Tensor t-structures, perversity functions and weight structures

We introduce the notion of tensor t-structures on the bounded derived categories of schemes. For a Noetherian scheme $X$ admitting a dualizing complex, Bezrukavnikov-Deligne, and then independently Gabber and Kashiwara have shown that given a monotone comonotone perversity function on $X$ one can construct a t-structure on $\mathbf{D}^b (X)$. We show that such t-structures are tensor t-structures and conversely every tensor t-structure on $\mathbf{D}^b (X)$ arises in this way. We achieve this by first characterising tensor t-structures in terms of Thomason-Cousin filtrations which generalises earlier results of Alonso, Jerem\'ias and Saor\'in, from Noetherian rings to schemes. We also show that for a smooth projective curve $C$, the derived category $\mathbf{D}^b (C)$ has no non-trivial tensor weight structures, this extends our earlier result on the projective line to higher genus curves.