Intersection K-theory

TL;DR

This paper introduces new filtrations on K-theory related to intersection cohomology, proposes definitions for intersection K-theory, and explores a conjectured decomposition theorem for certain maps.

Contribution

It defines two new filtrations on K-theory that relate to intersection cohomology and proposes novel definitions for intersection K-theory, including a conjecture and partial proofs.

Findings

Defined filtrations functorial under proper pushforward and pullback.

Proposed two definitions of intersection K-theory with cycle maps to intersection cohomology.

Conjectured a decomposition theorem for semismall surjective maps, proved in special cases.

Abstract

For a proper map $f:X\to S$ between varieties over $\mathbb{C}$ with $X$ smooth, we introduce increasing filtrations $\textbf{P}^{\leq \cdot}_f\subset P^{\leq \cdot}_f$ 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 ${}^pH^{\leq \cdot}_f(X)$. The filtrations $P^{\leq \cdot}_f$ and $\textbf{P}^{\leq \cdot}_f$ are functorial with respect to proper pushforward; $P^{\leq \cdot}_f$ is functorial with respect to pullback. We use the above filtrations to propose two definitions of (graded) intersection $K$-theory $\text{gr}^\cdot IK_\cdot(S)$ and $\text{gr}^\cdot \textbf{I}K_\cdot(S)$. Both have cycle maps to intersection cohomology $IH^{\cdot}(S)$. We conjecture a version of the decomposition theorem for semismall surjective maps and prove it in some particular cases.