I'm currently reading through Daniel Huybrecht's Lectures on K3 Surfaces and have come across (what seems at first) two different interpretations of a polarization. At the end of chapter 2 (§2.4 existence of K3 surfaces), Huybrechts defines a polarized K3 surface to be:
A projective K3 surface together with an ample line bundle L such that L is primitive, i.e. indivisible in $Pic(X)$, with $(L)^2 = 2d$.
Roughly seven pages later, Huybrechts defines a polarization of a Hodge structure (of weight $n$) to be a morphism of Hodge structures $\psi : V \otimes V \to \mathbb{Q}(-n)$ (also of weight $n$) such that $(v, w) \mapsto \psi(v, Cw)$ is a positive definite, symmetric bilinear form.
I understand the significance of these polarizations as it relates to whether a K3 surface is in fact an algebraic K3 surface, able to be embedded in $\mathbb{P}^N$, or a complex K3 surface. At least this is my impression from looking at these notes on abelian varieties and Lefschetz Theorem by Brian Conrad. I imagine the prototypical example for a K3 surfaces $X$ is something like:
Let the Hodge structure on $X$ be $V^* = H^*(X ; \mathbb{Z})$, $V^{p, q} = \bigoplus_{p + q = n} H^{p,q}(X; \mathbb{C}) = \bigoplus_{p+q = n} H^q(X, \Omega^p)$.
Let the polarization be the standard Hodge-Riemann pairing $Q(\alpha, \beta) = \langle L^{n - k} \alpha, \omega \rangle = \int_X \omega^{n - x} \wedge \alpha \wedge \beta$ where $\omega$ Kahler form. I'm guessing this is effectively the Riemann form $H$ one wants to consider for the Appell-Humbert theorem regarding $\mathcal{L}(H, \alpha) = \{ \textrm{something},\ \textrm{something},\ \textrm{theta}\ \textrm{functions}\}$
$\dots$ (don't understand this stuff)
- $\mathcal{L}^{\otimes 3}$ is very ample.
$\dots$ (where I'm trying to go)
- The moduli functor of K3 surfaces is the map $ T \mapsto \{ (f: X \to T, L) \} $ with $L$ primitive ample line bundle $L_X$ (s.t. $(L_X)^2 = 2d$) and we say $(f, L) \sim (f', L')$ iff $\exists$ isomorphism of $T$-schemes $\psi : X \to X'$ and another line bundle $L_0$ on $T$ with $\psi^* L' \cong L \otimes f^* L_0 $.
I'm having trouble making the jump to the language of line bundles here, which is proving to be a difficulty in later chapters. I understand why we want to call two K3 surfaces isomorphic if their Hermitian forms $Q(\alpha, \beta)$ are isomorphic (similar to an isometry in Riemannian geometry pulling back the bilinear form $g$). So defining the Moduli space in terms of isomorphism classes of $(X, Q)$ would make sense, but I just don't see the connection to ample line bundles $L$ here. Sorry for any gaps in my understanding here, I just began reading these notes earlier this quarter and am trying to finish all the exercises in Vakil at the same time.