An Introduction to Lawson Homology, II

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Review of Lawson homology and related theories Suslin’s Conjecture Correspondences Beilinson’s Theorem More on Suslin’s (strong) conjeture

An Introduction to Lawson Homology, II

Mark E. Walker University of Nebraska – Lincoln

December 2010

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Review of Lawson homology

ForX quasi-projective over C:

Z_r(X) =topological abelian group ofr-cycles on X LrHm(X) :=πm−2rZ_r(X)

There are maps:

H_m^M(X,Z(r))→LrHm(X)→H_m^BM(X(C),Z(r)) The left-hand map is an isomorphism withZ/n-coefficients.

The right-hand map is the topic of Suslin’s Conjecture (see below).

Additional comment: These are maps of (non-finitely

generated) MHS’s, whereH_m^M(X,Z(r))has the trivial MHS. The MHS ofL_rH_m(X) is induced by MHS onH∗^sing(C_r,e(X)).

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Review of Lawson homology

ForX quasi-projective over C:

Z_r(X) =topological abelian group ofr-cycles on X LrHm(X) :=πm−2rZ_r(X)

There are maps:

H_m^M(X,Z(r))→LrHm(X)→H_m^BM(X(C),Z(r)) The left-hand map is an isomorphism withZ/n-coefficients.

The right-hand map is the topic of Suslin’s Conjecture (see below).

Additional comment: These are maps of (non-finitely

generated) MHS’s, whereH_m^M(X,Z(r))has the trivial MHS.

The MHS ofL_rH_m(X)is induced by MHS on H∗^sing(C_r,e(X)).

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Related theories

I mentioned (but did not define)morphic cohomology, L^∗H^∗(X), yesterday.

There is also a version ofK-theory that uses algebraic equivalence, calledsemi-topological K-theory: Let Grass =`

nGrass(Cⁿ). ForX projective, K_q^semi(X) :=πq

Maps(X,Grass)^h+

whereh+denotes “homotopy theoretic group completion” of the homotopy-commutativeH-spaceMaps(X,Grass).

For example,K₀^semi(X) =K₀(X)/(alg. equiv.).

“Every formal property one might expect involvingL∗H∗, L^∗H^∗ andK_∗^semidoes indeed hold.”

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Some of the properties of K

_∗^semi

There are natural maps

Kn(X)→K_n^semi(X)→ku⁻ⁿ(X(C)).

K_n(X,Z/m)−→K^∼⁼ _n^semi(X,Z/m) form >0.

There is a Chern character isomorphism

ch:K_n^semi(X)_Q−→ ⊕^∼⁼ L^qH^2q−n(X,Q).

For X smooth, there is an Atiyah-Hirzebruch spectral sequence

E₂^p,q=L^−qH^p−q⇒K_−p−q^semi (X), which degenerates upon tensoring with Q.

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Review of Lawson homology and related theories Suslin’s Conjecture

Correspondences Beilinson’s Theorem More on Suslin’s (strong) conjeture

Suslin’s Conjecture for Lawson/morphic (co)homology

Conjecture (Suslin’s Conjecture — Lawson form) For a smooth, quasi-projective varietyX, the map

LrHm(X)→H_m^sing(X(C),Z(r))

is an isomorphism form≥d+r and a monomorphism for m=d+r−1.

“Suslin’s Conjecture = Bloch-Kato (really,

Beilinson-Lichtenbaum) withZ-coefficients (over C)”:

L^tHⁿ(X) ∼=^? Hⁿ_Zar(X, tr^≤tRπ∗Z), whereπ: (V ar/C)_analytic→(V ar/C)_Zar.

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(Thin) Evidence: Cases where Suslin’s Conjecture is known

For codimension one cycles. Proof is by explicit calculation ofL_dim(X)−1H∗(X).

For L0H∗ (by Dold-Thom Theorem) and L_dim(X₎H∗

(trivially). In particular, it’s known for all surfaces.

For special varieties, such as toric varieties, cellular varieties, linear varieties (that are smooth).

Certain hyper-surfaces of dim. 3[Voineagu]

With finite coefficients — i.e., Bloch-Kato is known [Voevodsky].

The cohomological form holds for π₀ [Bloch-Ogus]

Ld−tH_2(d−t)(X) =L^tH^2t(X)∼=H^2tZar(X, tr^≤tRπ∗Z)

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Finite generation of Lawson homology

Suslin’s Conjecture predicts, in particular, that

L_rH_m(X) is finitely generated form≥dim(X) +r−1.

The converse is also known: letCrH∗(X)be the “cone” of the map fromL_rH∗ toH_∗^BM, so that

· · · →C_rH_m(X)→L_rH_m(X)→H_m^BM(X)→C_rH_m−1(X)→ · · ·. Voevodsky’s Bloch-Kato⇒ CrHm(X,Z/n) = 0 for

m≥dim(X) +r−1, and henceCrHm(X,Z)is a divisible group in this range.

Thus, ifL_rH_m(X) is finitely generated for

m≥dim(X) +r−1, thenCrHm(X) = 0 in this range, and hence Suslin’s Conjecture holds.

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Behavior of correspondences on Lawson homology

A mapY → C_r(X) (for example, an inclusion) determines a cycle

Γ ^//

p

rel. dim. rIIIIIIIIYII$$×X

Y

and hence a map on cycles spaces

Γ∗ :Z₀(Y)→ Z_r(X) determined by

y7→Γ_y =p⁻¹(y)∈ Z_r(X).

Applyingπm−2r gives

Γ∗ :H_m−2r^sing (Y(C)) =L₀Hm−2r(Y)→L_rH_m(X).

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Lifting elements to singular cohomology

Given an element

α∈LrHm(X) =πm−2rZ_r(X) it lifts toα˜ ∈H_m−2r^sing (C_r,e(X)(C)) along the maps

H_m−2r^sing (C_r,e(X)(C))→H_m−2r^sing (Z_r(X))πm−2rZ_r(X).

Using singular Lefschetz,α˜ lifts to

˜˜

α∈H_m−2r^sing (Y(C))→LrHm(X) for someY ⊂ C_r,e(X) with dim(Y)≤m−2r.

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Lifting along correspondences

Thus we have a surjection M

dim(Y)≤m−2r

H_m−2r^sing (Y)LrHm(X)

where each mapH_m−2r^sing (Y)→L_rH_m(X) is the map associated to an equi-dimensional correspondence Γ : Y ^_ ^_ ^_^//X of rel. dim. r:

H_m−2r^sing (Y)∼=L0Hm−2r(Y)−→L^Γ^∗ _rHm(X).

Remark: This surjection can be used to understand MHS on LrHm(X).

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Lifting along correspondences

Thus we have a surjection M

dim(Y)≤m−2r

H_m−2r^sing (Y)LrHm(X)

where each mapH_m−2r^sing (Y)→L_rH_m(X) is the map associated to an equi-dimensional correspondence Γ : Y ^_ ^_ ^_^//X of rel. dim. r:

H_m−2r^sing (Y)∼=L0Hm−2r(Y)−→L^Γ^∗ _rHm(X).

Remark: This surjection can be used to understand MHS on LrHm(X).

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Lifting to smooth varieties, using Hodge theory

For such a correspondenceΓ, the composition H_m−2r^sing (Y)−→L^Γ^∗ _rH_m(X)→H_m^sing(X)

coincides with the map on singular cohomology induced byΓ:

Γ∗ :H_m−2r^sing (Y)→H_m^sing(X)

WhenX is smooth, lettingY˜ →Y be a resolution of singularities, Hodge theory gives:

im

H_m−2r^sing (Y)→H_m^sing(X)

=

im

H_m−2r^sing ( ˜Y)→H_m−2r^sing (Y)→H_m^sing(X)

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Characterizing image in singular homology

Proposition (Friedlander-Mazur)

ForX smooth and projective, the topological filtration is contained in the “correspondence” filtration: Every element of

F_r^topH_m^sing(X) := im LrHm(X)→H_m^sing(X(C)) is contained in the image of

Γ∗ :H_m−2r^sing (W(C))→H_m^sing(X(C)) whereW is smooth withdim(W)≤m−2r andΓ is a correspondence.

(SinceHm^sing(X(C))is f.g., a single pair W,Γsuffices.)

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Weak form of Suslin’s Conjecture

Recall Suslin’s conjecture predicts

LrHm(X)−→H^∼⁼ _m^sing(X(C)), form≥dim(X) +r.

Conjecture (Weak form of Suslin’s Conjecture (or Friedlander-Mazur Conjecture))

For a smooth, projective varietyX, the map LrHm(X)→H_m^sing(X(C)) is onto form≥dim(X) +r.

In particular, the weak Suslin conjecture predicts:

L_m−dim(X)H_m(X)H_m^sing(X)

is onto, for allm. (Ifm <dim(X), let Lm−dim(X):=L0.)

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Consequence of weak Suslin conjecture

Using the Proposition concerning the lifting of elements of im(LrHm→Hm^sing)along correspondences:

Proposition

Assume the weak form of Suslin’s Conjecture holds forX. Let d= dim(X).

Then for each integerm, there is a smooth, projective variety Y of dim 2d−m and a correspondenceΓ : Y ^_ ^_ ^_^//X of rel.

dim. m−dsuch that

Γ∗ :H_2d−m^sing (Y(C))H_m^sing(X(C)) is onto.

The existence of such aY,Γ turns out to be a very strong condition onX. In fact....

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Beilinson’s Theorem

Theorem (Beilinson)

The validity of all Grothendieck’s standard conjectures overC is equivalent to the following property: For each smooth, projectiveX, there is a Y andΓ as above such that

Γ∗ :H_2d−m^sing (Y(C))H_m^sing(X(C)) is onto.

Corollary (Beilinson)

The weak form of Suslin’s Conjecture is equivalent to the validity of all of Grothendieck’s standard conjectures overC.

Note: The⇐direction of the Corollary was originally shown by Friedlander-Mazur.

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Should we believe Suslin’s Conjecture?

Beilinson’s result makes it clear Suslin’s Conjecture is a very strong conjecture. Its weak form is equivalent to

Grothendieck’s standard conjectures.

Perhaps the strong form of Suslin’s conjecture is simply false.

The first unknown case occurs for1-cycles on a smooth projective3-dimensional variety X:

Question

For a smooth, projective3-dimensional variety X, is πm−2Z₁(X) =:L₁H_m(X)→H_m^sing(X(C)) one-to-one form≥3?

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Mixed Hodge structures for Lawson homology

Assume (for simplicity)X is projective. Then we have a surjection

M

Y,dim(Y)≤m−2r

H_m−2r^sing (Y(C))L_rH_m(X)

of (non f.g.) MHS’s (and where the maps are given by correspondences).

Thus,LrHm(X)has same Hodge type as H_m−2r^sing of a union of (highly singular) varieties of dimensionm−2r.

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MHS for L

₁

H

₃

(X ))

For example,

M

Y,dim(Y)=1

H₁^sing(Y(C))L1H3(X)

and soL₁H₃(X) has Hodge type: (0,0),(−1,0),(0,−1).

If we assumedim(X) = 3, then Suslin’s conjecture predicts L1H3(X)H₃^sing(X(C),Z(1))∼=H_sing³ (X,Z(2)) and the target has Hodge type(−1,0),(0,−1).

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A Conjecture concerning Hodge type

Conjecture

ForX smooth, projective of dimension 3, the Lawson group L1H3(X)has Hodge type (−1,0),(0,−1).

A proof (or counter-example) of just this conjecture would represent highly significant progress.

Note that the validity of this conjecture implies: Conjecture

ForX smooth, projective of dimension 3, the map H₃^M(X,Z(1))→L₁H₃(X) is a torsion map.

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A Conjecture concerning Hodge type

Conjecture

ForX smooth, projective of dimension 3, the Lawson group L1H3(X)has Hodge type (−1,0),(0,−1).

A proof (or counter-example) of just this conjecture would represent highly significant progress.

Note that the validity of this conjecture implies:

Conjecture

ForX smooth, projective of dimension 3, the map H₃^M(X,Z(1))→L₁H₃(X) is a torsion map.

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dim(X ) = 3, Y ⊂ X , dim(Y ) = 2, U = X \ Y

L₁H₄(Y)

∼=

dim(Y) = 2 //H₄^sing(Y)

L₁H₄(X)

GSConj⇒onto

SuslinConj⇒∼= //H₄^sing(X)

L₁H₄(U)

SuslinConj⇒∼= //H^sing₄ (U)

L₁H₃(Y)

∼=

dim(Y) = 2 //H₃^sing(Y)

L₁H₃(X)

SuslinConj⇒1-1 //H₃^sing(X)

L₁H₃(U) //H^sing₃ (U) 20 / 23

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Passing to function fields

LetC(X) := lim−→^U⊂XU. Proposition

LrHm(C(X)) = 0 ifm < d+r.

For example, L1H3(SpecC) = 0 ifdim(X) = 3.

Assuming Grothendieck Standard Conjectures:

L₁H₄(C(X))→H₄^sing(C(X))^SuslinC−→^⇒⁰L₁H₃(X)→H₃^sing(X).

Challenge

Find a good method of describing/constructing elements of

H_m^sing(C(X)) =H_sing^2d−m(C(X)).

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Toy examples

With finite coefficients:

H_m^sing(C(X),Z/n) =L_m−dim(X₎Hm(C(X),Z/n)

=H_M^2d−m(C(X),Z/n(2d−m))

=K_2d−m^Milnor(C(X))/n For H_sing¹ :

lim−→

Y

H_sing,Y¹ (X)→H_sing¹ (X)→H_sing¹ (C(X))

→lim−→

Y

H_sing,Y² (X)→H_sing² (X)

Since H_sing,Y¹ (X) = 0 andH_sing,Y² (X)∼=H_{2 dim(Y}^sing ₎(Y)= free abelian group on integral components ofY:

0→H_sing¹ (X)→H_sing¹ (C(X))→Z¹(X)hom∼0→0

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The End

Thanks for your attention!