On some invariants for algebraic cobordism cycles International Workshop on Motives, Tokyo February 15, 2016 Gereon Quick NTNU
On some invariants for algebraic cobordism
cycles
International Workshop on Motives, Tokyo February 15, 2016
Gereon QuickNTNU
Point of departure: Poincaré, Lefschetz, Hodge…
Let X⊂PN be a smooth projective complex variety.
Point of departure: Poincaré, Lefschetz, Hodge…
Let X⊂PN be a smooth projective complex variety.
Let zj be a local coordinate on X.
Point of departure: Poincaré, Lefschetz, Hodge…
Let X⊂PN be a smooth projective complex variety.
Let zj be a local coordinate on X. Write zj = xj + iyj with real coordinates xj and yj.
Point of departure: Poincaré, Lefschetz, Hodge…
Let X⊂PN be a smooth projective complex variety.
Let zj be a local coordinate on X. Write zj = xj + iyj with real coordinates xj and yj.
There are associated differential forms
dzj = dxj + idyj and dzj = dxj - idyj.-
Point of departure: Poincaré, Lefschetz, Hodge…
Let X⊂PN be a smooth projective complex variety.
Let zj be a local coordinate on X. Write zj = xj + iyj with real coordinates xj and yj.
There are associated differential forms
dzj = dxj + idyj and dzj = dxj - idyj.-
Every k-form 𝛼 on X can then be written as
𝛼 = ∑j fj dzj1∧…∧dzjp∧dzjp+1∧…∧dzjk.- -
Integrals over cycles:
𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -
Let’s write 𝛼 ∈ Ap,q(X) if 𝛼 is of the form
Integrals over cycles:
exactly p dzj’s for all j
𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -
Let’s write 𝛼 ∈ Ap,q(X) if 𝛼 is of the form
Integrals over cycles:
exactly p dzj’s for all j exactly q dzj’s for all j-
𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -
Let’s write 𝛼 ∈ Ap,q(X) if 𝛼 is of the form
Integrals over cycles:
exactly p dzj’s for all j exactly q dzj’s for all j-
𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -
Let’s write 𝛼 ∈ Ap,q(X) if 𝛼 is of the form
Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k.
Integrals over cycles:
exactly p dzj’s for all j exactly q dzj’s for all j-
𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -
Let’s write 𝛼 ∈ Ap,q(X) if 𝛼 is of the form
Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k. We can integrate 𝛼 over 𝛤: ∫ 𝜄*𝛼. 𝛤
Integrals over cycles:
exactly p dzj’s for all j exactly q dzj’s for all j-
𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -
Let’s write 𝛼 ∈ Ap,q(X) if 𝛼 is of the form
Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k. We can integrate 𝛼 over 𝛤: ∫ 𝜄*𝛼. 𝛤
If 𝛤 = Z happens to be an algebraic subvariety of X, say of complex dimension n, then ∫ 𝜄*𝛼 = 0 unless 𝛼 lies in An,n(X). Z
Hodge’s question:
Hodge’s question:This imposes a necessary condition on a topological cycle 𝜄: 𝛤 ⊂ X to be algebraic (homologous to an algebraic subvariety Z dimension n):
Hodge’s question:This imposes a necessary condition on a topological cycle 𝜄: 𝛤 ⊂ X to be algebraic (homologous to an algebraic subvariety Z dimension n):
𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤
Hodge’s question:This imposes a necessary condition on a topological cycle 𝜄: 𝛤 ⊂ X to be algebraic (homologous to an algebraic subvariety Z dimension n):
𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤
Hodge wondered: Is this condition also sufficient? ? ⇐ ?
Hodge’s question:This imposes a necessary condition on a topological cycle 𝜄: 𝛤 ⊂ X to be algebraic (homologous to an algebraic subvariety Z dimension n):
𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤
Hodge wondered: Is this condition also sufficient? ? ⇐ ?
Usually, we ask this question in cohomological terms…
Hodge’s question in terms of cohomology:
Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Z
Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Zits Poincaré dual complex cohomology class is represented by a form in Ap,p(X) with p = dimX - n.
Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Zits Poincaré dual complex cohomology class is represented by a form in Ap,p(X) with p = dimX - n.Such classes are said to be “Hodge-type (p,p)”. The subgroup of such classes is denoted by
Hp,p(X) ⊂ H2p(X;C).
Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Zits Poincaré dual complex cohomology class is represented by a form in Ap,p(X) with p = dimX - n.Such classes are said to be “Hodge-type (p,p)”. The subgroup of such classes is denoted by
Hp,p(X) ⊂ H2p(X;C).The Hodge Conjecture: The map
clH: Zp(X) → Hp,p(X) ∩ H2p(X;Z) is surjective.
Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Zits Poincaré dual complex cohomology class is represented by a form in Ap,p(X) with p = dimX - n.Such classes are said to be “Hodge-type (p,p)”. The subgroup of such classes is denoted by
Hp,p(X) ⊂ H2p(X;C).The Hodge Conjecture: The map
clH: Zp(X) → Hp,p(X) ∩ H2p(X;Z) is surjective.⊗Q
switch to Q-coefficients
×
A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g.
A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g. Let ω1, …, ωg be a basis of (holom.) 1-forms on C.
A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g. Let ω1, …, ωg be a basis of (holom.) 1-forms on C.
⌠⌡qω1
p⌠⌡qωg
p⎛⎝
⎞⎠, ...,
Then every pair of points p,q∈C defines a g-tuple of complex numbers
A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g. Let ω1, …, ωg be a basis of (holom.) 1-forms on C.
µ: Div0(C) → ℂg
⌠⌡qω1
p⌠⌡qωg
p⎛⎝
⎞⎠, ...,
Then every pair of points p,q∈C defines a g-tuple of complex numbers
A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g. Let ω1, …, ωg be a basis of (holom.) 1-forms on C.
µ: Div0(C) → ℂg
⌠⌡qω1
p⌠⌡qωg
p⎛⎝
⎞⎠, ...,
Then every pair of points p,q∈C defines a g-tuple of complex numbers
group of formal sums ∑i(pi-qi)
A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g. Let ω1, …, ωg be a basis of (holom.) 1-forms on C.
µ: Div0(C) → ℂg
⌠⌡qω1
p⌠⌡qωg
p⎛⎝
⎞⎠, ...,
Then every pair of points p,q∈C defines a g-tuple of complex numbers
group of formal sums ∑i(pi-qi) lattice of integrals
of ωj’s over loops
/Λ
A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g. Let ω1, …, ωg be a basis of (holom.) 1-forms on C.
µ: Div0(C) → ℂg
⌠⌡qω1
p⌠⌡qωg
p⎛⎝
⎞⎠, ...,
Then every pair of points p,q∈C defines a g-tuple of complex numbers
group of formal sums ∑i(pi-qi) lattice of integrals
of ωj’s over loops
/Λ
Jacobian variety of C
=: J(C)
A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g. Let ω1, …, ωg be a basis of (holom.) 1-forms on C.
µ: Div0(C) → ℂg
⌠⌡qω1
p⌠⌡qωg
p⎛⎝
⎞⎠, ...,
Then every pair of points p,q∈C defines a g-tuple of complex numbers
group of formal sums ∑i(pi-qi)
Jacobi Inversion Theorem: The map (Abel-Jacobi map)
is surjective.
lattice of integrals of ωj’s over loops
/Λ
Jacobian variety of C
=: J(C)
More canonically:
More canonically:
µ: Div0(C) → J(C) = H0(C; Ω1hol)∨/H1(C;Z)
⌠⌡qiω
pi
∑(pi-qi) ⟼ mod H1(C;Z).∑i⎝⎛ ⟼ ⎞
⎠ω
More canonically:
Abel’s Theorem: The kernel of this map is the subgroup of principal divisors.
µ: Div0(C) → J(C) = H0(C; Ω1hol)∨/H1(C;Z)
⌠⌡qiω
pi
∑(pi-qi) ⟼ mod H1(C;Z).∑i⎝⎛ ⟼ ⎞
⎠ω
Lefschetz’s proof for (1,1)-classes:
For simplicity, let X⊂PN be a surface.
Lefschetz’s proof for (1,1)-classes:
Let {Ct}t be a family of curves on X (parametrized over the projective line P1).
For simplicity, let X⊂PN be a surface.
Lefschetz’s proof for (1,1)-classes:
Let {Ct}t be a family of curves on X (parametrized over the projective line P1).
Associated to {Ct}t is the family of Jacobians
J := ⋃t J(Ct)
For simplicity, let X⊂PN be a surface.
Lefschetz’s proof for (1,1)-classes:
Let {Ct}t be a family of curves on X (parametrized over the projective line P1).
Associated to {Ct}t is the family of Jacobians
J := ⋃t J(Ct)
and a fibre space π: J → P1 (of complex Lie groups).
For simplicity, let X⊂PN be a surface.
Lefschetz’s proof for (1,1)-classes:
Let {Ct}t be a family of curves on X (parametrized over the projective line P1).
Associated to {Ct}t is the family of Jacobians
J := ⋃t J(Ct)
and a fibre space π: J → P1 (of complex Lie groups).
A “normal function” 𝜈 is a holomorphic section of π.
For simplicity, let X⊂PN be a surface.
Normal functions arise naturally:
Lefschetz’s proof continued:
Normal functions arise naturally: Let D be an algebraic curve on X. It intersects Ct in points p1(t),…, pd(t).
Lefschetz’s proof continued:
Normal functions arise naturally: Let D be an algebraic curve on X. It intersects Ct in points p1(t),…, pd(t).
Lefschetz’s proof continued:
Choose a point p0 on all Ct. Then ∑i pi(t) - dp0 is a divisor of degree 0 and defines a point
𝜈D(t) ∈ J(Ct).
Normal functions arise naturally: Let D be an algebraic curve on X. It intersects Ct in points p1(t),…, pd(t).
Lefschetz’s proof continued:
Hence D defines a normal function
𝜈D: t ↦ 𝜈D(t) ∈ J.
Choose a point p0 on all Ct. Then ∑i pi(t) - dp0 is a divisor of degree 0 and defines a point
𝜈D(t) ∈ J(Ct).
Poincaré’s Existence Theorem:
Every normal function 𝜈 arises as the normal function 𝜈D associated to an algebraic curve D.
Poincaré’s Existence Theorem:
Every normal function 𝜈 arises as the normal function 𝜈D associated to an algebraic curve D.
Poincaré’s Existence Theorem:
Then Lefschetz proved:
Every normal function 𝜈 arises as the normal function 𝜈D associated to an algebraic curve D.
Poincaré’s Existence Theorem:
• Every normal function 𝜈 defines a class 𝜂(𝜈)∈H2(X;Z) of Hodge type (1,1) such that 𝜂(𝜈D) = clH(D).
Then Lefschetz proved:
Every normal function 𝜈 arises as the normal function 𝜈D associated to an algebraic curve D.
Poincaré’s Existence Theorem:
• Every normal function 𝜈 defines a class 𝜂(𝜈)∈H2(X;Z) of Hodge type (1,1) such that 𝜂(𝜈D) = clH(D).
• Every class in H2(X;Z) of Hodge type (1,1) arises as 𝜂(𝜈) for some normal function 𝜈.
Then Lefschetz proved:
X a smooth projective complex variety with dimX=n.
Griffiths: in higher dimensions…
X a smooth projective complex variety with dimX=n.
Z⊂X a closed subvariety of codimension p.
Griffiths: in higher dimensions…
X a smooth projective complex variety with dimX=n.
Z⊂X a closed subvariety of codimension p.
Griffiths: in higher dimensions…
We assume that Z is the boundary of a differentiable chain Γ ∈ C2n-2p+1(X;Z).
X a smooth projective complex variety with dimX=n.
Z⊂X a closed subvariety of codimension p.
Griffiths: in higher dimensions…
Let ω be a smooth form of degree 2n-2p+1.
We assume that Z is the boundary of a differentiable chain Γ ∈ C2n-2p+1(X;Z).
X a smooth projective complex variety with dimX=n.
Z⊂X a closed subvariety of codimension p.
Then ⌠⌡Γω⟼ defines a map A2n-2p+1(X)→C.ω⎛⎝⎞⎠
Griffiths: in higher dimensions…
Let ω be a smooth form of degree 2n-2p+1.
We assume that Z is the boundary of a differentiable chain Γ ∈ C2n-2p+1(X;Z).
We would like this map to
Deligne, Griffiths, …:
We would like this map to
Deligne, Griffiths, …:
• descend to a map on cohomology,
We would like this map to
Deligne, Griffiths, …:
• descend to a map on cohomology,
• and to become independent of the choice of Γ.
We would like this map to
Deligne, Griffiths, …:
• descend to a map on cohomology,
• and to become independent of the choice of Γ.
Let ω’ be a form such that dω’= ω.
We would like this map to
Deligne, Griffiths, …:
• descend to a map on cohomology,
• and to become independent of the choice of Γ.
Let ω’ be a form such that dω’= ω.
Then Stokes’ formula tells us
⌠⌡Γω ⌠
⌡Zω’=
⌠⌡Γω ⌠
⌡Zω’=
The intermediate Jacobian of Griffiths and the Abel-Jacobi map:
⌠⌡Γω ⌠
⌡Zω’=
If ω is a form of degree (p’,2n-2p+1-p’) with p’ ≥ n-p+1, this integral vanishes.
The intermediate Jacobian of Griffiths and the Abel-Jacobi map:
⌠⌡Γω ⌠
⌡Zω’=
If ω is a form of degree (p’,2n-2p+1-p’) with p’ ≥ n-p+1, this integral vanishes.
We obtain a well-defined map⌠⌡Γ
⟼Z
µ: Zp(X)h → J2p-1(X) = Fn-p+1H2n-2p+1(X;C)∨/H2n-2p+1(X;Z)
for some Γ with Z=∂Γ
The intermediate Jacobian of Griffiths and the Abel-Jacobi map:
⌠⌡Γω ⌠
⌡Zω’=
If ω is a form of degree (p’,2n-2p+1-p’) with p’ ≥ n-p+1, this integral vanishes.
We obtain a well-defined map⌠⌡Γ
⟼Z
µ: Zp(X)h → J2p-1(X) = Fn-p+1H2n-2p+1(X;C)∨/H2n-2p+1(X;Z)
for some Γ with Z=∂Γ
J2p-1(X) is a complex torus and is called Griffiths’ intermediate Jacobian.
The intermediate Jacobian of Griffiths and the Abel-Jacobi map:
The Jacobian and Griffiths’ theorem:
The Jacobian and Griffiths’ theorem:
J2p-1(X) is, in general, not an abelian variety.
The Jacobian and Griffiths’ theorem:
J2p-1(X) is, in general, not an abelian variety.
But it varies homomorphically in families.
The Jacobian and Griffiths’ theorem:
J2p-1(X) is, in general, not an abelian variety.
But it varies homomorphically in families.
Have an induced a map:
Griffp(X):= Zp(X)h/Zp(X)alg → J2p-1(X)/J2p-1(X)alg
The Jacobian and Griffiths’ theorem:
J2p-1(X) is, in general, not an abelian variety.
But it varies homomorphically in families.
Griffith’s theorem: Let X⊂P4 be a general quintic hypersurface. There are lines L and L’ on X such that µ(L-L’) is a non torsion element in J3(X).
Have an induced a map:
Griffp(X):= Zp(X)h/Zp(X)alg → J2p-1(X)/J2p-1(X)alg
An interesting diagram:Let X be a smooth projective complex variety.
An interesting diagram:
Zp(X)Let X be a smooth projective complex variety.
An interesting diagram:
Zp(X) Z⊂XLet X be a smooth projective complex variety.
An interesting diagram:
Zp(X)
clH
Z⊂XLet X be a smooth projective complex variety.
An interesting diagram:
Hdg2p(X)
Zp(X)
clH
Z⊂XLet X be a smooth projective complex variety.
An interesting diagram:
Hdg2p(X)
Zp(X)
clH
Z⊂X
[Zsm]
−
Let X be a smooth projective complex variety.
An interesting diagram:
Hdg2p(X)
Zp(X)
clH
Zp(X)h=Kernel of clH ⊂ Z⊂X
[Zsm]
−
Let X be a smooth projective complex variety.
An interesting diagram:
Hdg2p(X)
Zp(X)
clH
Zp(X)h=Kernel of clH ⊂
Abel-Jacobimap µ
Z⊂X
[Zsm]
−
Let X be a smooth projective complex variety.
An interesting diagram:
Hdg2p(X)
Zp(X)
clH
Zp(X)h=Kernel of clH ⊂
Abel-Jacobimap µ
J2p-1(X)
Z⊂X
[Zsm]
−
Let X be a smooth projective complex variety.
An interesting diagram:
Hdg2p(X)
Zp(X)
clH
Zp(X)h=Kernel of clH ⊂
Abel-Jacobimap µ
J2p-1(X)
Z⊂X
[Zsm]
−
Let X be a smooth projective complex variety.
→ HD (X;Z(p)) →2p
An interesting diagram:
Hdg2p(X)
Zp(X)
clH
Zp(X)h=Kernel of clH ⊂
Abel-Jacobimap µ
J2p-1(X)
Z⊂X
[Zsm]
−
Let X be a smooth projective complex variety.
→ HD (X;Z(p)) →2p
Deligne cohomology combines topological with Hodge theoretic information
An interesting diagram:
Hdg2p(X)
Zp(X)
clH
Zp(X)h=Kernel of clH ⊂
Abel-Jacobimap µ
J2p-1(X)
Z⊂X
[Zsm]
−
Let X be a smooth projective complex variety.
→ 0→ HD (X;Z(p)) →2p
Deligne cohomology combines topological with Hodge theoretic information
An interesting diagram:
Hdg2p(X)
Zp(X)
clH
Zp(X)h=Kernel of clH ⊂
Abel-Jacobimap µ
J2p-1(X)
Z⊂X
[Zsm]
−
Let X be a smooth projective complex variety.
0 → → 0→ HD (X;Z(p)) →2p
Deligne cohomology combines topological with Hodge theoretic information
An interesting diagram:
Hdg2p(X)
Zp(X)
clHclHD
Zp(X)h=Kernel of clH ⊂
Abel-Jacobimap µ
J2p-1(X)
Z⊂X
[Zsm]
−
Let X be a smooth projective complex variety.
0 → → 0→ HD (X;Z(p)) →2p
Deligne cohomology combines topological with Hodge theoretic information
Another interesting map for smooth complex varieties:
Φ: Ω*(X) → MU *(X)2
Another interesting map for smooth complex varieties:
Φ: Ω*(X) → MU *(X)2
Another interesting map for smooth complex varieties:
algebraic cobordismof Levine and Morel
Φ: Ω*(X) → MU *(X)2
Another interesting map for smooth complex varieties:
algebraic cobordismof Levine and Morel
complex cobordism ofthe top. space X(C)
Φ: Ω*(X) → MU *(X)2
Another interesting map for smooth complex varieties:
algebraic cobordismof Levine and Morel
complex cobordism ofthe top. space X(C)
Recall: MUi(X(C)) is (roughly speaking) generated by maps f: M→X(C) of codimension i modulo the relation g-1(0) ∼ g-1(1) for maps g: M’→X(C)xR such that g is transversal to e0: X(C)→X(C)xR and e1: X(C)→X(C)xR.
Algebraic cobordism in a nutshell:
Algebraic cobordism in a nutshell:
Ωp(X) is generated (roughly speaking) by proj. maps f:Y→X of codimension p with Y smooth modulo …
Algebraic cobordism in a nutshell:
Ωp(X) is generated (roughly speaking) by proj. maps f:Y→X of codimension p with Y smooth modulo …
Levine’s and Pandharipande’s “double point relation”:
Algebraic cobordism in a nutshell:
Ωp(X) is generated (roughly speaking) by proj. maps f:Y→X of codimension p with Y smooth modulo …
Levine’s and Pandharipande’s “double point relation”:
π-1(0) ∼ π-1(∞) for projective morphisms π: Y’→XxP1 such that π-1(0) is smooth and π-1(∞)=A∪DB where A and B are smooth and meet transversally in D.
What can we say about the map Φ?
Ω*(X) MU2*(X)Φ
What can we say about the map Φ?
Ω*(X) MU2*(X)Φ[Y→X]
What can we say about the map Φ?
Ω*(X) MU2*(X)Φ[Y→X] [Y(C)→X(C)]⟼
What can we say about the map Φ?
• The image: Ω*(X) MU2*(X)Φ[Y→X] [Y(C)→X(C)]⟼
What can we say about the map Φ?
• The image:
CH*(X)
Ω*(X) MU2*(X)Φ[Y→X] [Y(C)→X(C)]⟼
What can we say about the map Φ?
• The image:
CH*(X)
Ω*(X) MU2*(X)Φ
Hdg2*(X) ⊆ H2*(X;Z)clH
[Y→X] [Y(C)→X(C)]⟼
What can we say about the map Φ?
• The image:
CH*(X)
Ω*(X) MU2*(X)Φ
Hdg2*(X) ⊆ H2*(X;Z)clH
[Y→X] [Y(C)→X(C)]⟼
What can we say about the map Φ?
• The image:
CH*(X)
Ω*(X) MU2*(X)Φ
Hdg2*(X) ⊆ H2*(X;Z)clH
[Y→X] [Y(C)→X(C)]⟼
What can we say about the map Φ?
• The image:
CH*(X)There is a “Hodge-theoretic” restriction for ImΦ.
Ω*(X) MU2*(X)Φ
Hdg2*(X) ⊆ H2*(X;Z)clH
[Y→X] [Y(C)→X(C)]⟼
What can we say about the map Φ?
• The image:
• The kernel:
CH*(X)There is a “Hodge-theoretic” restriction for ImΦ.
Ω*(X) MU2*(X)Φ
Hdg2*(X) ⊆ H2*(X;Z)clH
[Y→X] [Y(C)→X(C)]⟼
What can we say about the map Φ?
• The image:
• The kernel:
CH*(X)There is a “Hodge-theoretic” restriction for ImΦ.
Griffiths’ theorem suggests that Φ is not injective.
Ω*(X) MU2*(X)Φ
Hdg2*(X) ⊆ H2*(X;Z)clH
[Y→X] [Y(C)→X(C)]⟼
What can we say about the map Φ?
• The image:
• The kernel:
CH*(X)There is a “Hodge-theoretic” restriction for ImΦ.
Griffiths’ theorem suggests that Φ is not injective.
Question: Is there is an “Abel-Jacobi-invariant” which is able to detect elements in KerΦ?
Ω*(X) MU2*(X)Φ
Hdg2*(X) ⊆ H2*(X;Z)clH
[Y→X] [Y(C)→X(C)]⟼
What can we say about ImΦ?
Ω*(X) → MU2*(X)
What can we say about ImΦ?
Ω*(X) → MU2*(X)HdgMU2*(X) ⊂
What can we say about ImΦ?
Ω*(X) → MU2*(X)HdgMU2*(X) ⊂
This map is not surjective, …
What can we say about ImΦ?
Ω*(X) → MU2*(X)
… but the classical topological arguments do not work!
HdgMU2*(X) ⊂
This map is not surjective, …
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
Ω*(X)⊗L*Z MU2*(X)⊗L*Z
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
Ω*(X)⊗L*Z MU2*(X)⊗L*Z
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
CH*(X)
Ω*(X)⊗L*Z MU2*(X)⊗L*Z
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH
Ω*(X)⊗L*Z MU2*(X)⊗L*Z
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH
Ω*(X)⊗L*Z MU2*(X)⊗L*Z
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH
Totaro
Ω*(X)⊗L*Z MU2*(X)⊗L*Z
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH
Totaro
Ω*(X)⊗L*Z MU2*(X)⊗L*Z
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
CH*(X)
Levine-Morel ≈
Hdg2*(X) ⊆ H2*(X;Z)clH
Totaro
Ω*(X)⊗L*Z MU2*(X)⊗L*Z
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
CH*(X)
Levine-Morel ≈ ≉ in general
Hdg2*(X) ⊆ H2*(X;Z)clH
Totaro
Ω*(X)⊗L*Z MU2*(X)⊗L*Z
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
CH*(X)
Levine-Morel ≈ ≉ in general
Atiyah-Hirzebruch: clH is not surjective.
Hdg2*(X) ⊆ H2*(X;Z)clH
Totaro
Ω*(X)⊗L*Z MU2*(X)⊗L*Z
Ω*(X) MU2*(X)Φ
Atiyah-Hirzebruch and Totaro:
CH*(X)
Levine-Morel ≈ ≉ in general
Atiyah-Hirzebruch: clH is not surjective.
Hdg2*(X) ⊆ H2*(X;Z)clH
This argument does not work for Φ.
Totaro
Kollar’s examples: (see also Soulé-Voisin)
Kollar’s examples: (see also Soulé-Voisin)
Let X⊂P4 a very general hypersurface of degree d.
Kollar’s examples: (see also Soulé-Voisin)
Let X⊂P4 a very general hypersurface of degree d.
Lefschetz’s hyperplane theorem:
Kollar’s examples: (see also Soulé-Voisin)
Let X⊂P4 a very general hypersurface of degree d.
Lefschetz’s hyperplane theorem:
• H2(X;Z) is torsion-free and generated by the class h=c1(𝓞X(1)) of a hyperplane section on X,
Kollar’s examples: (see also Soulé-Voisin)
Let X⊂P4 a very general hypersurface of degree d.
Lefschetz’s hyperplane theorem:
• H2(X;Z) is torsion-free and generated by the class h=c1(𝓞X(1)) of a hyperplane section on X,
• H4(X;Z) is torsion-free and generated by a class α such that ∫ α∙h=1, and all classes are Hodge classes.
X
Kollar’s examples: H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
Kollar’s examples:
Assume now d=p3 for a prime p≥5.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
Kollar’s examples:
Kollar: Then any algebraic curve C on X has degree divisible by p.
Assume now d=p3 for a prime p≥5.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
Kollar’s examples:
Kollar: Then any algebraic curve C on X has degree divisible by p.
Assume now d=p3 for a prime p≥5.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
Consequence:
Kollar’s examples:
Kollar: Then any algebraic curve C on X has degree divisible by p.
Assume now d=p3 for a prime p≥5.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
Consequence: Let C be a curve on X and [C]∈H4(X;Z) be its cohomology class.
Kollar’s examples:
Kollar: Then any algebraic curve C on X has degree divisible by p.
Assume now d=p3 for a prime p≥5.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
Consequence: Let C be a curve on X and [C]∈H4(X;Z) be its cohomology class.
Then [C]= kα for some k and ∫ [C]∙h=k.X
Kollar’s examples:
Kollar: p divides the degree of any curve on X.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
Kollar’s examples:
Kollar: p divides the degree of any curve on X.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
k = ∫ [C]∙hX
Kollar’s examples:
Kollar: p divides the degree of any curve on X.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
k = ∫ [C]∙hX
= number of intersection points of C with h
Kollar’s examples:
Kollar: p divides the degree of any curve on X.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
k = ∫ [C]∙hX
= number of intersection points of C with h
= degree of C
Kollar’s examples:
Kollar: p divides the degree of any curve on X.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
k = ∫ [C]∙hX
= number of intersection points of C with h
= degree of C ⟹ p divides k
Kollar’s examples:
Kollar: p divides the degree of any curve on X.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
k = ∫ [C]∙hX
= number of intersection points of C with h
= degree of C ⟹ p divides k
In particular, α is not algebraic (for k cannot be 1).
Kollar’s examples:
Kollar: p divides the degree of any curve on X.
H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X
k = ∫ [C]∙hX
= number of intersection points of C with h
= degree of C ⟹ p divides k
In particular, α is not algebraic (for k cannot be 1).
But dα is algebraic (for ∫ dα∙h = d = ∫ h2∙h ⇒ dα=h2). XX
Consequences for Φ: Ω*(X) → MU2*(X):
Consequences for Φ: Ω*(X) → MU2*(X):
Let X⊂P4 be a very general hypersurface as above.
Consequences for Φ: Ω*(X) → MU2*(X):
Let X⊂P4 be a very general hypersurface as above.
Then MU4(X) ↠ H4(X;Z) is surjective, and thus Kollar’s argument implies that Φ is not surjective (on Hodge classes).
Consequences for Φ: Ω*(X) → MU2*(X):
Let X⊂P4 be a very general hypersurface as above.
Then MU4(X) ↠ H4(X;Z) is surjective, and thus Kollar’s argument implies that Φ is not surjective (on Hodge classes).
These examples are “not topological”:there is a dense subset of hypersurfaces Y⊂P4 such that the generator in H4(Y;Z) is algebraic.
What about the kernel?
Φ: Ω*(X) → MU *(X)2
[Y→X] [Y(C)→X(C)]⟼
What about the kernel?
Φ: Ω*(X) → MU *(X)2
[Y→X] [Y(C)→X(C)]⟼
Φ(Y)=0 creates a new equivalence relation.
What about the kernel?
Φ: Ω*(X) → MU *(X)2
[Y→X] [Y(C)→X(C)]⟼
Φ(Y)=0 creates a new equivalence relation.
top. cobordismrelation
What about the kernel?
Φ: Ω*(X) → MU *(X)2
[Y→X] [Y(C)→X(C)]⟼
Φ(Y)=0 creates a new equivalence relation.
double point relation
top. cobordismrelation
What about the kernel?
Φ: Ω*(X) → MU *(X)2
[Y→X] [Y(C)→X(C)]⟼
Φ(Y)=0 creates a new equivalence relation.
double point relation
top. cobordismrelation
⊆
What about the kernel?
Φ: Ω*(X) → MU *(X)2
[Y→X] [Y(C)→X(C)]⟼
Φ(Y)=0 creates a new equivalence relation.
double point relation
top. cobordismrelation
⊆ algebraic equivalence
What about the kernel?
Φ: Ω*(X) → MU *(X)2
[Y→X] [Y(C)→X(C)]⟼
Φ(Y)=0 creates a new equivalence relation.
double point relation
top. cobordismrelation
⊆ algebraic equivalence
of A. Krishna and J. Park
What about the kernel?
Φ: Ω*(X) → MU *(X)2
[Y→X] [Y(C)→X(C)]⟼
Φ(Y)=0 creates a new equivalence relation.
double point relation
top. cobordismrelation
⊆ algebraic equivalence
⊆
of A. Krishna and J. Park
A new diagram:
Let X be a smooth projective complex variety.
A new diagram:
Ωp(X)
Let X be a smooth projective complex variety.
A new diagram:
Ωp(X) [Y→X]
Let X be a smooth projective complex variety.
A new diagram:
Ωp(X) [Y→X]
Let X be a smooth projective complex variety.
HdgMU(X)2p
A new diagram:
Ωp(X) [Y→X]
[Y(C)→X(C)]
−
Let X be a smooth projective complex variety.
HdgMU(X)2p
A new diagram:
Ωp(X)
Φ
[Y→X]
[Y(C)→X(C)]
−
Let X be a smooth projective complex variety.
HdgMU(X)2p
A new diagram:
Ωp(X)
Φ
[Y→X]
[Y(C)→X(C)]
−
Let X be a smooth projective complex variety.
MUD (p)(X) →2p HdgMU(X)2p
A new diagram:
Ωp(X)
Φ
[Y→X]
[Y(C)→X(C)]
−
Let X be a smooth projective complex variety.
MUD (p)(X) →2p HdgMU(X)2p
combines topol. cobordism with Hodge theoretic information (jt. with M. Hopkins)
A new diagram:
Ωp(X)
Φ
[Y→X]
[Y(C)→X(C)]
−
Let X be a smooth projective complex variety.
MUD (p)(X) →2p HdgMU(X)2p → 0
combines topol. cobordism with Hodge theoretic information (jt. with M. Hopkins)
A new diagram:
Ωp(X)
Φ
[Y→X]
[Y(C)→X(C)]
−
Let X be a smooth projective complex variety.
0 →JMU (X) →2p-1 MUD (p)(X) →2p HdgMU(X)2p → 0
combines topol. cobordism with Hodge theoretic information (jt. with M. Hopkins)
A new diagram:
Ωp(X)
Φ
[Y→X]
[Y(C)→X(C)]
−
Let X be a smooth projective complex variety.
0 →JMU (X) →2p-1 MUD (p)(X) →2p HdgMU(X)2p
complex torus ≈ MU2p-1(X)⊗R/Z
→ 0
combines topol. cobordism with Hodge theoretic information (jt. with M. Hopkins)
A new diagram:
Ωp(X)
Φ
Kernel of Φ ⊂ [Y→X]
[Y(C)→X(C)]
−
Let X be a smooth projective complex variety.
0 →JMU (X) →2p-1 MUD (p)(X) →2p HdgMU(X)2p
complex torus ≈ MU2p-1(X)⊗R/Z
→ 0
combines topol. cobordism with Hodge theoretic information (jt. with M. Hopkins)
A new diagram:
Ωp(X)
Φ
Kernel of Φ ⊂
“Abel-Jacobimap” µMU
[Y→X]
[Y(C)→X(C)]
−
Let X be a smooth projective complex variety.
0 →JMU (X) →2p-1 MUD (p)(X) →2p HdgMU(X)2p
complex torus ≈ MU2p-1(X)⊗R/Z
→ 0
combines topol. cobordism with Hodge theoretic information (jt. with M. Hopkins)
A new diagram:
Ωp(X)
ΦΦD
Kernel of Φ ⊂
“Abel-Jacobimap” µMU
[Y→X]
[Y(C)→X(C)]
−
Let X be a smooth projective complex variety.
0 →JMU (X) →2p-1 MUD (p)(X) →2p HdgMU(X)2p
complex torus ≈ MU2p-1(X)⊗R/Z
→ 0
combines topol. cobordism with Hodge theoretic information (jt. with M. Hopkins)
Examples:
The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes:
Ωp(X)
MU2p(X)CHp(X)
ΦµMU
JMU (X)2p-1
Examples:
The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes:
Ωp(X)
MU2p(X)CHp(X)
∃ α ∊
ΦµMU
JMU (X)2p-1
Examples:
The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes:
Ωp(X)
MU2p(X)CHp(X)
∃ α ∊
0ΦµMU
JMU (X)2p-1
Examples:
The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes:
Ωp(X)
MU2p(X)CHp(X)
∃ α ∊
0 0ΦµMU
JMU (X)2p-1
Examples:
The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes:
Ωp(X)
MU2p(X)CHp(X)
∃ α ∊
0 0≠0ΦµMU
JMU (X)2p-1
The Abel-Jacobi map:
The Abel-Jacobi map:
Given α ∈ Ωp(X), by Rost’s degree formula (due to Levine-Morel)
with Zi⊂X of codim ci≥0, Zi a res. of singularities and 𝛾i ∈ π2p+2 MU and
α = ∑i=1𝛾i[Zi→X] k
~~
ci
The Abel-Jacobi map:
Given α ∈ Ωp(X), by Rost’s degree formula (due to Levine-Morel)
with Zi⊂X of codim ci≥0, Zi a res. of singularities and 𝛾i ∈ π2p+2 MU and
α = ∑i=1𝛾i[Zi→X] k
~~
ci
Assume Φ(α)=0. Then there are chains (modulo torsion)Γi ∈ C2n-2p+1-2ci(X;Z) such that
The Abel-Jacobi map:
Given α ∈ Ωp(X), by Rost’s degree formula (due to Levine-Morel)
with Zi⊂X of codim ci≥0, Zi a res. of singularities and 𝛾i ∈ π2p+2 MU and
α = ∑i=1𝛾i[Zi→X] k
~~
ci
Assume Φ(α)=0. Then there are chains (modulo torsion)Γi ∈ C2n-2p+1-2ci(X;Z) such that
“Image of α = ∂(∑i=1𝛾iΓi) ∈ C2n-2p-2∗(X;Z)⊗π2∗MU”k
The Abel-Jacobi map:
Taking integrals modulo MU2n-2p+1(X) yields a map
n = dim X
Fn-p+1H2n-2p+1(X;π2∗MU⊗C)∨ MU2n-2p+1(X)/Ker(Φ) →
α ⌠⌡Γi
⟼ mod MU2n-2p+1(X).
≈ JMU2p-1(X)
∑i=1 𝛾i k
Thank you!