Diagonal Arithmetics. An introduction : Chow groups.

Diagonal Arithmetics.

An introduction : Chow groups.

Alena Pirutka

CNRS, École Polytechnique

May 25, 2015, IMPA, Rio de Janeiro

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

Objects of interest

k a field, X/k of finite type, equidimensional, d =dim(X).

(often considerX a projective variety, i.e. defined by homogeneous polynomials with

coefficients ink).

Look at all V ⊂X irreducibles of dimension d −i.

A cycleis a formal linear combination of such V’s: the group of cyclesof codimension i is

Zⁱ(X) =Zd−i(X) =⊕ZV

Denote[V] :=1·V ∈Zⁱ(X) the cycle corresponding to V, we call such cycleprime.

Some properties :

Objects of interest

k a field, X/k of finite type, equidimensional, d =dim(X).

(often considerX a projective variety, i.e.

defined by homogeneous polynomials with

coefficients ink).

Look at all V ⊂X irreducibles of dimension d −i.

A cycleis a formal linear combination of such V’s: the group of cyclesof codimension i is

Zⁱ(X) =Zd−i(X) =⊕ZV

Denote[V] :=1·V ∈Zⁱ(X) the cycle corresponding to V, we call such cycleprime.

Some properties :

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

Objects of interest

k a field, X/k of finite type, equidimensional, d =dim(X).

(often considerX a projective variety, i.e.

defined by homogeneous polynomials with

coefficients ink).

Look at all V ⊂X irreducibles of dimension d −i.

A cycleis a formal linear combination of such V’s: thegroup of cycles of codimension i is

Zⁱ(X) =Z_d−i(X) =⊕ZV

Denote[V] :=1·V ∈Zⁱ(X) the cycle corresponding to V, we call such cycleprime.

Some properties :

Objects of interest

k a field, X/k of finite type, equidimensional, d =dim(X).

(often considerX a projective variety, i.e.

defined by homogeneous polynomials with

coefficients ink).

Look at all V ⊂X irreducibles of dimension d −i.

A cycleis a formal linear combination of such V’s: thegroup of cycles of codimension i is

Zⁱ(X) =Z_d−i(X) =⊕ZV

Denote[V] :=1·V ∈Zⁱ(X) the cycle corresponding to V, we call such cycleprime.

Some properties :

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

Objects of interest

k a field, X/k of finite type, equidimensional, d =dim(X).

(often considerX a projective variety, i.e.

defined by homogeneous polynomials with

coefficients ink).

Look at all V ⊂X irreducibles of dimension d −i.

A cycleis a formal linear combination of such V’s: thegroup of cycles of codimension i is

Zⁱ(X) =Z_d−i(X) =⊕ZV

Denote[V] :=1·V ∈Zⁱ(X) the cycle corresponding to V, we call such cycleprime.

Some properties :

I (push-forward) forf :X →Y proper define f∗ :Z_i(X)→Z_i(Y) by

f∗([V]) =0 if dim f(V)<i and

f∗([V]) =a·f(V) ifdim f(V) =i, where ais the degree of the field extension k(V)/k(f(V)).

I (pull-back) Iff :X →Y is flat of relative dimension n, f^∗:Z_i(Y)→Z_i+n(X),f^∗([W]) = [f⁻¹(W)],W ⊂Y.

I (intersections) V ⊂X andW ⊂X intersect properly if all irreducible components of V ×_X W have codimension

codim_XV+codim_XW. One then definesV ·W as the sum of these components (with some multiplicities!).

I For X/Csmooth projective one has a cycle class map

cⁱ :Zⁱ(X)→H²ⁱ(X,Z), givingZⁱ(X)⊗Q→Hdgⁱ(X) where Hdgⁱ(X) =H²ⁱ(X,Q)∩H^i,i(X) (the Hodge classes). The Hodge conjecture predicts that this last map should be surjective.

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

I (push-forward) forf :X →Y proper define f∗ :Z_i(X)→Z_i(Y) by

f∗([V]) =0 if dim f(V)<i and

f∗([V]) =a·f(V) ifdim f(V) =i, where ais the degree of the field extension k(V)/k(f(V)).

I (pull-back) Iff :X →Y is flat of relative dimension n, f^∗:Z_i(Y)→Z_i+n(X),f^∗([W]) = [f⁻¹(W)],W ⊂Y.

I (intersections) V ⊂X andW ⊂X intersect properly if all irreducible components of V ×_X W have codimension

codim_XV+codim_XW. One then definesV ·W as the sum of these components (with some multiplicities!).

I For X/Csmooth projective one has a cycle class map

cⁱ :Zⁱ(X)→H²ⁱ(X,Z), givingZⁱ(X)⊗Q→Hdgⁱ(X) where Hdgⁱ(X) =H²ⁱ(X,Q)∩H^i,i(X) (the Hodge classes). The Hodge conjecture predicts that this last map should be surjective.

I (push-forward) forf :X →Y proper define f∗ :Z_i(X)→Z_i(Y) by

f∗([V]) =0 if dim f(V)<i and

f∗([V]) =a·f(V) ifdim f(V) =i, where ais the degree of the field extension k(V)/k(f(V)).

I (pull-back) Iff :X →Y is flat of relative dimension n, f^∗:Z_i(Y)→Z_i+n(X),f^∗([W]) = [f⁻¹(W)],W ⊂Y.

I (intersections) V ⊂X andW ⊂X intersect properly if all irreducible components of V ×_X W have codimension

codim_XV+codim_XW. One then definesV ·W as the sum of these components (with some multiplicities!).

I For X/Csmooth projective one has a cycle class map

cⁱ :Zⁱ(X)→H²ⁱ(X,Z), givingZⁱ(X)⊗Q→Hdgⁱ(X) where Hdgⁱ(X) =H²ⁱ(X,Q)∩H^i,i(X) (the Hodge classes). The Hodge conjecture predicts that this last map should be surjective.

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

I (push-forward) forf :X →Y proper define f∗ :Z_i(X)→Z_i(Y) by

f∗([V]) =0 if dim f(V)<i and

f∗([V]) =a·f(V) ifdim f(V) =i, where ais the degree of the field extension k(V)/k(f(V)).

I (pull-back) Iff :X →Y is flat of relative dimension n, f^∗:Z_i(Y)→Z_i+n(X),f^∗([W]) = [f⁻¹(W)],W ⊂Y.

I (intersections) V ⊂X andW ⊂X intersect properly if all irreducible components of V ×_X W have codimension

codim_XV+codim_XW. One then definesV ·W as the sum of these components (with some multiplicities!).

I For X/Csmooth projective one has a cycle class map

cⁱ :Zⁱ(X)→H²ⁱ(X,Z), givingZⁱ(X)⊗Q→Hdgⁱ(X) where Hdgⁱ(X) =H²ⁱ(X,Q)∩H^i,i(X) (the Hodge classes). The Hodge conjecture predicts that this last map should be surjective.

Equivalence relations

Z^∗(X) :=⊕Zⁱ(X) is is too huge!

Question: what cycles should one consider as equivalent?

∼an equivalence relation on algebraic cycles isadequate if

I ∼is compatible with addition of cycles;

I for any X/k smooth projective andα, β ∈Z^∗(X) one can find α⁰ ∼α andβ⁰ ∼β such that α⁰ andβ⁰ intersect properly (i.e. all components have rightcodimension)

I if X,Y/k are smooth projective,pr_X (resp. pr_Y) X ×Y →X (resp. Y) is the first (resp. second) projection and

α∈Z^∗(X),β=pr_X⁻¹(α) andγ∈Z^∗(X ×Y) intersectingβ properly, then α∼0⇒pr_Y∗(β·γ)∼0.

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

Equivalence relations

Z^∗(X) :=⊕Zⁱ(X) is is too huge!

Question: what cycles should one consider as equivalent?

∼an equivalence relation on algebraic cycles isadequate if

I ∼is compatible with addition of cycles;

I for any X/k smooth projective andα, β ∈Z^∗(X) one can find α⁰ ∼α andβ⁰ ∼β such that α⁰ andβ⁰ intersect properly (i.e. all components have rightcodimension)

I if X,Y/k are smooth projective,pr_X (resp. pr_Y) X ×Y →X (resp. Y) is the first (resp. second) projection and

α∈Z^∗(X),β=pr_X⁻¹(α) andγ∈Z^∗(X ×Y) intersectingβ properly, then α∼0⇒pr_Y∗(β·γ)∼0.

Equivalence relations

Z^∗(X) :=⊕Zⁱ(X) is is too huge!

Question: what cycles should one consider as equivalent?

∼an equivalence relation on algebraic cycles isadequate if

I ∼is compatible with addition of cycles;

I for any X/k smooth projective andα, β ∈Z^∗(X) one can find α⁰ ∼α andβ⁰ ∼β such that α⁰ andβ⁰ intersect properly (i.e.

all components have rightcodimension)

I ifX,Y/k are smooth projective,pr_X (resp. pr_Y) X ×Y →X (resp. Y) is the first (resp. second) projection and

α∈Z^∗(X),β=pr_X⁻¹(α) andγ ∈Z^∗(X ×Y) intersectingβ properly, thenα ∼0⇒pr_Y∗(β·γ)∼0.

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

The finest adequate equivalence relation

I For X =C a curve and i =1 define : Xa_jP_j ∼_rat 0 iff

Xa_jP_j =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by XajYj ∼_rat 0 if X

ajYj =div(f)

for some functionf onW ⊂X of dimensiond −i+1(better: the normalization of W).

I Chow groups : CHⁱ(X) =Zⁱ(X)/∼_rat;

I for Y ⊂X an integral subvariety of codimensioni write [Y]∈CHⁱ(X) for the class of Y. More generally, forY a subscheme ofX (not necessarily reduced ni irreducible) one can define[Y]∈CHⁱ(X).

The finest adequate equivalence relation

I For X =C a curve and i =1 define : Xa_jP_j ∼_rat 0 iff X

a_jP_j =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by XajYj ∼_rat 0 if

XajYj =div(f)

for some functionf onW ⊂X of dimensiond −i+1(better: the normalization of W).

I Chow groups : CHⁱ(X) =Zⁱ(X)/∼_rat;

I for Y ⊂X an integral subvariety of codimensioni write [Y]∈CHⁱ(X) for the class of Y. More generally, forY a subscheme ofX (not necessarily reduced ni irreducible) one can define[Y]∈CHⁱ(X).

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

The finest adequate equivalence relation

I For X =C a curve and i =1 define : Xa_jP_j ∼_rat 0 iff X

a_jP_j =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by XajYj ∼_rat 0 if X

ajYj =div(f) for some function f onW ⊂X of dimensiond −i+1

(better: the normalization of W).

I Chow groups : CHⁱ(X) =Zⁱ(X)/∼_rat;

I for Y ⊂X an integral subvariety of codimensioni write [Y]∈CHⁱ(X) for the class of Y. More generally, forY a subscheme ofX (not necessarily reduced ni irreducible) one can define[Y]∈CHⁱ(X).

The finest adequate equivalence relation

I For X =C a curve and i =1 define : Xa_jP_j ∼_rat 0 iff X

a_jP_j =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by XajYj ∼_rat 0 if X

ajYj =div(f)

for some function f onW ⊂X of dimensiond −i+1(better:

the normalization of W).

I Chow groups : CHⁱ(X) =Zⁱ(X)/∼_rat;

I for Y ⊂X an integral subvariety of codimensioni write [Y]∈CHⁱ(X) for the class of Y. More generally, forY a subscheme ofX (not necessarily reduced ni irreducible) one can define[Y]∈CHⁱ(X).

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

The finest adequate equivalence relation

I For X =C a curve and i =1 define : Xa_jP_j ∼_rat 0 iff X

a_jP_j =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by XajYj ∼_rat 0 if X

ajYj =div(f)

for some function f onW ⊂X of dimensiond −i+1(better:

the normalization of W).

I Chow groups : CHⁱ(X) =Zⁱ(X)/∼_rat;

I for Y ⊂X an integral subvariety of codimensioni write [Y]∈CHⁱ(X) for the class of Y. More generally, forY a subscheme ofX (not necessarily reduced ni irreducible) one can define[Y]∈CHⁱ(X).

The finest adequate equivalence relation

I For X =C a curve and i =1 define : Xa_jP_j ∼_rat 0 iff X

a_jP_j =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by XajYj ∼_rat 0 if X

ajYj =div(f)

for some function f onW ⊂X of dimensiond −i+1(better:

the normalization of W).

I Chow groups : CHⁱ(X) =Zⁱ(X)/∼_rat;

I for Y ⊂X an integral subvariety of codimensioni write [Y]∈CHⁱ(X) for the class of Y. More generally, for Y a subscheme ofX (not necessarily reduced ni irreducible) one can define[Y]∈CHⁱ(X).

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

The finest adequate equivalence relation

I For X =C a curve and i =1 define : Xa_jP_j ∼_rat 0 iff X

a_jP_j =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by XajYj ∼_rat 0 if X

ajYj =div(f)

for some function f onW ⊂X of dimensiond −i+1(better:

the normalization of W).

I Chow groups : CHⁱ(X) =Zⁱ(X)/∼_rat;

I for Y ⊂X an integral subvariety of codimensioni write [Y]∈CHⁱ(X) for the class of Y. More generally, for Y a subscheme ofX (not necessarily reduced ni irreducible) one can define[Y]∈CHⁱ(X).

Functorial properties for Chow groups

I (push-forward)f :X →Y proper then f∗ induces f∗ :CHi(X)→CHi(Y);

I (pull-back) f :X →Y flat of relative dimensionn, thenf^∗ inducesf^∗:CH_i(Y)→CH_i+n(X). If X,Y are smooth, by a more difficult construction one defines

f^∗:CHⁱ(Y)→CHⁱ(X);

I if K/k is a finite field extension of degreem,π :X_K →X, then the compositionπ∗◦π^∗:CH_i(X)→CH_i(X_K)→CH_i(X) is the multiplication by m.

I (cycle class) forX smooth projective, we have CHⁱ(X)→Hdgⁱ(X) (k =C).

I (localisation sequence) τ :Z ⊂X closed,j :U ⊂X the complement. Then we have an exact sequence

CH_i(Z)→^τ∗ CH_i(X) ^j

∗

→CH_i(U)→0.

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

Functorial properties for Chow groups

I (push-forward)f :X →Y proper then f∗ induces f∗ :CHi(X)→CHi(Y);

I (pull-back) f :X →Y flat of relative dimensionn, thenf^∗ inducesf^∗:CH_i(Y)→CH_i+n(X). If X,Y are smooth, by a more difficult construction one defines

f^∗:CHⁱ(Y)→CHⁱ(X);

I if K/k is a finite field extension of degreem,π :X_K →X, then the compositionπ∗◦π^∗:CH_i(X)→CH_i(X_K)→CH_i(X) is the multiplication by m.

I (cycle class) forX smooth projective, we have CHⁱ(X)→Hdgⁱ(X) (k =C).

I (localisation sequence) τ :Z ⊂X closed,j :U ⊂X the complement. Then we have an exact sequence

CH_i(Z)→^τ∗ CH_i(X) ^j

∗

→CH_i(U)→0.

Functorial properties for Chow groups

I (push-forward)f :X →Y proper then f∗ induces f∗ :CHi(X)→CHi(Y);

I (pull-back) f :X →Y flat of relative dimensionn, thenf^∗ inducesf^∗:CH_i(Y)→CH_i+n(X). If X,Y are smooth, by a more difficult construction one defines

f^∗:CHⁱ(Y)→CHⁱ(X);

I ifK/k is a finite field extension of degreem,π :X_K →X, then the compositionπ∗◦π^∗:CHi(X)→CHi(XK)→CHi(X) is the multiplication by m.

I (cycle class) forX smooth projective, we have CHⁱ(X)→Hdgⁱ(X) (k =C).

I (localisation sequence) τ :Z ⊂X closed,j :U ⊂X the complement. Then we have an exact sequence

CH_i(Z)→^τ∗ CH_i(X) ^j

∗

→CH_i(U)→0.

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

Functorial properties for Chow groups

I (push-forward)f :X →Y proper then f∗ induces f∗ :CHi(X)→CHi(Y);

I (pull-back) f :X →Y flat of relative dimensionn, thenf^∗ inducesf^∗:CH_i(Y)→CH_i+n(X). If X,Y are smooth, by a more difficult construction one defines

f^∗:CHⁱ(Y)→CHⁱ(X);

I ifK/k is a finite field extension of degreem,π :X_K →X, then the compositionπ∗◦π^∗:CHi(X)→CHi(XK)→CHi(X) is the multiplication by m.

I (cycle class) forX smooth projective, we have CHⁱ(X)→Hdgⁱ(X) (k =C).

I (localisation sequence) τ :Z ⊂X closed,j :U ⊂X the complement. Then we have an exact sequence

CH_i(Z)→^τ∗ CH_i(X) ^j

∗

→CH_i(U)→0.

Correspondences

LetX,Y/k be smooth projective varieteies. Then

I Any map f :X →Y givesf∗ :CH_i(X)→CH_i(Y)

I More : any α∈CH∗(X ×Y) givesα∗ :CH∗(X)→CH∗(Y): if γ∈CH∗(X), thenα∗(γ) =pr_Y∗(α·pr_X^∗(γ)),i.e. α∗ is the composition

CH∗(X)→CH∗(X ×Y)→^·α CH∗(X ×Y)→CH∗(Y).

I On cohomology: anyα∈CHⁱ(X ×Y) gives

α∗:H^∗(X,Q)→H^∗(Y,Q) : α∗(γ) =pr_Y∗(cⁱ(α)∪pr_X^∗(γ)), γ ∈H^∗(X,Q).

I An important example : consider ∆_X ⊂X ×X the diagonal. Then [∆X]∗ is the identity map.

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

Correspondences

LetX,Y/k be smooth projective varieteies. Then

I Any map f :X →Y givesf∗ :CH_i(X)→CH_i(Y)

I More : any α∈CH∗(X ×Y) givesα∗ :CH∗(X)→CH∗(Y):

ifγ ∈CH∗(X), thenα∗(γ) =pr_Y∗(α·pr_X^∗(γ)),i.e. α∗ is the composition

CH∗(X)→CH∗(X ×Y)→^·α CH∗(X ×Y)→CH∗(Y).

I On cohomology: anyα∈CHⁱ(X ×Y) gives

α∗:H^∗(X,Q)→H^∗(Y,Q) : α∗(γ) =pr_Y∗(cⁱ(α)∪pr_X^∗(γ)), γ ∈H^∗(X,Q).

I An important example : consider ∆_X ⊂X ×X the diagonal.

Then [∆_X]∗ is the identity map.

Other classical equivalence relations

Letα∈Zⁱ(X)

I (algebraic) α∼_alg 0 if there exists a smooth projective curve C and two pointsc₁,c₂ ∈C(k) andβ ∈Zⁱ(X×C) such that α=τ_c^∗₁β−τ_c^∗₂β, where τci is the inclusion ofci inC.

I (homological) α∼_hom 0 ifcⁱ(α) =0 (over C, over k take another (Weil) cohomology)

I (numerical) α∼_num 0 if for anyβ ∈Z^d−i(X) one hasα·β (is well-defined!) is zero.

one has{α∼_rat 0} ⊂ {α∼_alg 0} ⊂ {α∼_hom0} ⊂ {α∼_num0}.

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.

Plan of the lectures

General question : What one can do by studying zero-cycles ?

I (Bloch-Srinivas) triviality of CH0 and equivalence relations;

I (Voisin, Colliot-Thélène – Pirutka, Beauville, Totaro, Hassett-Kresch-Tschinkel) universal triviality ofCH₀ and stable rationality.

Plan of the lectures

General question : What one can do by studying zero-cycles ?

I (Bloch-Srinivas) triviality of CH0 and equivalence relations;

I (Voisin, Colliot-Thélène – Pirutka, Beauville, Totaro, Hassett-Kresch-Tschinkel) universal triviality ofCH₀ and stable rationality.

Alena Pirutka CNRS, École Polytechnique

Diagonal Arithmetics. An introduction : Chow groups.