---
date: 2022-02-07 12:13
modification date: Monday 7th February 2022 12:13:17
title: weak and hard Lefschetz theorems
aliases: [weak Lefschetz, hard Lefschetz, Lefschetz decomposition, Lefschetz decomposition theorem]

---

Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #AG 
- Refs:
	- #todo/add-references
- Links:
	- [right derived pushforward](Unsorted/pushforward.md)

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# Lefschetz theorems

# Hard Lefschetz

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

![](attachments/Pasted%20image%2020220424184636.png)

![](attachments/Pasted%20image%2020220424184725.png)
![](attachments/Pasted%20image%2020220424184738.png)

## Application: Signatures

For a [[Kahler]] surface:

![](attachments/Pasted%20image%2020220424184755.png)
![](attachments/Pasted%20image%2020220424184808.png)
![](attachments/Pasted%20image%2020220424184826.png)

See [Hodge index theorem](Unsorted/Hodge%20index%20theorem.md)

# Weak Lefschetz
![](attachments/Pasted%20image%2020220207121313.png)

# Hard Lefschetz


For [symplectic](Unsorted/symplectic.md) manifolds $(X, \omega)$ of real dimension $2n$: define a map $L: H_\dR(X) \to \Sigma^{2}H_{\dR}(X)$ by $[\alpha] \mapsto [\omega \wedge \alpha]$.
Then the iterates $L^i$ restrict to $L^i: H^{n-i}_\dR(X) \to H_\dR^{n+i}(X)$, and Hard Lefschetz states that this is an isomorphism for compact [Kahlers](Unsorted/Kahler.md).


For smooth complex [projective varieties](Unsorted/projective%20(schemes).md) of complex dimension $n$, replace $\omega$ with the class of a hyperplane.