Let $X$ be a projective variety with ample sheaf $\mathcal{O}_X(1)$. Then $H^*(\oplus_n \mathcal{O}_X(n))$ is a graded algebra via the cup product: $H^i(\mathcal{O}(n)) \otimes H^j(\mathcal{O}(m)) \to H^{i+j}(\mathcal{O}(n) \otimes \mathcal{O}(m)) \cong H^{i+j}(\mathcal{O}(n+m))$. Remark that each individual homogeneous component is a graded abelian group (via $n$). Does this "bigraded" cohomology ring have a name, and is it studied in the literature?
If $X,Y$ are projective varieties such that the corresponding "bigraded" cohomology rings are isomorphic, do we then have $X \cong Y$? If this is false, what about the special case $Y=\mathbb{P}^d$ (here the cohomology ring is quite simple).