If $\varphi$ and $\psi$ are two plurisubharmonic weights on an algebraic manifold $X$, then we have $\mathcal{J}(\varphi+\psi)\subseteq \mathcal{J}(\varphi)\mathcal{J}(\psi)$.
My question is, if we only allow algebraic functions in such ideals, that is, replace $\mathcal{J}(\varphi)$ by $\mathcal{J}(\varphi) \bigcap \mathcal{O}$, where $\mathcal{O}$ is the algebraic structure sheaf of $X$. Do we still have $\mathcal{J}(\varphi+\psi)\subseteq \mathcal{J}(\varphi)\mathcal{J}(\psi)$?
It seems to me that GAGA tells you also that if $\mathcal G$ is a coherent analytic sheaf of $\mathcal O^{an}_X$-modules over $X^{an}$ then there exists a coherent algebraic sheaf $\mathcal F$ of $\mathcal O_X$-modules and an isomorphism $\mathcal F^{an}\simeq\mathcal G$ (provided $X^{an}$ is Hausdorff and compact); here, $\mathcal F^{an}$ is defined to be $f_X^{-1}\mathcal F\otimes_{f_X^{-1}\mathcal O_X}\mathcal O^{an}_X$, where $f_X$ is the continuous inclusion map $f_X\colon X^{an}\to X$. –  diverietti Nov 29 '11 at 17:45
yes, it's true when $X$ is projective. So the question might follow by GAGA principle... Thank you! That sounds reasonable. –  Zhengyu Hu Nov 30 '11 at 1:46