TITLE: Relationship between Line Bundles with isomorphic ring of sections
QUESTION [5 upvotes]: Suppose two positive holomorphic line bundles $L_1 \to X_1, L_2\to X_2$ over two projective complex manifold $X_1, X_2$ have isomorphic ring of sections $R=R_1=R_2$ where $R_i=\oplus_{m=0}^\infty\Gamma(X_i,mL_i)$. Isomorphism as graded ${\mathbb C}$- algebras.
Is there any relationship betweeen $X_1$ and $X_2$? Eg, some morphism between them? How about relationship to $Proj R$?
Thanks.

REPLY [4 votes]: To expand on the answer above: as B. Cais says, if the line bundles are ample (which I think follows from positivity by Kodaira), we have a canonical isomorphism $\mathrm{Proj} R_i\cong X_i$.  Thus, if the graded rings $R_i$ are isomorphic, then the induced map of Proj's gives an isomorphism $R_1\cong R_2$ carrying one line bundle to the other.