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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.

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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.

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  • $\begingroup$ Why do you need ampleness? $\endgroup$ – user100272 Jan 14 '18 at 10:44
  • $\begingroup$ Ampleness is equivalent to saying that the obvious rational map is an isomorphism $\mathrm{Proj} R_i \cong X_i$. $\endgroup$ – Ben Webster Jan 15 '18 at 1:07
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If $X$ is a smooth projective algebraic variety of dimension $d$ over a field and $L$ is an ample line bundle on $X$, then $R=\bigoplus_{m=0}^{\infty} H^0(X,mL)$ is a graded $k$-algebra of dimension $d+1$ and one has $X\simeq \mathrm{Proj}(R)$.

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