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 associated 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 cohomology 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 singular 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 computations of K-theory via a K-theoretic version of the decomposition theory, and in representation theory, for example in the construction of representations of vertex algebras using (framed) Uhlenbeck spaces [4]. The Goresky–MacPherson construction 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 a∈AV

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 : 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 perverse 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 theorem. 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 {Sa| a ∈ I} be a stratiﬁcation of S such that fa : f (Sa) → Sa is o S a locally trivial ﬁbration, where Sa = Sa − b∈I (Sa ∩ Sb) . Let A ⊂ I be the set of o relevant strata, that is, those strata such that for xa ∈ Sa: 1 dim f −1(x ) = (dim S − dim S ) . a 2 a o 0 For xa ∈ 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). a∈A We conjecture the analogous statement in K-theory.

Conjecture 1.3. Let f : X → S be a semismall map and consider {Sa| a ∈ I} 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. a∈A 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 ∈ 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. Thus i gr IK0(S)Q is a candidate for intersection Chow groups of S. Corti–Hanamura already defined intersection Chow groups (or Chow motives) in [9], [10] inspired by the decomposition theorem. One proposed definition assumes conjectures of Grothendieck and Murre and proves a version of the decomposition theorem for Chow groups; the other approach defines a perverse-type filtration on Chow groups by induction on level i of the filtration and via correspondences involving all varieties W → S with certain properties for the perverse filtration in cohomology. The advantage in our definition is that one can control the correspondences used to 6i 6i define Pf and Pf and allows for computations, see Subsection 4.4 and Theorem 5.3. For varieties S with a semismall resolution f : X → S with L a local system satisfying the Assumption in Subsection 1.1, de Cataldo–Migliorini [5] proposed a definition of Chow motives ICH (S, L) and proved a version of the decomposition theorem for semismall maps.

It is an important problem to find a definition of the perverse filtration on K·(X) · which recovers the above definition when passing to gr K·(X). A natural such definition will also provide a definition of equivariant intersection K-theory with applications to geometric representation theory, for example in understanding the K-theoretic version of [4]. However, our approach uses functoriality of the perverse · filtration in an essential way for which it is essential to pass to gr K·(X). There are proposed definitions of intersection K-theory in particular cases. Cautis [7], Cautis–Kamnitzer [8] have an approach for categorification of intersection sheaves for certain subvarieties of the affine Grassmannian. Eberhardt defined intersection K-theoretic sheaves for varieties with certain stratifications [11]. In [19], we proposed a definition of intersection K-theoretic for good moduli spaces which has applications to the structure theory of Hall algebras of Kontsevich–Soibelman [20]. Friedlander–Ross [13] developed an approach of intersecting algebraic cycles on singular varieties using motivic complexes. Edidin–Satriano [12] studied intersection of cycles on (possibly singular) GIT quotients. We plan to compare some of these intersection K-theoretic/ Chow groups in future work.

1.6. Acknowledgements. I thank the Institute of Advanced Studies for support during the preparation of the paper. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1926686. INTERSECTION K-THEORY 6

2. Preliminary material 2.1. Notations and conventions. All schemes considered in this paper are finite type quasi-projective over C. The definition of the filtration in Subsection 3.1 works over any field, but to define intersection K-theory we use resolution of singularities, and the construction works over any field of characteristic zero. A variety is an irreducible reduced scheme. For S a scheme, let DbCoh(S) be the derived category of bounded complexes of coherent sheaves and Perf(S) its subcategory of bounded complexes of locally free sheaves on S. The functors used in the paper are derived; we sometimes drop R or L from notation, for example we write f∗ instead of Rf∗. When S is smooth, the two categories coincide. Define b G·(S) = K·(D Coh(S))

K·(S) = K·(Perf(S)).

b b For Y a subvariety of X, let D CohY (X) be the subcategory of D Coh(X) of complexes supported on Y , and deﬁne

b GY,·(X) := K· D CohY (X) .

When X is smooth, we also use the notation KY,·(X) for the above. We will usually drop the subscript · from the notation. Singular and intersection cohomology are used only with rational coeﬃcients.

2.2. Filtrations in K-theory. A reference for the following is [14], especially Sec- i tion 5 in loc. cit. Let F G·(S) be the ﬁltration on G·(S) by sheaves with support of codimension > i; it induces a ﬁltration on K·(S). The associated graded will be · · denoted by gr G·(S), gr K·(S). A morphism f : X → Y of smooth varieties induces maps: ∗ i i f : F K·(Y ) → F K·(X) ∗ i i f : gr K·(Y ) → gr K·(X).

dim Further, let Fi G·(S) be the ﬁltration on G·(S) by sheaves with support of dimension 6 i; it induces a ﬁltration on K·(S). The associated graded will be denoted by gr·G·(S), gr·K·(S). A proper morphism f : X → Y of schemes induces maps: dim dim f∗ : Fi G·(X) → Fi G·(Y )

f∗ : griG·(X) → griG·(Y ).

There are similar ﬁltrations and associated graded on GY (X) for Y,→ X a subva- d−i riety. If X is smooth of dimension d, then griGY (X) = gr GY (X). a Proposition 2.1. Let S −→ Spec C be a variety of dimension d. Then   ∗ M M a , ιT ∗ : G0(Spec C) ⊕ gr·G0(T ) gr·G0(S), T $S T $S INTERSECTION K-THEORY 7 where the sum is taken over all proper subvarieties T of S. Proof. For i < d, the map M M ιT ∗ : griG0(T ) griG0(S) T $S T $S

i is surjective by deﬁnition of the ﬁltration Fdim. Finally, the following map is an isomorphism ∗ ∼ a : G0(Spec C) −→ grdG0(S). Proposition 2.2. Let S be a singular variety of dimension d, and let f : X → S be a resolution of singularities. The following map is surjective:

f∗ : griG0(X) griG0(S). Proof. We use induction on d. By Proposition 2.1, the following is an isomorphism ∼ ∼ f∗ : grdG0(X) −→ grdG0(S) −→ G0(Spec C). For V $ S a subvariety, consider g a resolution of singularities as follows: Y X g f V S.

The surjectivity of f∗ for i < d follows using Proposition 2.1 and the induction hypothesis. 2.3. The perverse ﬁltration in cohomology. Let S be a scheme over C. Let b Dc(S) be the derived category of bounded complexes of constructible sheaves [6, Section 2]. Consider the perverse t-structure P6i, P>i on this category. There i∈Z are functors: p 6i b 6i τ :Dc(S) → P ,

p >i b >i τ :Dc(S) → P

b b such that for F ∈ Dc(S) there is a distinguished triangle in Dc(S):

[1] pτ 6iF → F → pτ >i+1F −→ . b · For a proper map f : X → S and F ∈ Dc(X), the perverse ﬁltration on H (X,F ) is deﬁned as the image of

p 6i · p 6i · · Hf (X,F ) := H (S, τ Rf∗F ) → H (S, Rf∗F ) = H (X,F ).

For F = ICX , the decomposition theorem implies that

p 6i · IHf (X) ,→ IH (X). Let f : X → S be a generically ﬁnite morphism from X smooth, let U be a smooth −1 0 open subset of X such that f (U) → U is smooth, and let L = R f∗Qf −1(U). INTERSECTION K-THEORY 8

−1 For V,→ S, denote by XV := f (V ). Let AV be the set of irreducible compo- a a nents of XV . Let cV be the codimension on XV ,→ X. Further, consider a resolution a of singularities πV such that: a YV a πV ιa a V XV X. a a a Let gV := fπV : YV → V . Then   p 0 M M p >ca a a τ 6 Rf IC = ker Rf IC → τ V Rg IC a [c ] . ∗ X  ∗ X V ∗ YV V  V $S a∈AV Deﬁne the subspace   p 0 M M p −ca a a p 0 τ 6 Rf IC = image τ 6 V Rg IC a [−c ] → τ 6 Rf IC . e ∗ X  V ∗ YV V ∗ X  V $S a∈AV By a computation of Corti–Hanamura [10, Proposition 1.5, Theorem 2.4], we have that:

p 60 p 60 (1) ICS(L) = τ Rf∗ICX τe Rf∗ICX . Further, consider a more general morphism f : X → S with X smooth. Let V $ S p i be a subvariety. For i ∈ Z, denote by H (Rf∗ICX )V the direct sum of simple p i summands of H (Rf∗ICX ) with support equal to V . A computation of Corti– Hanamura [10, Proposition 1.5] shows that:

p i M p i+ca a (2) H (Rf IC ) ,→ H V Rg IC a . ∗ X V V ∗ YV a∈AV

3. The perverse filtration in K-theory 3.1. Definition of the filtration P 06·. Let f : X → S be a proper map between varieties. We define an increasing filtration

06i · · Pf gr G·(X) ⊂ gr G·(X). · It induces a ﬁltration on gr K·(X). We use the notations from Subsection 2.3. Let Y,→ X be a subvariety and let T −→π S be a map generically ﬁnite onto its image from T smooth. Consider the diagram:

p T × X X q f T π S.

For a correspondence Γ ∈ grdim X−sGT ×S Y,0(T × X), deﬁne ∗ · ·−s ΦΓ := p∗(Γ ⊗ q (−)) : gr Ki(T ) → gr GY,i(X). INTERSECTION K-THEORY 9

· We usually drop the shift by s in the superscript of gr GY (X). We also drop the · subscript on relative K-theory. We deﬁne the subspace of gr GY (X): 06i · · Pf,T := spanΓ (ΦΓ : gr K·(T ) → gr GY (X)) 06i 06i Pf := span Pf,T for all maps π as above , where the dimension of the correspondence satisﬁes i + dim X − dim T (3) s. 2 >

· We also deﬁne a quotient of gr GY (X):

06i · · 0>i · Pf gr GY (X) ,→ gr GY (X) Pf gr GY (X).

3.2. Functoriality of the ﬁltration P 06·. Proposition 3.1. 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

∗ 06i−c · 06i · h : Pf gr K·(X) → Pg gr K·(Y ). Proof. Let T → S be a generically ﬁnite map onto its image with T smooth. It suﬃces to show that

∗ 06i−c · 06i · h : Pf,T gr K·(X) → Pg,T gr K·(Y ). Consider the diagram:

Y h X

pY pX Y × T eh X × T

qY qX T

Let Θ ∈ grdim X−sGT ×S X,0(T × X) be a correspondence such that

i > 2s − dim X + dim T. For j ∈ Z, we have that:

j ΦΘ j−s gr K·(T ) gr K·(X)

h∗ Φ he∗Θ j−s gr K·(Y ). INTERSECTION K-THEORY 10

To see this, we compute:

h∗Φ (F ) = h∗p (Θ ⊗ q∗ F ) = p h∗(Θ ⊗ q∗ F ) = p (h∗Θ ⊗ q∗ F ) = Φ (F ). Θ X∗ X Y ∗e X Y ∗ e Y eh∗Θ ∗ The correspondence eh Θ ∈ grdim Y −sGT ×S Y (T × Y ) satisﬁes i + c > 2s − dim Y + dim T, and this implies the desired conclusion. Proposition 3.2. Let X and Y be varieties with proper maps

Y h X

g f S Let c = dim X − dim Y . There are induced maps

06i−c 06i h∗ : Pg gr·G·(Y ) → Pf gr·G·(X). Proof. Let T → S be a generically ﬁnite map onto its image from T smooth. We ﬁrst explain that 06i−c 06i h∗ : Pg,T gr·G·(Y ) → Pf,T gr·G·(X). We use the notation from the proof of Theorem 3.1. Consider a correspondence

Γ ∈ grdim Y −sGT ×S Y,0(T × Y ) such that i > 2s − dim Y + dim T. For j ∈ Z, we have that:

ΦΓ grdim T −jK·(T ) grdim Y −j+sG·(Y )

h∗ Φ he∗Γ grdim Y −j+sG·(X).

To see this, we compute:

∗ ∗ ∗ ∗ h∗pY ∗(Γ ⊗ qY F ) = pX∗eh∗(Γ ⊗ eh qX F ) = pX∗(eh∗Γ ⊗ qX F ). The correspondence

eh∗Γ ∈ grdim Y −sGT ×S X (T × X) = grdim X−(c+s)GT ×S X (T × X) satisﬁes i + c > 2(s + c) − dim X + dim T, and thus the conclusion follows. We continue with some further properties of the ﬁltration P 06·. The following is immediate: Proposition 3.3. Let f : X → S be a proper map. Let U be an open subset of S, −1 XU := f (U), ι : XU ,→ X, and fU : XU → U. Then ι∗ : P 06igr G (X) → P 06igr G (X ). f · · fU · · U INTERSECTION K-THEORY 11

Proposition 3.4. Let f : X → S be a proper map from X smooth and consider j e ∈ gr K0(X). Then

06i a 06i+2j a+j e · Pf gr K·(X) ⊂ Pf gr K·(X).

Proof. Let T → S be a generically ﬁnite map onto its image with T smooth and let

Θ ∈ graGT ×S X,0(T × X). Let p : T × X → X be the natural projection. Then

∗ p (e) · Θ ∈ gra−jGT ×S X,0(T × X).

For x ∈ gr·K·(T ), we have that

e · ΦΘ(x) = Φp∗(e)·Θ(x), and the conclusion thus follows. Proposition 3.5. Let X and Y be smooth varieties with proper maps

Y h X

g f S such that h is surjective. Let c = dim X − dim Y . Then 06i 06i+c h∗ Pf gr·K·(Y )Q = Pf gr·K·(X)Q

∗ · 06i+c · ∗ 06i · h gr K(X)Q ∩ Pg gr K·(Y )Q = h Pf gr K·(X)Q.

If there exists X0 → Y such that the induced map X0 → X is birational, then the above isomorphisms hold integrally.

Proof. The statement and its proof are similar to [10, Proposition 3.11]. Let i : X0 → Y be a map such that hi : X0 → X is generically ﬁnite and surjective. Then, by Proposition 3.2:

06i+c 0 i∗ 06i h∗ 06i+c Pf gr·K·(X ) −→ Pf gr·K·(Y ) −→ Pf gr·K·(X).

0 The map h∗i∗ : gr·K·(X ) → gr·K·(X) is multiplication by the degree of the map hi, so is an isomorphism rationally; it is an isomorphism integrally if X0 → X has degree 1. The pullback statement is similar.

3.3. The ﬁltration P 6·. Let f : X → S be a proper map from X smooth. Let −1 V,→ S be a subvariety, and let AV the set of irreducible components of f (V ). For a a an irreducible component XV , consider a resolution of singularities πV as follows:

πa ιa a V a V XgV XV X a f fV a ffV V S. INTERSECTION K-THEORY 12

a a a a a Let cV be the codimension of XV in X. Denote by τV = ιV πV . Consider a subvariety Y,→ X. Deﬁne

0>i+ca 6i · \ \ a∗ 06i · V · a Pf gr GY (X) := ker τV : Pf gr GY (X) → P a gr K·(XgV ) . ffV V $S a∈AV

a The deﬁnition is independent of the resolutions πV chosen. For two diﬀerent reso- a 0a lutions XgV , XgV , there exists W such that

W π π0

a 0a XgV XgV

a XV

0 a where the maps π and π are successive blow-ups along smooth subvarieties of XgV 0a 0a 0a a 0a 0 and XgV , respectively. Let τV : XgV → X as above. Then τV π = τV π . By Proposition 3.5,

0>i+ca a∗ 06i · V · a ker τV : Pf gr GY (X) → P a gr K·(XgV ) = ffV

0>i+ca ∗ a∗ 06i · V · ker π τV : Pf gr GY (X) → P a gr K·(W ) = ffV

0>i+ca 0a∗ 06i · V · 0a ker τV : Pf gr GY (X) → P a gr K·(XgV ) ffV

Theorem 3.6. 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 : Pf gr K·(X) → Pg gr K·(Y )

6i−c 6i h∗ : Pg gr·K·(Y ) → Pf gr·K·(X).

Proof. The functoriality of h∗ follows from Proposition 3.1 and induction on dimension of S. We discuss the statement for h∗. We use induction on the dimension of S. The 06i 6i case of S = Spec (C) is clear as Pf = Pf . We use the notation from the be- a ginning of Subsection 3.3. Let V be a subvariety of S. Let XV be an irreducible −1 a a component of f (V ) with a resolution of singularities XgV → XV . Let B be the INTERSECTION K-THEORY 13

a set of irreducible component of YV over XV . For b ∈ B, consider a resolution of b b singularities YfV → YV such that

L hb F b B V a b∈B YfV XgV L b τ a B τV V Y h X.

Consider the cartesian diagram

der eh a YV XgV a τ τV Y h X.

der The scheme YV is quasi-smooth, see [17] for a deﬁnition, and reldim eh = reldim h. b der b der For b ∈ B, there is a map pb : YfV → YV . Let db = dim YfV − dim YV and deﬁne b∗ db b eb = det b /h τ a ∈ gr K0 Yf , LτV V L V V where by Lτ we denote the cotangent complex of the map τ. By a version of the excess intersection formula, the following diagram commutes:

h∗ gr·K·(Y ) gr·K·(X)

L b∗ B τV

(4) L b τ a∗ B gr·K·(YfV ) V L B eb L hb L b B V ∗ a B gr·K·(YfV ) K·(XgV ).

We ignore shifts in the gradings above. Consider the diagram

L hb F b B V B YfV tB pb

der eh a YV XgV L τ b a B V τ τV Y h X.

Then ! ! X b b∗ X ∗ ∗ X ∗ hV ∗ eb · τV = eh∗pb∗ (eb · pb τ ) = eh∗ pb∗eb · τ . b∈B b∈B b∈B INTERSECTION K-THEORY 14

It suﬃces to show that

X 0 der (5) pb∗eb = 1 ∈ gr K0 YV . b∈B

cl The underlying scheme YV has irreducible component indexed by B birational b F b der b b to YfV . There exist open sets W = b∈B W ⊂ YV , U ⊂ YfV whose complements have codimension > 1 and such that W b,cl = U b. After possibly shrinking the open sets, we can assume that for any b ∈ B:

b b b U = W × der Yf YV V h^ i OW b = OU b E[1]; d ,

b where E is a vector bundle on U of dimension db and the diﬀerential E → OU b is b b,cl b d b zero. Let ib : W → W = U and let εb := ib∗(1) ∈ gr b K0 U be the Euler 0 b class of E. Then pb∗ (εb) = 1 ∈ gr K0(W ) and the restriction map sends db b db b res : gr K0 YfV → gr K0 U

eb 7→ εb.

0 der ∼ L 0 b Back to proving (5), we have that gr K0 YV = b∈B gr K0 W . Consider the diagram db b res db b gr K0 YfV gr K0 U

pb∗ pb∗

0 der res 0 b gr K0 YV gr K0(W ), where the horizontal maps are restriction to open sets maps. Then

0 b res pb∗(eb) = pb∗ (εb) = 1 in gr K0(W ). The diagram (4) thus commutes. The conclusion now follows from Propositions 3.2 and 3.4. 6i 3.4. Towards the ﬁltration Pf . We continue with the notation from Subsection 3.1. Let X be a smooth variety with a proper map f : X → S. Let T −→π S a generically ﬁnite map onto its image from T smooth. We say that Γ is a T -quasi-smooth scheme if Γ is a derived scheme with maps

X0 ι t p Γ X q f T S INTERSECTION K-THEORY 15 such that ι is a closed immersion in a smooth variety X0 (i.e. the cotangent complex cl Lι is a vector bundle on Γ), t is smooth, and q is surjective. The conditions of the maps ι and t imply that Γ is quasi-smooth, see [17] for a deﬁnition. Let

gr·Kq (T × X) ⊂ gr·K (T × X) T ×S X T ×S X be the subspace generated by classes [Γ] for T -quasi-smooth schemes.

Proposition 3.7. Let h be a proper map:

Y h X

g f S.

There are induced maps

h : gr Kq (T × Y ) → gr Kq (T × X). ∗ · T ×S Y · T ×S X If h is surjective, then there are induced maps

h∗ : gr·Kq (T × X) → gr·Kq (T × Y ). T ×S X T ×S Y Proof. We discuss the statement about pullback. Consider the diagram:

t Θ Y 0 Y Y

r h0 h t Γ X0 X X q f T S,

cl where Γ is a quasi-smooth scheme with q is surjective, tX is smooth, and the upper squares are cartesian. Then the map Θ ,→ Y is a closed immersion and tY is smooth. The map h is surjective, so rcl :Θcl → Γcl is surjective, and thus (qr)cl :Θcl → T is surjective as well. We next discuss the statement about pushforward. Consider

Y 0 ι t p Γ Y q g T S such that ι is a closed immersion, t is smooth, and qcl is surjective. The map Y 0 → X is a proper map of smooth quasi-projective varieties, so we can choose X0 with maps

0 0 Y 0 −→ι X0 −→t X INTERSECTION K-THEORY 16 such that ι0 is a closed immersion and t0 is smooth. Then

X0 ι0ι t0 hp Γ X q f T S

0 0 cl such that ι ι is a closed immersion, t is smooth, and q is surjective.

Consider a diagram

X0 ι t p (6) Γ X q f T π S as above, with t a smooth map and with ι a closed immersion. Let

T ×S X = Z1 ∪ Z2, where Z1 is the union of irreducible components of T ×S X dominant over T and Z2 o is the union of the other irreducible components. Denote by Z1 := Z1 − (Z1 ∩ Z2). 0 0 0 Similarly deﬁne Z1 and Z2 for T ×S X . Let b = reldim q and a = b+dim T = dim Γ.

0 0 0 Proposition 3.8. The class [Γ] ∈ graKT ×S X (T × X ) is not supported on Z2. Proof. Let ` be an ft-ample divisor; it also induces a g-ample class. Denote by 0 pr1 : T × X → T . Then

b pr1∗ [Γ] · ` = d[T ] ∈ grdim T K·(T ) for d a non-zero integer. Let η be the generic point of T ; by abuse of notation, we denote by η its image in S. It suﬃces to show the analogous result when restricting b 0 to η, and d is the intersection number ` · Γη in Xη. 0 Further, let x ∈ gr K 0 (T × X ). We have that a Z2

b pr1∗(x · ` ) = 0 ∈ grdim T K·(T )

b because the support on x·` is not dominant over T . The conclusion thus follows.

Proposition 3.9. Let T −→π X be a generically ﬁnite map from T smooth with image V . Let a > dim X . Then gr Kq (T ×X) = 0. Further, gr Kq (T ×X) V a T ×S X dim XV T ×S X is generated by irreducible components of T ×S X dominant over T of dimension XV . INTERSECTION K-THEORY 17

Proof. Suppose we are in the setting of (6) and let s : X → X0 be a section of t. Assume that t ι [Γ] = p [Γ] 6= 0 ∈ gr Kq (T × X). ∗ ∗ ∗ a T ×S X Then there exists a non-zero x ∈ gr Kq (T × X) such that a T ×S X q 0 p [Γ] = s (x) ∈ gr K 0 (T × X ). ∗ ∗ a T ×S X Consider the diagram

0 res 0 0 gr K 0 (T × X ) gr K 0o (T × X − Z ) a T ×S X a Z1 2

s∗ s∗ res gr K (T × X) gr K o (T × X − Z ). a T ×S X a Z1 2

By Proposition 3.8, we have that res(x) 6= 0 ∈ gr K o (T × X − Z ). We have that a Z1 2 o dim Z1 = dim XV , and the conclusion follows from here.

6i 6i 6i 3.5. The perverse filtration Pf . We now define a smaller filtration Pf ⊂ Pf . We use the notation from Subsection 3.1. Let X be a smooth variety with a proper map f : X → S and let T −→π S be a generically finite map onto its image from T smooth. Consider a subvariety Y,→ X. · Define the subspaces of gr GY (X): 06i · · Pf,T := spanΓ (ΦΓ : gr K·(T ) → gr GY (X)) 06i 06i Pf,V := span Pf,T for all maps π as above V , where Γ ∈ gr Kq (T × X) and dim X−s T ×S Y,0 i + dim X − dim T s. 2 > Using the notation from Subsection 3.3, define

0>i+ca 6i · \ \ a∗ 06i · V · a Pf gr GY (X) := ker τV : Pf gr GY (X) → P a gr K·(XgV ) . ffV V $S a∈AV

a The deﬁnition is independent of the resolutions XgV chosen, see Subsection 3.3. Theorem 3.10. 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 06i−c 06i h∗ : Pg gr·K·(Y ) → Pf gr·K·(X)

6i−c 6i h∗ : Pg gr·K·(Y ) → Pf gr·K·(X). INTERSECTION K-THEORY 18

If h is surjective, then there are induced maps ∗ 06i−c · 06i · h : Pf gr K·(X) → Pg gr K·(Y )

∗ 6i−c · 6i · h : Pf gr K·(X) → Pg gr K·(Y ). Proof. The functoriality follow as in Propositions 3.1, 3.2, and Theorem 3.6, using Proposition 3.7. 3.6. Properties of the perverse ﬁltration. Consider a proper map f : X → S with X smooth. Deﬁne the defect of semismallness of f by

s := s(f) = dim X ×S X − dim X. Further, define s0 = max (dim X + dim S − 4, dim X). It is known [6, Section 1.6] that the perverse filtration in cohomology satisfies

p 6−s−1 p 6s · Hf (X) = 0 and Hf (X) = H (X). We prove an analogous result in K-theory: Theorem 3.11. For f as above, 0 6−s −1 · 6−s−1 · Pf gr K·(X) = Pf gr K·(X) = 0

6s · 6s · · Pf gr K0(X) = Pf gr K0(X) = gr K0(X). Proposition 3.12. Let f : X → S be a surjective map from X smooth and consider a subvariety Z,→ X of codimension > 2. Then there exists a subvariety ι : Y,→ X of codimension 1 such that Z ⊂ Y and fι : Y → S is surjective. Proof. It suﬃces to pass to an open subset of Z, and we can thus assume that Z,→ X is given by a regular closed immersion with functions f1, ··· , fr with r > 2. Pick f ∈ (f1, ··· , fr) such that Z(f) is surjective onto S. Proposition 3.13. Let f : X → S be a proper surjective map from X smooth of relative dimension d. Then

06d · · Pf gr K0(X) = gr K0(X). Proof. We use induction on d. Assume that f is generically ﬁnite. Consider the ∼ correspodence ∆ = X,→ X ×S X: ∆ ∼ X ∼ f f X S.

060 · · This implies that Pf gr K·(X) = gr K·(X). Consider a general f. Let ι : Z,→ X be a subvariety of codimension > 2. By Proposition 3.12, there exists Y,→ X of codimension 1 such that Z ⊂ Y and Y → S is surjective. Let Y 0 → Y be a resolution of singularities and denote the resulting map by g : Y 0 → S. By induction,

06d−1 · 0 · 0 Pg gr K0(Y ) = gr K0(Y ). INTERSECTION K-THEORY 19

By Proposition 2.2, 0 image (ι∗ : gr·G0(Z) → gr·K0(X)) ⊂ image g∗ : gr·K0(Y ) → gr·K0(X) . Finally, assume that Z,→ Y has codimension 1. By Proposition 2.1, it suﬃces to show that 06d image (grdim Z G0(Z) → grdim Z G0(X)) ⊂ Pf because griG0(Z) for i < dim Z is generated by varieties of smaller dimension than Z. If Z → S is surjective, then it has relative dimension d − 1 and we can treat it as above. If Z → S is not surjective, let W ⊂ S be its image. Choose a resolution of singularities T → W and a smooth variety Γ with surjective maps p and q:

p Γ Z X q f T W S.

Then [Γ] ∈ gr Kq (T × X) and its image Φ is in P06dgr·K (X). Then dim X−1 T ×S X Γ f 0

06d · image (grdim Z G0(Z) → grdim Z K0(X)) ⊂ image ΦΓ ⊂ Pf gr K0(X). The conclusion now follows from Proposition 2.1. 0 6−s −1 · Proof of Theorem 3.11. We ﬁrst show that Pf gr K·(X) = 0. Consider a map π : T → X generically ﬁnite onto its image V ⊂ S with T smooth and consider a correspondence

Γ ∈ grdim X−bGT ×S X (T × S).

Then dim X − b 6 dim T ×S X 6 max (dim X, dim X + dim T − 2), and so b > min (0, − dim T + 2) . By the bound (3), it suﬃces to show that −s0 − 1 + dim X − dim T < min (0, − dim T + 2) 2 max (dim X − dim T − 1, dim X + dim T − 5) < s0, which is true because 0 6 dim T 6 dim S. 6−s−1 · We next explain that Pf gr K·(X) = 0. We keep the notation from the previous paragraph. Let [Γ] ∈ gr Kq (T × S). By Proposition 3.9, we dim X−b T ×S X have that b > dim X − dim XV . It suﬃces to show that −s − 1 + dim X − dim T < dim X − dim X 2 V

2 dim XV − dim V 6 s − dim X = dim X ×S X, which is true because 2 dim XV − dim V 6 dim XV ×V XV 6 dim X ×S X. INTERSECTION K-THEORY 20

6s · · We next show that Pf gr K0(X) = gr K0(X). We can assume that f is surjective of relative dimension d. Use the notation from Subsection 3.3. We have that

0>s+ca 6s · \ \ a∗ 06s · V · a Pf gr K0(X) := ker τV : Pf gr K0(X) → P a gr K0(XgV ) . ffV V $S a∈AV We claim that

a a a reldim (XgV → V ) = reldim (XV → V ) 6 s + cV . Indeed, a a dim XV − dim V 6 (dim X ×S X − dim X) + (dim X − dim XV ) a a a 2 dim XV − dim V 6 dim XV ×V XV 6 dim X ×S X,

a 0>s+cV · a which is true. By Proposition 3.13, this implies that P a gr K0(XgV ) = 0. Fur- ffV 06s · · thermore, s > d, so Proposition 3.13 implies that Pf gr K0(X) = gr K0(X), and 6s · · 6s · · thus Pf gr K0(X) = gr K0(X). This also implies that Pf gr K0(X) = gr K0(X). 3.7. Examples of perverse ﬁltration in K-theory.

3.7.1. Let X be a smooth variety of dimension d, and let f : X → Spec C. Then ( grjK (X) if j b i+d c, 6i j · 6 2 P gr K·(X) = f 0 otherwise.

3.7.2. Let X be a smooth variety and let E be a vector bundle on X of rank d + 1. 2 Let Y := PX (E). Denote by ~ := c1(OY (1)) ∈ gr K0(Y ). Consider the projection map f : Y → X. We have that s(f) = d. For i 6 d, there exists an isomorphism M a−2j ∼ 06i a gr K·(X) = Pf gr K·(Y ) i+d 06j6b 2 c X j ∗ x0, ··· , x i+d 7→ ~ f (xj). b 2 c i+d j6b 2 c

The condition for P 6i is checked using projective bundles over varieties of smaller dimension, and we obtain that

M a−2j ∼ 6i a gr K·(X) = Pf gr K·(Y ). i+d 06j6b 2 c

3.7.3. Let X be a smooth variety and let Z be a smooth subvariety of codimension d + 1. Consider the blow-up diagram for Y = BlZ X:

E ι Y p f j Z X. INTERSECTION K-THEORY 21

2 Let ~ := c1(OE(1)) ∈ gr K0(E). We have that s(f) = d − 1. For i 6 d − 1, there is an isomorphism:

a ε M a−2−2j ∼ 6i a gr K(X) ⊕ gr K·(Z) = Pf gr K·(Y ) i+d 06j6b 2 c−1 ∗ X j ∗ x, z0, ··· , z i+d 7→ f (x) + ι∗ ~ q (zj) . b 2 c−1 i+d j6b 2 c−1 Here ε is 0 if i < 0 and is 1 otherwise. This follows from the computation in Subsection 3.7.2 and Proposition 4.4. 6· 6· One can check that in the above examples, we have that Pf = Pf .

3.8. Compatibility with the perverse filtration in cohomology. Consider a 06i 6i · · proper map f : X → S with X smooth. Define filtrations Pf ,Pf on H (X),H (X)alg as in Subsections 3.1 and 3.5. We have that 6i · 6i · · 6i · · image ch : Pf gr K0(X)Q → Pf gr H (X) = Pf gr H (X)alg.

p 6i p 6i We use the notation Hf (X)full for the cohomology of summands of τ Rf∗ICX with support S.

Proposition 3.14. There exist natural inclusions

6i · 6i · p 6i Pf H (X) ⊂ Pf H (X) ⊂ Hf (X)

6i · 6i · p 6i Pf H (X)alg ⊂ Pf H (X)alg ⊂ Hf (X)alg. Thus the cycle map restricts to

6i · p 6i ch : Pf gr K0(X)Q → Hf (X)alg

6i · p 6i ch : Pf gr K0(X)Q → Hf (X)alg.

Proof. Let π : T → S be a generically ﬁnite map with T smooth. Consider a correspondence

Γ ∈ grdim X−sKT ×S X,0(T × X) such that i + dim X − dim T s. 2 > The correspondence Γ induces a map of constructible sheaves on S:

ΦΓ Rπ∗QT [−2s] −−→ Rf∗QX .

ΦΓ Rp∗ICT [dim X − dim T − 2s] −−→ Rf∗ICX .

If π is not surjective, Rπ∗ICT has summands with support W $ S. If π is surjective, the complex Rπ∗ICT has summands ICS(L) of full support and of perverse INTERSECTION K-THEORY 22 degree zero, and other summands with support W $ S. The perverse degree of the sheaf with support S in the image of ΦΓ is

dim X − dim T − 2s 6 i.

06i · Thus Pf H (X) contains cohomology of sheaves ICS(L)[j] with j 6 i which appear as summands of Rf∗ICX and of other sheaves with support W $ S. Thus

0>i · p >i Pf H (X) Hf (X)full. Using the notation in Subsection 3.3, we have that

0>i+ca 6i · \ \ a∗ 06i · V · a Pf H (X) := ker τV : Pf H (X) → P a H (XgV ) . ffV V $S a∈AV In particular,

>i+ca 6i · \ \ a∗ 06i · p V a Pf H (X) ⊂ ker τV : Pf H (X) → H a (XgV )full . ffV V $S a∈AV

6i · p 6i Using (2), we obtain that Pf H (X) ⊂ Hf (X).

6i · 6i · p 6i Remark. We expect equalities Pf H (X)alg = Pf H (X)alg = Hf (X)alg in the above proposition.

4. Intersection K-theory 4.1. Definition of intersection K-theory. Let S be a variety and let L be a local system on an open smooth subset U of S such that there exists a generically finite proper map f : X → S such that X is smooth, f −1(U) → U is smooth, and 0 L = R f∗Qf −1(U). Recall the notation of Subsection 3.3. Define   i · M M i · i · P 6 gr K (X) := image P 6 gr K a (X) → P 6 gr K (X) ef ·  f XV f ·  V $S a∈AV   6i · M M i · i · P gr K (X) := image P6 gr K a (X) → P6 gr K (X) . e f ·  f XV f ·  V $S a∈AV 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 4.1. 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. INTERSECTION K-THEORY 23

We start with some preliminary results. Let f : X → S be a proper map with X smooth. Let Z be a smooth subvariety of X with normal bundle N, Y = BlZ X, and E = PZ (N) the exceptional divisor

E ι Y p π j Z X.

Consider the proper maps

E ι Y π X g h f S.

Let X0 ,→ X be a closed subset, and denote its preimages in Y , Z, E by Y 0, Z0, E0 respectively. Denote by

0 gr·KY 0 (Y ) = ker (π∗ : gr·KY 0 (Y ) → gr·KX0 (X)) . Proposition 4.2. Let T → S be a map with T smooth which is generically ﬁnite onto its image. Then ∗ 0 0 0 0 gr·KT ×S Y (T × Y ) = π gr·KT ×S X (T × X) ⊕ gr·KT ×S E (T × Y ) q ∗ q q 0 gr K 0 (T × Y ) = π gr K 0 (T × X) ⊕ gr K 0 (T × Y ) . · T ×S Y · T ×S X · T ×S E Proof. Let c + 1 be the codimension of Z in X. Denote by O(1) the canonical 2 line bundle on E and let ~ = c1(O(1)) ∈ gr K0(E). There is a semi-orthogonal decomposition [3, Theorem 4.2]:

b D ∗ b ∗ b ∗ b E D (Y ) = π D (X), ι∗ p D (Z) ⊗ O(−1) , ··· , ι∗ p D (Z) ⊗ O(−c) , which implies that

j ∗ j M k ∗ j−2−2k gr K·(Y ) = π gr K·(X) ⊕ ι∗ ~ · p gr K·(Z) . 06k6c−1 0 0 Using the analogous decomposition for Y −Y = BlZ−Z0 (X−X ) and the localization sequence in K-theory [21, V.2.6.2], we obtain that

j ∗ j M k ∗ j−2−2k gr KY 0 (Y ) = π gr KX0 (X) ⊕ ι∗ ~ · p gr KZ0 (Z) . 06k6c−1 In particular, we have that

j ∗ j M k ∗ j−2−2k 0 0 0 gr KT ×S Y (T ×Y ) = π gr KT ×S X (T ×X)⊕ ι∗ ~ · p gr KT ×S Z (T × Z) 06k6c−1 and thus that ∗ 0 0 0 0 gr·KT ×S Y (T × Y ) = π gr·KT ×S X (T × X) ⊕ gr·KT ×S E (T × Y ) . INTERSECTION K-THEORY 24

By Proposition 3.7, we also have that

q ∗ q q 0 gr K 0 (T × Y ) = π gr K 0 (T × X) ⊕ gr K 0 (T × Y ) . · T ×S Y · T ×S X · T ×S E

An immediate corollary of Proposition 4.2 is:

Corollary 4.3. We continue with the notation from Proposition 4.2. There are decompositions

06i · ∗ 06i · 06i · Pg gr KY 0 (Y ) = π Pf gr KX0 (X) ⊕ Pg gr KE0 (Y )

06i · ∗ 06i · 06i · Pg gr KY 0 (Y ) = π Pf gr KX0 (X) ⊕ Pg gr KE0 (Y ).

We next prove:

Proposition 4.4. We continue with the notation from Proposition 4.2. There are decompositions

6i · ∗ 6i · 6i · Pg gr KY 0 (Y ) = π Pf gr KX0 (X) ⊕ Pg gr KE0 (Y )

6i · ∗ 6i · 6i · Pg gr KY 0 (Y ) = π Pf gr KX0 (X) ⊕ Pg gr KE0 (Y ).

Proof. We use the notation from Subsection 3.3. For V $ S, let AV be the set of −1 a irreducible components of f (V ). Let XV be such a component. a a −1 If X ⊂ Z, then there is only one irreducible component Y = a (N) of g (V ) V V PXV above it. a a −1 If XV is not in Z, then there is one component YV of g (V ) birational to a b X . The other components are b (N), where W is an irreducible component of V PWV V a XV ∩ Z. Denote by Ba the set of such components. For a ∈ A and b ∈ Ba, consider resolutions of singularities such that

a a YfV YV Y

a a XgV XV X

a XV ∩ Z

b b WgV WV .

a a Denote by τ the maps as in Subsection 3.3, for example τV : XgV → X, and by µ the map

τ a b b µ a V (7) τV : WgV −→ XgV −→ X. INTERSECTION K-THEORY 25

We consider the proper maps

a a a fV : XgV → XV → V a a a gfV : YfV → YV → V

b b b fV : WgV → WV → V

b gfV : P b (N) → PW b (N) → V. WgV V Denote by a a a cV = codim (XV in X) = codim (YV in Y )

b b cV = codim WV in X

0b c = codim b (N) in Y V PWV the codimensions as in Subsection 3.3. By Proposition 3.5, we have that

0>i+ca a∗ ∗ 06i · V · a ∼ (8) ker τV : π Pf gr K·(X) → P a gr K· YfV = gfV

0>i+ca a∗ 06i · V · a ker τV : Pf gr K·(X) → P a gr K·(XgV ) . ffV By Proposition 3.5 and Proposition 3.1 for the map µ in (7), we have that

0>i+c0b b∗ ∗ 06i · V · ∼ (9) ker τ : π P gr K·(X) → P gr K· (N) = V f b PWgb gfV V

0>i+cb ker τ b∗ : P 06igr·K (X) → P V gr·K (Wgb ) ⊃ V f · b · V ffV

0>i+ca a∗ 06i · V · a ker τV : Pf gr K·(X) → P a gr K·(XgV ) . ffV

−1 Let BV be the set of irreducible components of g (V ). For d ∈ BV , denote by d d d d S gfV : YfV → V and let cV := codim YV in Y . We have that BV = A ∪ a∈A Ba. The statements in (8) and (9) imply that \ \ 0>i+cd π : ker τ d∗ : π∗P 06igr·K (X) → P V gr·K (Yfd) =∼ ∗ V f · d · V gfV V $S d∈BV

0>i+ca \ \ a∗ 06i · V · a ker τV : Pf gr K·(X) → P a gr K·(XgV ) . ffV V $S a∈AV Using Corollary 4.3, we obtain that

6i · ∗ 6i · 6i · 0 Pg gr KY 0 (Y ) = π Pf gr KX0 (X) ⊕ Pg gr KE0 (Y ) .

i The analogous statement for P6 follows similarly. INTERSECTION K-THEORY 26

Proof of Theorem 4.1. Any two such varieties f : X → S and f 0 : X0 → S are birational, so by [1] there is a smooth variety W such that

W π π0

X X0

f f 0 S and the maps π and π0 are successive blow-ups along smooth subvarieties of X and X0, respectively. It thus suﬃces to show that (10) 60 · 60 · ∼ 60 · 60 · Pf gr K·(X) Pef gr K·(X) ∩ ker f∗ = Pg gr K·(Y ) Peg gr K·(Y ) ∩ ker g∗ , where π : Y → X is the blow up along smooth subvariety Z,→ X and

Y π X f g S.

By Proposition 4.4, we have that

6i · ∗ 6i · 6i · 0 Pg gr K·(Y ) = π Pf gr K·(X) ⊕ Pg gr KE(Y )

6i · ∗ 6i · 6i · 0 Peg gr K·(Y ) = π Pef gr K·(X) ⊕ Pg gr KE(Y ) . Taking the quotients we thus obtain the isomorphism (10). The analogous statement for IK is similar. 4.2. Cycle map for intersection K-theory. Let S be a variety and consider a local system L on an open smooth subset U of S such that there exists a map f : X → S as in Subsection 4.1.

j 2j Proposition 4.5. The cycle map ch : gr K0(X)Q → H (X) induces cycle maps independent of the map f : X → S as in Subsection 4.1: j 2j ch : gr IK0(S, L)Q → IH (S, L) j 2j ch : gr IK0(S, L)Q → IH (S, L).

06i · Proof. Deﬁne P H a (X) as in Subsection 3.1 and denote by f XV   0 · M M 0 i · · 0 · P 6 H (X) := image P 6 H a (X) → H (X) ∩ P 6 H (X). ef  f XV  f V $S a∈AT

p 60 p 60 p 60 Denote by Hef (X) ⊂ Hf (X) the sum of summands of τ Rf∗ICX with support strictly smaller than S. By Proposition 3.14, the cycle map respects the perverse INTERSECTION K-THEORY 27

ﬁltrations in K-theory and cohomology 60 j 60 2j p 60 ch : Pf gr K0(X)Q → Pf H (X) ,→ Hf (X)

60 j 60 2j p 60 ch : Pef gr K0(X)Q → Pef H (X) ,→ Hef (X). Taking the quotient and using (1), we obtain a map

j 2j ch : gr IK0(S, L)Q → IH (S, L). The proof that the above cycle map is independent of the map f chosen follows as in Theorem 4.1. The argument for IK is similar. 4.3. Further properties of intersection K-theory. Intersection cohomology satisﬁes the following properties, the second one explaining its name [9, Motivation]: i BM • The natural map H (S) → H2d−i(S) factors through i i BM H (S) → IH (S) → H2d−i(S). • There is a natural intersection map

i j BM IH (S) ⊗ IH (S) → H2d−i−j(S) which is non-degenerate for i + j = 2d. We prove analogous, but weaker versions of the above properties in K-theory. Proposition 4.6. (a) There are natural maps i gr IK·(S) → grd−iG·(S) i gr IK·(S) → grd−iG·(S). (b) There are natural intersection maps i j gr IK·(S) ⊗ gr IK·(S) → grd−i−jG·(S) i j gr IK·(S) ⊗ gr IK·(S) → grd−i−jG·(S). Proof. Let f : X → S be a resolution of singularities. We discuss the claims for IK·, the ones for IK· are similar. We construct maps as above using f; they are independent by f by an argument as in Theorem 4.1. i f∗ (a) There is a natural map gr K·(X) = grd−iG·(X) −→ grd−iG·(S), and we thus obtain a map i 60 i 60 i gr IK·(S) = Pf gr K·(X) Pef gr K·(X) ∩ ker f∗ → grd−iG·(S). (b) Consider the composite map

∗ f 60 i 60 j i+j ∆ i+j ∗ Pf gr K·(X)Pf gr K·(X) → gr K·(X ×X) −−→ gr K·(X) −→ grd−i−jG·(S). The subspaces 60 i 60 i Pef gr K·(X) ∩ ker f∗ Pf gr K·(X) 60 i 60 i Pf gr K·(X) Pef gr K·(X) ∩ ker f∗ INTERSECTION K-THEORY 28

∗ ∗ are in the kernel of f∗∆ = ∆ (f∗ f∗). We thus obtain the desired map. 4.4. Computations of intersection K-theory.

· · · 4.4.1. If S is smooth, then gr IK·(S) = gr IK·(S) = gr K·(S).

4.4.2. Let f : X → S be a small resolution of singularities. Then · · gr IK0(S) = gr K0(X).

Let T −→π S be a generically surjective ﬁnite map from T smooth. By Proposition 3.9, gr Kq (T × X) is generated by the irreducible components of T × X dim X T ×S X S dominant over S. This means that the cycles in gr Kq (T × X) supported on a T ×S X the exceptional locus have a < dim X, and thus they have perverse degree > 1, see (3). Next, say that T −→π S has image V S. Let [Γ] ∈ gr Kq (T × X). By $ dim X−a T ×S X Proposition 3.9, a 6 dim X − dim XV . Its perverse degree i satisﬁes i + dim X − dim V dim X − dim X , 2 > V and thus that i > dim X + dim V − 2 dim XV > 1. 0 6 · 60 · · This means that Pe f gr K·(X) = 0. By Theorem 3.11, Pf gr K0(X) = gr K0(X), · · and thus gr IK0(S) = gr K0(X).

4.4.3. Let S be a surface. Consider a resolution of singularities f : X → S. Let a B be the set of singular points of S. For each p in B, let Ap = {Cp } be the set of −1 irreducible components of Xp := f (p). For each such curve, consider the diagram

ga a p Cp X

a f hp p S.

Consider the maps a a a∗ 1 mp := gp∗hp : K·(p) → gr K·(X) a a a∗ 1 ∆p := hp∗g : gr K·(X) → K·(p). We claim that   60 · M M a 1 Pe f gr K·(X) = image  mp : K·(p) → gr K·(X) . p∈B a∈Ap

0 The correspondences which contribute to P6 are in gr Kq (T × X) for e f 2−s T ×S X π : T → S a generically ﬁnite map onto its image V $ S with T smooth. By INTERSECTION K-THEORY 29

Proposition 3.9, 2 − dim V s dim X − dim X . 2 > > V So the map T → S is the inclusion of a point p ,→ S for p ∈ B and Γ is in gr1GXp (X). Further, for p, q ∈ B, a ∈ Ap, b ∈ Aq:

b a ∆qmp = δpqδab id. This means that:

0 M M a M |Ap| ∼ 6 1 mp : K·(p) = Pe f gr K·(X). p∈B a∈Ap p∈B

The map f is semismall, so by Theorem 3.11 we obtain a form of the decomposition theorem for the map f:

· ∼ · M |Ap| gr K0(X) = gr IK0(S) ⊕ K0(p) . p∈B

See Section 5 for further discussions of the decomposition theorem for semismall maps.

4.4.4. Let Y be a smooth projective variety of dimension d and let L be a line bundle on Y . Consider the cone S = CY L and its resolution of singularities

f X := TotY L −→ S. Let o be the vertex of the cone X. There is only one ﬁber with nonzero dimension

Y ι X g f o S. ∼ Using the correspondence X = ∆ ,→ X ×S X, we see that

060 · · Pf gr K·(X) = gr K·(X).

−1 For V $ S, the irreducible components of f (V ) are fV : W → V birational to V and, if V contains o, the fiber Y . As above, we have that P 060gr·G (W ) = gr·G (W ), fV · · 6i so the conditions in defining Pf are automatically satisfied for these irreducible components. We thus have that

60 · ∗ · 0>1 · Pf gr K·(X) = ker ι : gr K·(X) → Pg gr K·(Y ) . By the computation in Subsection 3.7.1,

( j d+1 0>1 j gr K·(Y ) if j > b 2 c, P gr K·(Y ) = g 0 otherwise. INTERSECTION K-THEORY 30

∗ j j The map ι : gr K·(X) → gr K·(Y ) is an isomorphism, so we have that ( grjK (Y ) if j b d+1 c, 60 j · 6 2 P gr K·(X) = f 0 otherwise.

60 · ∼ Further, Pef gr K·(X) is generated by the cycles over Xo = Y of codimension be- d−1 tween 0 and b 2 c. The map i i+2 ∼ i+2 ι∗ : gr K·(Y ) → gr K·(X) = gr K·(Y )

2 is multiplication by the class ~ := c1(L|Y ) ∈ gr K0(Y ). As a vector space, we thus have that ( j j−2 d+1 j gr K·(Y ) ~ gr K·(Y ) if j 6 b 2 c, gr IK·(S) = 0 otherwise. The answer in cohomology is similar, see [6, Example 2.2.1].

5. The decomposition theorem for semismall maps We will be using the notation from Subsection 1.4. For a, b ∈ A, we write b < a if Sb $ Sa. Denote by ιba : Xb ,→ Xa. For a ∈ A, deﬁne ! 0 6 · M 60 · 60 · Pe f gr KXa (X) = image ιba∗ : Pf gr KXb (X) → Pf gr KXa (X) . b

Conjecture 5.1. Let f : X → S be a semismall map and consider {Sa| a ∈ I} a stratiﬁcation as in Subsection 1.4, denote by A ⊂ I the set of relevant strata. For a ∈ A, consider generically ﬁnite maps πa : Ta → Sa with Ta is smooth such that −1 o o 0 o πa (Sa) → Sa is smooth and R f∗QSa = La. For each a, there exists a rational map Xa 99K Ta, and let Γa be the closure of its graph

ιa Γa Xa X

fa f

πa Ta Sa S,

The correspondence Γa induces an isomorphism

0 60 j−ca 6 j−ca ∼ (11) ιa∗ΦΓa : Pπa gr K·(Ta)Q Pe πa gr K·(Ta)Q = 0 60 j 6 j ιa∗ Pf gr KXa (X)Q Pe f gr KXa (X)Q and a decomposition

M j−ca ∼ j gr IK·(Sa,La)Q = gr K·(X)Q a∈A X (xa)a∈A 7→ ιa∗ΦΓa (xa). a∈A INTERSECTION K-THEORY 31

In relation to (11), we propose the following:

Conjecture 5.2. Let f : X → S be a surjective map of relative dimension d with X is smooth. Let U be a smooth open subset of S such that f −1(U) → U is smooth. −1 For y ∈ U, π1(U, y) acts on the irreducible components of f (y) of top dimension; let L be the associated local system. If L satisﬁes the assumption on local systems in Subsection 4.1, then there is an isomorphism

−d 6−d j 6 j ∼ j Pf gr K·(X)Q Pe f gr K·(X)Q = gr IK·(S, L)Q.

The analogous statement in cohomology follows from the decomposition theorem. In this section, we prove the following:

Theorem 5.3. We use the notation of Conjecture 5.1. Assume that the maps πa : Ta → Sa are small. Then Conjecture 5.1 holds for K0.

We ﬁrst note a preliminary result.

Proposition 5.4. Consider varieties S and X, and a smooth variety Y with surjective maps f : X → S of relative dimension d and g : Y → S of relative dimension 0. Assume there exists an open subset U of S and a map h such that:

g−1(U) h f −1(U)

g f U

Denote also by h the rational map h : X 99K Y . Consider a resolution of singularities π : X0 → X such that there exists a regular map h0 as follows:

X0 π h0 X h f g Y S.

Let Γ be the closure of the graph of h in Y × X and let Γ0 be the graph of h0 in Y × X0. Then the following diagram commutes:

· ΦΓ0 · 0 gr K·(Y ) gr K·(X )

π∗ ΦΓ · gr G·(X). INTERSECTION K-THEORY 32

Proof. Consider the maps:

p0 Y × X0 X0

π0 π p Y × X X q f g Y S.

· Let x ∈ gr K·(Y ). We want to show that:

0 0 0∗ ∗ ∗ π∗p∗(Γ ⊗ π q (x)) = p∗(Γ ⊗ q (x)). It suﬃces to show that

0 0 (12) π∗Γ = Γ in gr·G(X × Y ). Both Γ and Γ0 have dimension equal to the dimension of X. The map π0 :Γ0 → Γ is birational, so the cone of 0 OΓ → π∗OΓ0 is supported on a proper set of Γ, which implies the claim of (12).

Proof of Theorem 5.3. Let a ∈ A and consider the diagram:

τa

ha Xa X

fa f

πa Ta Sa S, where the map τa is a resolution of singularities. Let Γa be the closure of the natural rational map Xa 99K Ta. By Proposition 5.4 and Theorem 3.10, the map ΦΓa factors as: h∗ j a j τa∗ j j+ca ΦΓa : gr K·(Ta) −→ gr K·(Ya) −−→ gr G·(Xa) → gr KXa (X).

By Theorem 3.11, the map ΦΓa factors as:

h∗ Φ : grjK (T ) = P60grjK (T ) −→a P6−da grjK (Y ) −−→τa∗ P6−da grjK (X ) → Γa 0 a ha 0 a faτa 0 a fa 0 a

60 j+ca 60 j+ca j+ca P gr KXa,0(X) → Pf gr K0(X) = gr K0(X).

We thus obtain a map of vector spaces

0 M M j−ca M 60 j 6 j (13) ΦΓa : gr K0(Ta) → ιa∗ Pf gr KXa,0(X) Pe f gr KXa,0(X) a∈A a∈A a∈A j → gr K0(X). INTERSECTION K-THEORY 33

A theorem of de Cataldo–Migliorini [5, Theorem 4.0.4] says that there is an isomorphism:

M M j−ca ∼ j ΦΓa : gr K0(Ta)Q −→ gr K0(X)Q. a∈A a∈A Combining with (13), we see that in this case

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