
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3 votes
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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 ...
Andrey Feldman's user avatar
15 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 ...
asv's user avatar
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7 votes
1 answer
605 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}(...
weissss's user avatar
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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 ...
ChiHong Chow's user avatar
4 votes
0 answers
170 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, ...
Larue's user avatar
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13 votes
0 answers
707 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 ...
user avatar
6 votes
1 answer
201 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 ...
L.F. Cavenaghi's user avatar
2 votes
0 answers
328 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 ...
user avatar
11 votes
1 answer
796 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 ...
user avatar
6 votes
0 answers
256 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 ...
user avatar
2 votes
0 answers
122 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 ...
sam's user avatar
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3 votes
0 answers
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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-...
Andrea Ricolfi's user avatar