MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 4 down vote accepted

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.

share|cite|improve this answer

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 $X\simeq \mathrm{Proj}(R)$.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.