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created: 2022-04-05T23:42
updated: 2024-01-01T23:13
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date: 2022-01-15 21:49
modification date: Sunday 13th February 2022 14:29:29
title: Kahler
aliases: [hyperkahler, kahler manifold, "hyper-kahler"]

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- Tags 
	- #physics #geomtop/differential-geometry #AG/complex 
- Refs:
	- <https://people.maths.ox.ac.uk/brantner/hodge.pdf> #resources/notes 
- Links:
	- [supersymmetry](Unsorted/superalgebra.md)
	- [almost complex structure](Unsorted/almost%20complex%20structure.md)
	- [Calabi-Yau](Unsorted/Calabi-Yau.md)
	- [Kahler problem](Kahler%20problem.md)
	- [Dolbeaut cohomology](Dolbeaut%20cohomology)
	- [Kobayashi-Hitchin correspondence](Kobayashi-Hitchin%20correspondence)
	- [Higgs bundle](Unsorted/Higgs%20bundle.md)
	- [nonabelian Hodge theorem](nonabelian%20Hodge%20theorem)
	- [Hodge index theorem](Unsorted/Hodge%20index%20theorem.md)
	- The package:
		- [pure Hodge structure](Unsorted/pure%20Hodge%20structure.md) and [Hodge theory](Unsorted/Hodge%20theory%20MOC.md)
		- [Lefschetz hyperplane theorem](Unsorted/Lefschetz%20hyperplane%20theorem.md)
		- [hard Lefschetz](Unsorted/weak%20and%20hard%20Lefschetz%20theorems.md)

---

# Kahler

![](attachments/Pasted%20image%2020220424181938.png)

Relation to [Hodge theory](Unsorted/Hodge%20theory%20MOC.md):
![](attachments/Pasted%20image%2020220424181951.png)


![](attachments/Pasted%20image%2020220422233026.png)![](attachments/Pasted%20image%2020220422233039.png)

These can all fail for singular varieties, leading to [intersection cohomology](intersection%20cohomology.md).

![](attachments/Pasted%20image%2020220319200239.png)

![](attachments/Pasted%20image%2020220213223211.png)

![](attachments/Pasted%20image%2020220213223108.png)

## Examples

![](attachments/Pasted%20image%2020220213223243.png)

![](attachments/Pasted%20image%2020220422233301.png)
![](attachments/Pasted%20image%2020220422233310.png)

![](attachments/Pasted%20image%2020220422233320.png)
# hyperkahler

For orbifolds, a hyperkahler resolution ensures that [Gromov-Witten invariants](Gromov-Witten%20invariants) vanish.

![attachments/Pasted image 20210627224711.png](attachments/Pasted%20image%2020210627224711.png)

![attachments/Pasted image 20210627224733.png](attachments/Pasted%20image%2020210627224733.png)

![attachments/Pasted image 20210627224754.png](attachments/Pasted%20image%2020210627224754.png)

![attachments/Pasted image 20210627224818.png](attachments/Pasted%20image%2020210627224818.png)
![](attachments/Pasted%20image%2020220213224010.png)

## Examples
The only compact hyperkahler [fourfold](fourfold) is the [K3 surface](K3%20surfaces.md).

![](attachments/Pasted%20image%2020220503233952.png)


# Lefschetz decomposition and Hard Lefschetz

![](attachments/Pasted%20image%2020220424184442.png)
![](attachments/Pasted%20image%2020220424184510.png)
![](attachments/Pasted%20image%2020220424184548.png)![](attachments/Pasted%20image%2020220424184622.png)

See [hard Lefschetz](Unsorted/weak%20and%20hard%20Lefschetz%20theorems.md)