---
created: 2023-03-31T12:07
updated: 2024-01-01T23:13
---

---
date: 2022-01-15 21:49
modification date: Wednesday 9th February 2022 12:13:13
title: Calabi-Yau
aliases: [Calabi-Yau, CY, CYs, Calabi-Yaus]

---


---

- Tags 
	- #AG #AG/deformation-theory
- Refs:
	- #todo/add-references
- Links:
	- [Enriques-Kodaira Classification](Enriques-Kodaira%20Classification.md)
	- [canonical bundle](Unsorted/canonical%20bundle.md)

---

## Calabi-Yau

![](attachments/2023-02-08-abcd.png)
See [gauged linear sigma model](Unsorted/GLSM.md)
![](attachments/Pasted%20image%2020221110215552.png)

![](attachments/Pasted%20image%2020220422224805.png)

![](attachments/Pasted%20image%2020220213223913.png)

- Various definitions:
	- A compact [Kahler manifold](Kahler%20manifold) with vanishing [first Chern class](Unsorted/Chern%20class.md) $c_1 = 0$ which is [Ricci-flat](Unsorted/Ricci-flat.md).
	- A smooth proper variety $X\in \Var\slice k$ with trivial [canonical bundle](Unsorted/canonical%20bundle.md), so $\omega_X \da \Extpower^{\dim X}\Omega^1_{X\slice k} \cong\OO_X$.
	When $k=\CC$, the trivialization must be holomorphic and not just topological!
	- A [riemannian manifold](Unsorted/riemannian%20manifold.md) $X$ of even real dimension $\dim_\RR(X) = 2n$ with [Holonomy](Holonomy) $\holonomy(X) \subseteq \SU_n \subset \Orth_{2n}(\RR)$.


## Motivations

- Setting for [Unsorted/mirror symmetry](Unsorted/mirror%20symmetry.md) : the symplectic geometry of a Calabi-Yau is "the same" as the complex geometry of its mirror.

- Applications: Physicists want to study $G_2$ manifolds (an exceptional Lie group, automorphisms of octonions), part of $M\dash$theory uniting several [superstring theories](Unsorted/Superstring%20theory.md), but no smooth or complex structures. 
  - Indirect approach: compactify add one small $S^1$ dimension to a 10 dimensional space to compactify, yielding an 11 dimensional space. 
    - Why 10 dimensions: 4 from spacetime and 6 from a "small" [Calabi-Yau threefold](Unsorted/Calabi-Yau%20manifold.md).

- Yau, Fields Medal 1982: There are [Ricci-flat](Unsorted/Ricci-flat.md) but non-flat (nontrivial [holonomy](holonomy)) projective complex manifolds of dimensions $\geq 2$.

![](attachments/Pasted%20image%2020220213222739.png)

## Examples/Classification

- Examples:
	- $\dim X = 1$: [Elliptic Curves](Projects/2022%20Advanced%20Qual%20Projects/Elliptic%20Curves/Elliptic%20Curves.md).
	- $\dim X = 2$: [K3 surfaces](K3%20surfaces.md).

- Compact classification for $\CC\dash$dimension:
  - Dimension 1: 1 type, all elliptic curves (up to homeomorphism)
  - Dimension 2: 1 type, [K3 Surface](K3%20Surface.md)
  - Dimension 3: (threefolds) #open/conjectures that there is a bounded number, but unknown.
    At least 473,800,776!


![](attachments/Pasted%20image%2020220318185120.png)

Appearance in [mirror symmetry](Unsorted/mirror%20symmetry.md):
![](attachments/Pasted%20image%2020220322095658.png)

![](attachments/Pasted%20image%2020220422224843.png)

# Manifolds

![](2023-03-31-10.png)