# Polarizations in algebraic and symplectic geometry

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:

1. 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?

2. 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.

• The top answer to this question addresses the historical origins of the notion of a polarization of an Abelian variety, which is apparently due to Weil: mathoverflow.net/questions/177208/… So to answer question 1, I believe the story is the reverse of what you conjectured: as far as I know, the notion of a polarized abelian variety predates the establishment of symplectic geometry as a genuine field of study.... Oct 23 '20 at 0:58
• ...However, given that a polarization is almost enough to determine a canonical Kähler form, one would hope that if there is any justice in the world then the two concepts coincide. (I say "almost" because you may need to take a tensor power of the line bundle to get an embedding, and then there is no canonical choice of power; once you have an embedding, pull back the Fubini-Study form from projective space; and I don't know if it's easy to describe the change in the Kähler structure under a Veronese re-embedding) Oct 23 '20 at 1:01
• Existence of Polarisation is the main problem in Symplectic Geometry. It has important application in quantization of physical systems.
– user160903
Oct 23 '20 at 6:51
• @TabesBridges: so as far as I understood the answer of the linked question it will stay Weil's sectret at all why he called them "polarizations" and there is most probably no direct analogy from other scientific fields motivating to justify the choice of this name? Nov 19 '20 at 17:43
• @HassanJolany: Yes, but the main concern of my question is why the "polarization" is called "polarization" in symplectic geometry & study of abelian varieties. So it is more focused on the origin of name itself. Nov 19 '20 at 17:46