This is a part of material I do not understand from "Analytic Theory of Abelian Varieties" by Swinnerton-Dyer.
Let $A=\mathbb{C}^n/\Lambda$ be an abelian variety with positive-definite Hermitian form $H$ associated to divisor $a_1$ where $\Lambda$ is a rank $2n$ lattice in $\mathbb{C}^n$. A polarization is a subset $P$ of positive non-degenerate divisors of $A$ satisfying the 2 properties below:
- For any $a,b\in P$ there exist $n_a,n_b\in \mathbb{Z}$ s.t. $n_aa$ and $n_bb$ have the same positive definite Hermitian form integral multiples of $H$. In particular $a_1\in P$.
- $P$ is maximal against condition 1.
Suppose $a,b\in P$. Then $n_aa$ and $n_bb$ have the same Hermitian Riemann forms. I could replace $a_1$ by $n_aa$. Call $D:=n_aa$, $E:=n_bb$. Then $D-E$ is linear equivalent to $(D-D_x)$ for some $x\in A$ where $D_x$ is the translation of $D$ by $x$. Then I see $D-E\sim (D-D_x)\implies E\sim D_x$ where $\sim$ denotes linear equivalence. Thus I see $L(D)\cong L(D_x)$ via $f(z)\to f(z-x)$ and $L(D_x)\cong L(E)$. Then I see $A\xrightarrow{|L(D)|}P^N,A\xrightarrow{|L(E)|}P^N$ have the image translated by $x$ as all sections of $L(D)$ in $L(E)$ are translated by $x$.
$\textbf{Q:}$ The book says the maps induced by different elements of polarization differ up to Veronese embedding and translation. Where does Veronese embedding come from here? It seems I want to say that if $L(a_1)$ gives rise to projective embedding, then $L(na_1)$ are given by degree $n$ monomials of $L(a_1)$'s sections.
