In context of Abelian varieties there are a couple of equivalent ways to
introduce the *polarization* of a algebraic variety. One way is to
choose a line bundle $\mathcal{L}$ which satisfies certain identities.

Seemingly the historical motivation for the name "*polarization*" in theory of algebraic varieties
is that the Chern class of this line bundle
$\mathcal{L}$ has similar properties like the Chern class of Kähler form
in complex analytic geometry. Indeed the name "*polarization*" occurs also
in symplectic geometry and a *Kähler polarization* of a symplectic manifold
by definition a polarization by a compatible Kähler manifold structure. Where
the last *polarization* is in symplectic sense:

Let $(X, \omega)$ be a symplectic manifold. A polarization polarization in symplectic sense of $(X, \omega)$ is a choice of involutive Lagrangian subbundle $\mathcal{P} \subset T_{\mathbb{C}} X$ of of the complexified tangent bundle of $X$.

Questions:

Does the name

*polarization*for Abelian varieties arise and is just "transfered" from the terminology for symplectic manifolds or are the "polarizations" in algebraic geometry and symplectic geometry not really related to each other?Let us focus on

*polarizations*in theory of symplectic manifolds. The choice if a polarization plays eg an important role in geometric quantizations. Does the name polarization in symplectic geometry have some geometrical & intuitive meaning. I understand the definition but I would like to know what was the motivation behind the name polarization there.

Naively one associates the polarization with certain physical phenomena from optics. Does the motivation behind the choice of the name polarization in symplectic geometry have a physical origin?

adds: I had a look at the thread linked by Tabes Bridges but the discussion there not provides a satisfying answer of my question at all except that this name firstly was used to study abelian varieties and later it was token by symplectic/differential geometers, that is it have been established in reverse order to that what I conjectured above as Tabes Bridges remarked in his comment. According to an answer I found there the term "polarization" was coined by Andre Weil but there is no explanation why he choosed this name. I cannot imagine that he choosed this name ad hoc "randomly" without having any association with something in mind. I'm very curious if there is something "more" behind his choice of this term.

whythe "polarization" is called "polarization" in symplectic geometry & study of abelian varieties. So it is more focused on the origin of name itself. $\endgroup$4more comments