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INTERSECTION K-THEORY Contents 1. Introduction 1 2. Preliminary INTERSECTION K-THEORY TUDOR PADURARIU˘ Abstract. For a proper map f : X ! S between varieties over C with X 6· 6· · smooth, we introduce increasing filtrations Pf ⊂ Pf on gr K·(X), the associ- ated graded on K-theory with respect to the codimension filtration, both sent by p 6· the cycle map to the perverse filtration on cohomology Hf (X). The filtrations 6· 6· 6· Pf and Pf are functorial with respect to proper pushforward; Pf is functorial with respect to pullback. We use the above filtrations to propose two definitions of intersection K- · · theory gr IK·(S) and gr IK·(S). Both have cycle maps to intersection cohomol- ogy IH·(S). We conjecture a version of the decomposition theorem for semismall surjective maps and prove it in some particular cases. Contents 1. Introduction 1 2. Preliminary material 6 3. The perverse filtration in K-theory 8 4. Intersection K-theory 22 5. The decomposition theorem for semismall maps 30 References 33 1. Introduction For a complex variety X, intersection cohomology IH·(X) coincides with singu- lar cohomology H·(X) when X is smooth and has better properties than singular cohomology when X is singular, for example it satisfies Poincar´eduality and the Hard Lefschetz theorem. Many applications of intersection cohomology, for example in representation theory [18], [6, Section 4] are through the decomposition theorem of Beilinson{Bernstein{Deligne{Gabber [2]. A construction of intersection K-theory is expected to have applications in com- putations of K-theory via a K-theoretic version of the decomposition theory, and in representation theory, for example in the construction of representations of ver- tex algebras using (framed) Uhlenbeck spaces [4]. The Goresky{MacPherson con- struction of intersection cohomology [16] does not generalize in an obvious way to K-theory. 1.1. The perverse filtration and intersection cohomology. For S a variety · · over C, intersection cohomology IH (S) is a subquotient of H (X) for any resolution 1 INTERSECTION K-THEORY 2 of singularities π : X ! S. More generally, let L be a local system on an open smooth subscheme U of S satisfying the following 0 Assumption: L = R f∗Qf −1(U) for a generically finite map f : X ! S from X smooth such that f −1(U) ! U is smooth. The decomposition theorem implies that IH·(S; L) is a (non-canonical) direct sum- mand of H·(X). Consider the perverse filtration p 6i · p 6i · · Hf (X) := H S; τ Rf∗ICX ,! H (S; Rf∗ICX ) = H (X): −1 For V,! S, denote by XV := f (V ). Let AV be the set of irreducible components a a a of XV and let cV be the codimension on XV ,! X. For any component XV , consider a resolution of singularities a YV a πV ιa a V XV X: a a a Let gV := fπV : YV ! V . Define i−ca p 6i M a a p 6 V a p 6i He := ιV ∗πV ∗ H a (YV ) ⊂ H ; f;V gV f a2AV p 6i M p 6i p 6i Hef := Hef;V ⊂ Hf : V $S The decomposition theorem implies that · p 60 · p 60 · IH (S; L) = Hf H (X) Hef H (X): 1.2. Perverse filtration in K-theory. Inspired by the above characterization of intersection cohomology via the perverse filtration, we propose two K-theoretic 6i 6i · perverse filtrations Pf ⊂ Pf on gr K·(X) for a proper map f : X ! S of complex · varieties with X smooth. Here, the associated graded gr K·(X) is with respect to the codimension of support filtration on K·(X) [14, Definition 3.7, Section 5.4]. 6i · The precise definition of the filtration Pf gr K·(X) is given in Subsection 3.3; roughly, it is generated by (subspaces of) images · · Γ : gr K·(T ) ! gr K·(X) induced by correspondences Γ on X × T of restricted dimension, see (3), for T a smooth variety with a generically finite map onto a subvariety of S. These subspaces a satisfy conditions when restricted to the subvarieties YV from Subsection 1.1. 6i · The definition of filtration Pf gr K·(X) is given in Subsection 3.5. We further impose that Γ is a quasi-smooth scheme surjective over T . This futher restricts the possible dimension of the cycles Γ, see Proposition 3.9, and allows for more computations. INTERSECTION K-THEORY 3 Theorem 1.1. Let f : X ! S be a proper map with X smooth. Then the cycle · · map ch : gr K0(X)Q ! H (X) respects the perverse filtration 6i · 6i · ch p 6i Pf gr K0(X)Q ⊂ Pf gr K0(X)Q −! Hf (X): Perverse filtrations in K-theory have the following functorial properties. Let X and Y be smooth varieties with c = dim X − dim Y . Consider proper maps Y h X g f S: There are induced maps 6i−c 6i h∗ : Pg gr·K·(Y ) ! Pf gr·K·(X); 6i−c 6i h∗ : Pg gr·K·(Y ) ! Pf gr·K·(X); ∗ 6i−c · 6i · h : Pf gr K·(X) ! Pg gr K·(Y ): If h is surjective, then there is also a map ∗ 6i−c · 6i · h : Pf gr K·(X) ! Pg gr K·(Y ): Let S be a singular scheme, a local system L, and a smooth variety X as in 0 60 · 6 · p 6i Subsection 1.1. We define Pef gr K·(X) and Pe f gr K·(X) similarly to He (X). Inspired by the discussion in cohomology from Subsection 1.1, define · 60 · 60 · gr IK·(S; L) := Pf gr K·(X) Pef gr K·(X) \ ker f∗ 0 · 60 · 6 · gr IK·(S; L) := Pf gr K·(X) Pe f gr K·(X) \ ker f∗ : · · Theorem 1.2. The definitions of gr IK·(S; L) and gr IK·(S; L) do not depend on the choice of the map f : X ! S with the properties mentioned above. Further, there are cycle maps j 2j ch : gr IK0(S; L)Q ! IH (S; L) j 2j ch : gr IK0(S; L)Q ! IH (S; L): 1.3. Properties of the perverse filtration intersection K-theory. The per- verse filtration in K-theory and intersection K-theory have similar properties to their counterparts in cohomology. For a map f : X ! S, let s := dim X×S X−dim X be the defect of semismallness. In Theorem 3.11, we show that 6−s−1 · Pf gr K0(X) = 0; 6s · 6s · · Pf gr K0(X) = Pf gr K0(X) = gr K0(X): INTERSECTION K-THEORY 4 This implies that · · · gr IK·(S) = gr IK·(S) = gr K·(S) for S smooth, · · gr IK0(S) = gr K0(X) if S has a small resolution f : X ! S: For more computations of perverse filtrations in K-theory and intersection K- theory, see Subsections 3.7 and 4.4. In cohomology, there are natural maps i i BM H (S) ! IH (S) ! H2d−i(S) i j BM IH (S) ⊗ IH (S) ! H2d−i−j(S): i BM The composition in the first line is the natural map H (S) ! H2d−i(S). The second map is non-degenerate for cycles of complementary dimensions. In Subsection 4.3 we explain that there exist natural maps griIK·(S) ! griG(S) i j gr IK·(S) × gr IK·(S) ! grd−i−jG·(S) and their analogues for IK. The above filtration on G-theory is by dimension of supports, see [14, Section 5.4]. 1.4. The decomposition theorem for semismall maps. As mentioned above, many applications of intersection cohomology are based on the decomposition the- orem. When the map f : X ! S is semismall, the statement of the decomposition theorem is more explicit, which we −1 o o now explain. Let fSaj a 2 Ig be a stratification of S such that fa : f (Sa) ! Sa is o S a locally trivial fibration, where Sa = Sa − b2I (Sa \ Sb) : Let A ⊂ I be the set of o relevant strata, that is, those strata such that for xa 2 Sa: 1 dim f −1(x ) = (dim S − dim S ) : a 2 a o 0 For xa 2 Sa, the monodromy group π1(Sa; xa) acts on the set of irreducible compo- −1 nents of f (xa) of top dimension; let La be the corresponding local system. Let −1 ca be the codimension of Xa = f (Sa) in X. The decomposition theorem for the map f : X ! S says that there exists a canonical decomposition [6, Theorem 4.2.7]: j M j−ca H (X) = IH (Sa;La): a2A We conjecture the analogous statement in K-theory. Conjecture 1.3. Let f : X ! S be a semismall map and consider fSaj a 2 Ig a stratification as above, and let A ⊂ I be the set of relevant strata. There is a decomposition for any integer j: j M j−ca gr K·(X)Q = gr IK·(Sa;La)Q: a2A INTERSECTION K-THEORY 5 See Conjecture 5.1 for a more precise statement. In Theorem 5.4, we check the above conjecture for K0 under the extra condition that for any a 2 A, there are small maps πa : Ta ! Sa satisfying the Assumption in Subsection 1.1. The proof of the above result is based on a theorem of de Cataldo{Migliorini [5, Section 4]. In Subsection 4.4.4, we prove the statement for K0 when f : X ! S is a resolution of singularities of a surface. i i 1.5. Past and future work. When X is smooth, gr K0(X)Q = CH (X)Q.
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