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.
62 questions with no upvoted or accepted answers
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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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14
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857
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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 $\mathcal{O}_{\mathbb{P}^1}(-1)^{\...
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13
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0
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743
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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 ...
user21574
12
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731
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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
12
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0
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729
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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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12
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0
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580
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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 S^...
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836
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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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9
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402
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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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9
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519
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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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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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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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7
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760
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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
7
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295
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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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6
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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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6
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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
6
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509
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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
5
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300
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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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247
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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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5
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319
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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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5
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628
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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
5
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260
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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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159
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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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4
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200
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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 ...
- 2,054
4
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182
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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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4
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254
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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
4
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581
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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
4
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answers
67
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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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4
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328
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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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118
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Decomposition of forms in $\operatorname{SU}(4)$-manifold
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let $(X,\Omega,\omega,J)$ be a manifold with an $\SU(4)$ structure. Since $\SU(4)\subset\Spin(7)$, $X$ also has a $\Spin(7)$-structure. I ...
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209
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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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3
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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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3
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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 ...
- 1,841
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170
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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 ...
- 1,307
3
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216
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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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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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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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3
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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 ...
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3
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189
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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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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", ...
3
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334
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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 map ...
- 31
2
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101
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Conjecture on the moduli space of stable sheaves on Calabi-Yau threefolds
Let $X$ be a smooth projective Calabi-Yau threefold over $\mathbb{C}$. Let $M_{X}(r,c_1,c_2)$ denote the moduli space of Gieseker-stable sheaves on $X$ with Mukai vector $(r,c_1,c_2)$.
Is the ...
2
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0
answers
129
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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 ...
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2
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200
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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 ...
- 1,841
2
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0
answers
151
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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 ...
- 1,841
2
votes
0
answers
481
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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 ...
- 936
2
votes
0
answers
507
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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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2
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0
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147
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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 ...
- 1,379
2
votes
0
answers
349
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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 ...
user21574
2
votes
0
answers
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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2
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0
answers
167
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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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