---
title: polarization
aliases:
  - polarized
  - principal polarization
  - principally polarized
  - PPAV
  - primitive polarization
  - quasipolarization
  - polarized
  - polarized variety
  - numerical polarization
created: 2022-04-22T15:18
updated: 2024-01-10T11:51
---

---
- Links:
	- [abelian variety](Unsorted/abelian%20variety.md)
	- [torsor](Unsorted/torsor.md)

---

# polarization

Polarized variety: $(X, L)$ with $L\in \Pic(X)^\amp$.

## For AVs

![[2023-03-28.png]]
![[2023-03-28-1.png]]

See [[isoduality]].

## [@Hun96]

Let $V$ be an abelian variety over $\mathbb{C}, \operatorname{End}(V)$ the endomorphism ring and
$$
\operatorname{End}_{\mathbb{Q}}(V)=\operatorname{End}(V) \otimes_{\mathbf{Z}} \mathbb{Q}
$$
the endomorphism algebra. A polarization, i.e., a linear equivalence class of ample divisors giving a projective embedding of $V$, gives rise to a positive involution on $\operatorname{End}_{\mathbb{Q}}(V)$, the so-called Rosati involution:
$$
\begin{aligned}
\varrho: \operatorname{End}_{\mathbb{Q}}(V) & \longrightarrow \operatorname{End}_{\mathbb{Q}}(V) \\
\phi & \mapsto \phi^{\varrho} .
\end{aligned}
$$

This can be defined, for example, by viewing the polarization as an isogeny $\mathcal{C}$ : $V \rightarrow V^{\vee}$, where $V^{\vee}$ is the dual abelian variety. The degree of this isogeny is the degree of the polarization. Then for any $\alpha \in \operatorname{End}_{\mathbb{Q}}(V)$, one sets $\alpha^{\ell}:=\mathcal{C}^{-1} \circ \alpha^{\vee} \circ \mathcal{C}$ (see $[\mathrm{Mi}], \S 17$ for a presentation along these lines). The Rosati involution gives rise to a positive definite bilinear form $(\alpha, \beta) \mapsto \operatorname{Tr}\left(\alpha \circ \beta^{\varrho}\right)$ (loc. cit. 17.3).
## For K3s

- **Definition**: Let $X$ be a [K3 surface](K3%20surfaces.md) over a field $k$. The self-intersection index $(\mathcal{L}, \mathcal{L})_X$ of a line bundle $\mathcal{L}$ on $X$ will be called its **degree**. 
- A line bundle $\mathcal{L}$ on $X$ is called **primitive** if $\mathcal{L} \otimes \bar{k}$ is is not a positive power of a line bundle on $X_{\bar{k}}$.

![](attachments/2023-03-12polar.png)
![](attachments/2023-03-12prim-1.png)

![](attachments/2023-02-24polarization.png)

![](attachments/Pasted%20image%2020220531021814.png)

![](attachments/Pasted%20image%2020220422151846.png)


![](attachments/Pasted%20image%2020220504223900.png)

![](attachments/Pasted%20image%2020220504223916.png)

# Principal polarization

![](attachments/Pasted%20image%2020220502171337.png)
![](attachments/Pasted%20image%2020220502171509.png)

![](attachments/2023-03-11poolar.png)

# For topological vector spaces

![](attachments/2023-03-11polar1.png)
![](attachments/2023-03-11polar2.png)

Any self-adjoint Dirac operator splits $V$ into $V^+ \oplus V^-$.
See [determinant bundle](determinant%20bundle).

## Examples

![](attachments/2023-03-11polarex.png)


In [Floer Theory](Unsorted/Floer%20Theory.md):

![](attachments/Pasted%20image%2020220422151943.png)
![](attachments/Pasted%20image%2020220422151958.png)
![](attachments/Pasted%20image%2020220422152035.png)