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

## 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_{r}H_{m}(X) is induced by MHS onH∗^{sing}(C_{r,e}(X)).

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

## 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_{r}H_{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(C^{n}). 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,K_{0}^{semi}(X) =K_{0}(X)/(alg. equiv.).

“Every formal property one might expect involvingL∗H∗,
L^{∗}H^{∗} andK_{∗}^{semi}does indeed hold.”

## 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^{q}H^{2q−n}(X,Q).

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

E_{2}^{p,q}=L^{−q}H^{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^{t}H^{n}(X) ∼=^{?} H^{n}_{Zar}(X, tr^{≤t}Rπ∗Z),
whereπ: (V ar/C)_{analytic}→(V ar/C)_{Zar}.

Review of Lawson homology and related theories Suslin’s Conjecture

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

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

For codimension one cycles. Proof is by explicit
calculation ofL_{dim(X)−1}H∗(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 π_{0} [Bloch-Ogus]

Ld−tH_{2(d−t)}(X) =L^{t}H^{2t}(X)∼=H^{2t}Zar(X, tr^{≤t}Rπ∗Z)

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Review of Lawson homology and related theories Suslin’s Conjecture

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

## Finite generation of Lawson homology

Suslin’s Conjecture predicts, in particular, that

L_{r}H_{m}(X) is finitely generated form≥dim(X) +r−1.

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

· · · →C_{r}H_{m}(X)→L_{r}H_{m}(X)→H_{m}^{BM}(X)→C_{r}H_{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_{r}H_{m}(X) is finitely generated for

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

## 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∗ :Z_{0}(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)) =L_{0}Hm−2r(Y)→L_{r}H_{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.

## 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_{r}H_{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}^{∗} _{r}Hm(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_{r}H_{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}^{∗} _{r}Hm(X).

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

## Lifting to smooth varieties, using Hodge theory

For such a correspondenceΓ, the composition
H_{m−2r}^{sing} (Y)−→L^{Γ}^{∗} _{r}H_{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}^{top}H_{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.)

## 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....

## 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−2Z_{1}(X) =:L_{1}H_{m}(X)→H_{m}^{sing}(X(C))
one-to-one form≥3?

## 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_{r}H_{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.

17 / 23

## MHS for L

_{1}

## H

_{3}

## (X ))

For example,

M

Y,dim(Y)=1

H_{1}^{sing}(Y(C))L1H3(X)

and soL_{1}H_{3}(X) has Hodge type: (0,0),(−1,0),(0,−1).

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

## 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
H_{3}^{M}(X,Z(1))→L_{1}H_{3}(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
H_{3}^{M}(X,Z(1))→L_{1}H_{3}(X) is a torsion map.

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

L_{1}H_{4}(Y)

∼=

dim(Y) = 2 //H_{4}^{sing}(Y)

L_{1}H_{4}(X)

GSConj⇒onto

SuslinConj⇒∼= //H_{4}^{sing}(X)

L_{1}H_{4}(U)

SuslinConj⇒∼= //H^{sing}_{4} (U)

L_{1}H_{3}(Y)

∼=

dim(Y) = 2 //H_{3}^{sing}(Y)

L_{1}H_{3}(X)

SuslinConj⇒1-1 //H_{3}^{sing}(X)

L_{1}H_{3}(U) //H^{sing}_{3} (U)
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## Passing to function fields

LetC(X) := lim−→^{U⊂X}U.
Proposition

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

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

Assuming Grothendieck Standard Conjectures:

L_{1}H_{4}(C(X))→H_{4}^{sing}(C(X))^{SuslinC}−→^{⇒}^{0}L_{1}H_{3}(X)→H_{3}^{sing}(X).

Challenge

Find a good method of describing/constructing elements of

H_{m}^{sing}(C(X)) =H_{sing}^{2d−m}(C(X)).

## 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_{sing}^{1} :

lim−→

Y

H_{sing,Y}^{1} (X)→H_{sing}^{1} (X)→H_{sing}^{1} (C(X))

→lim−→

Y

H_{sing,Y}^{2} (X)→H_{sing}^{2} (X)

Since H_{sing,Y}^{1} (X) = 0 andH_{sing,Y}^{2} (X)∼=H_{2 dim(Y}^{sing} _{)}(Y)=
free abelian group on integral components ofY:

0→H_{sing}^{1} (X)→H_{sing}^{1} (C(X))→Z^{1}(X)hom∼0→0

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## The End

Thanks for your attention!