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.



Relationship between Line Bundles with isomorphic ring of sections
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.


