2021/03/10 by Tudor Pădurariu, Pădurariu, Tudor
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)
paper · pdf · doi:10.48550/arxiv.2103.06223
openalex publication_date 2021/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a proper map f:X→ S between varieties over ℂ with X smooth, we introduce increasing filtrations P≤ ⋅f⊂ P≤ ⋅f on gr^⋅ K_⋅(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≤ ⋅f(X). The filtrations P≤ ⋅f and P≤ ⋅f are functorial with respect to proper pushforward; P≤ ⋅f is functorial with respect to pullback. We use the above filtrations to propose two definitions of (graded) intersection K-theory gr^⋅ IK_⋅(S) and gr^⋅ IK_⋅(S). Both have cycle maps to intersection cohomology IH⋅(S). We conjecture a version of the decomposition theorem for semismall surjective maps and prove it in some particular cases.