# $X$ closed compact Riemann surface, $Jac(X)$ Jacobian of $X$. Relation between embedding $X$ and $Jac(X)$ into projective space

$X$ is a closed compact Riemann surface. $\operatorname{Jac}(X)$ is the Jacobian of $X$. One can always embed $X$ into projective space by twisting line bundle over $X$ to an ample line bundle by raising high power enough.

$\operatorname{Jac}(X)$ is an abelian group. $\operatorname{Jac}(X)$ admits a projective embedding if it has a positive definite hermitian form/admits a divisor whose associated $\theta$ line bundle is globally generated.(This is basically saying the condition of admitting the projective embedding.)

For $X$ being elliptic curve and Riemann surface, $\operatorname{Jac}(X)\cong X$. Since all elliptic curve are non-singular for closed compact Riemann surface, I see both $X$ and $\operatorname{Jac}(X)$ can be projectivized and this is if and only if relationship.

$\textbf{Q1:}$ What is the relationship between $X$ projectivizable and $\operatorname{Jac}(X)$ projectivizable?(For closed compact Riemann surface $X$ is certainly projectivizable into $CP^3$.) I do not know whether this should also imply $\operatorname{Jac}(X)$ projectivizable. The book says there are some lattice which do not admit positive definite hermitian forms and hence it cannot be embedded into projective space.

$\textbf{Q2:}$ What is the relationship between meromorphic sections on $\operatorname{Jac}(X)$ and meromorphic sections on $X$?

Ref. Analytic Theory of Abelian Varieties by Swinnerton.

• For any compact Riemann surface, the Jacobian is projective. For general lattices, the key phrase to look for is Riemann bilinear relations. – Mohan Jun 15 '18 at 17:07
• @Mohan Ah. Thanks. I have completely missed that part. – user45765 Jun 15 '18 at 17:15