Intersection K-theory

Abstract

For a proper map $f:X\to S$ between varieties over $\mathbb{C}$ with $X$ smooth, we introduce increasing filtrations ${\text{P}}_f^{\le \cdot }\subset P_f^{\le \cdot }$ on ${\text{gr}}^{\cdot }{K}_{\cdot }(X)$, the associated graded on $K$-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 ${\text{P}}_f^{\le \cdot }$ 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 ${\text{gr}}^{\cdot }\text{I}{K}_{\cdot }(S)$. Both have cycle maps to intersection cohomology $I{H}^{\cdot }(S)$. We conjecture a version of the decomposition theorem for semismall surjective maps and prove it in some particular cases.

35 pages, minor revisions