Intersection K-theory

For a proper map $f:X\to S$ between varieties over $\mathbb{C}$ with $X$ smooth, we introduce increasing filtrations $$\mathbf{\text{P}}$^{\leq \cdot}f\subset P^{\leq \cdot}f$on$$\text{gr}$^\cdot K\cdot(X)$,theassociatedgradedon$K$-theory with respect to the codimension filtration, both sent by the cycle map to the perverse filtration on cohomology ${}^p{H}_f^{\le \cdot }(X)$. The filtrations $P_f^{\le \cdot }$ and ${\mathbf{\text{P}}}_f^{\le \cdot }$ are functorial with respect to proper pushforward; $P_f^{\le \cdot }$ is functorial with respect to pullback. We use the above filtrations to propose two definitions of (graded) intersection $K$-theory ${\text{gr}}^{\cdot }I{K}_{\cdot }(S)$ and $$\text{gr}$^\cdot $\mathbf{\text{I}}$K\cdot(S)$\mathrm{.}Bothhavecyclemapstointersectioncohomology$IH^{\cdot}(S)$. We conjecture a version of the decomposition theorem for semismall surjective maps and prove it in some particular cases.