On topological invariants for algebraic cobordism27th Nordic Congress of Mathematicians,

Celebrating the 100th anniversary of Institut Mittag-Leffler

Gereon QuickNTNU

joint work with Michael J. Hopkins

Point of departure: Poincaré, Lefschetz, Hodge…

Let X⊂PN be a smooth projective complex variety. Point of departure: Poincaré, Lefschetz, Hodge…

𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -For a differential form 𝛼 write 𝛼 ∈ Ap,q(X) if

Let X⊂PN be a smooth projective complex variety. Point of departure: Poincaré, Lefschetz, Hodge…

exactly p dzj’s for all j

𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -For a differential form 𝛼 write 𝛼 ∈ Ap,q(X) if

Let X⊂PN be a smooth projective complex variety. Point of departure: Poincaré, Lefschetz, Hodge…

exactly p dzj’s for all j exactly q dzj’s for all j-

𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -For a differential form 𝛼 write 𝛼 ∈ Ap,q(X) if

Let X⊂PN be a smooth projective complex variety. Point of departure: Poincaré, Lefschetz, Hodge…

exactly p dzj’s for all j exactly q dzj’s for all j-

𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -For a differential form 𝛼 write 𝛼 ∈ Ap,q(X) if

Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k.

Let X⊂PN be a smooth projective complex variety. Point of departure: Poincaré, Lefschetz, Hodge…

exactly p dzj’s for all j exactly q dzj’s for all j-

𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -For a differential form 𝛼 write 𝛼 ∈ Ap,q(X) if

Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k. We can integrate 𝛼 over 𝛤: ∫ 𝜄*𝛼. 𝛤

Let X⊂PN be a smooth projective complex variety. Point of departure: Poincaré, Lefschetz, Hodge…

exactly p dzj’s for all j exactly q dzj’s for all j-

𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -For a differential form 𝛼 write 𝛼 ∈ Ap,q(X) if

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

Let X⊂PN be a smooth projective complex variety. Point of departure: Poincaré, Lefschetz, Hodge…

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 of 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 of 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 of 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 of dimension n):

𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤

Hodge wondered: Is this condition also sufficient? ? ⇐ ?

clH: Zp(X) → H2p(X;Z)

Hodge’s question:This imposes a necessary condition on a topological cycle 𝜄: 𝛤 ⊂ X to be “algebraic” (homologous to an algebraic subvariety Z of dimension n):

𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤

Hodge wondered: Is this condition also sufficient? ? ⇐ ?

clH: Zp(X) → H2p(X;Z)Hp,p(X) ∩

Hodge’s question:This imposes a necessary condition on a topological cycle 𝜄: 𝛤 ⊂ X to be “algebraic” (homologous to an algebraic subvariety Z of dimension n):

𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤

Hodge wondered: Is this condition also sufficient? ? ⇐ ?

clH: Zp(X) → H2p(X;Z)

The Hodge Conjecture: The map

is surjective.Hp,p(X) ∩

Hodge’s question:This imposes a necessary condition on a topological cycle 𝜄: 𝛤 ⊂ X to be “algebraic” (homologous to an algebraic subvariety Z of dimension n):

𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤

Hodge wondered: Is this condition also sufficient? ? ⇐ ?

clH: Zp(X) → H2p(X;Z)⊗Q

switch to Q-coefficients

×The Hodge Conjecture: The map

is surjective.Hp,p(X) ∩

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 (Abel-Jacobi) map

is surjective.

lattice of integrals of ωj’s over loops

/Λ

Jacobian variety of C

=: J(C)

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:

Griffiths: Higher dimensions

X a smooth projective complex variety with dimX=n.

Z⊂X a subvariety of codimension p which is the boundary of a differentiable chain Γ.

Griffiths: Higher dimensions

X a smooth projective complex variety with dimX=n.

Z⊂X a subvariety of codimension p which is the boundary of a differentiable chain Γ.

Then ⌠⌡Γω|→ ∈ Fn-p+1H2n-2p+1(X;C)∨.ω⎛⎝⎞⎠

Griffiths: Higher dimensions

X a smooth projective complex variety with dimX=n.

Z⊂X a subvariety of codimension p which is the boundary of a differentiable chain Γ.

Then ⌠⌡Γω|→ ∈ Fn-p+1H2n-2p+1(X;C)∨.ω⎛⎝⎞⎠

Griffiths: Higher dimensions

But the value depends on the choice of Γ.

X a smooth projective complex variety with dimX=n.

The intermediate Jacobian of Griffiths and the Abel-Jacobi map:

The intermediate Jacobian of Griffiths and the Abel-Jacobi map:

We obtain a well-defined map⌠⌡Γ

⟼Z

µ: Zp(X)h → 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:

We obtain a well-defined map⌠⌡Γ

⟼Z

µ: Zp(X)h → Fn-p+1H2n-2p+1(X;C)∨/H2n-2p+1(X;Z)

for some Γ with Z=∂Γ

≈ H2p-1(X;Z)⊗R/Z

The intermediate Jacobian of Griffiths and the Abel-Jacobi map:

We obtain a well-defined map⌠⌡Γ

⟼Z

µ: Zp(X)h → Fn-p+1H2n-2p+1(X;C)∨/H2n-2p+1(X;Z)

for some Γ with Z=∂Γ

≈ H2p-1(X;Z)⊗R/Z

= J2p-1(X)

The intermediate Jacobian of Griffiths and the Abel-Jacobi map:

We obtain a well-defined map⌠⌡Γ

⟼Z

µ: Zp(X)h → 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.

≈ H2p-1(X;Z)⊗R/Z

= J2p-1(X)

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)

Ωp(X) is generated by projective maps f:Y→X of codimension p with Y smooth variety modulo Levine’s and Pandharipande’s “double point relation”:

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)

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.

Ωp(X) is generated by projective maps f:Y→X of codimension p with Y smooth variety modulo Levine’s and Pandharipande’s “double point relation”:

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:

Z*(X)/rat.eq = CH*(X)

Ω*(X) MU2*(X)Φ[Y→X] [Y(C)→X(C)]⟼

What can we say about the map Φ?

• The image:

Z*(X)/rat.eq = 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:

Z*(X)/rat.eq = 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:

Z*(X)/rat.eq = 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:

Z*(X)/rat.eq = 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:

Z*(X)/rat.eq = 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:

Z*(X)/rat.eq = 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:

Z*(X)/rat.eq = 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)]⟼

Ω*(X) MU2*(X)Φ

The image:

Ω*(X) MU2*(X)Φ

The image:HdgMU2*(X)

∩

Ω*(X) MU2*(X)Φ

The image: not surjective, but … HdgMU2*(X)

∩

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ

The image: not surjective, but … HdgMU2*(X)

∩

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ

CH*(X)

The image: not surjective, but … HdgMU2*(X)

∩

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ

CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH

The image: not surjective, but … HdgMU2*(X)

∩

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ

CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH

The image: not surjective, but … HdgMU2*(X)

∩

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ

CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH

The image: not surjective, but … HdgMU2*(X)

∩

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ

CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH

Totaro

The image: not surjective, but … HdgMU2*(X)

∩

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ

CH*(X)

Levine-Morel ≈

Hdg2*(X) ⊆ H2*(X;Z)clH

Totaro

The image: not surjective, but … HdgMU2*(X)

∩

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ

CH*(X)

Levine-Morel ≈ ≉ in general

Hdg2*(X) ⊆ H2*(X;Z)clH

Totaro

The image: not surjective, but … HdgMU2*(X)

∩

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ

CH*(X)

Levine-Morel ≈ ≉ in general

Atiyah-Hirzebruch: clH is not surjective.

Hdg2*(X) ⊆ H2*(X;Z)clH

Totaro

The image: not surjective, but … HdgMU2*(X)

∩

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ

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

The image: not surjective, but … HdgMU2*(X)

∩

Kollar’s examples: (see also Soulé-Voisin et. al.)

Kollar’s examples: (see also Soulé-Voisin et. al.)Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5.

Kollar’s examples: (see also Soulé-Voisin et. al.)Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Z∙h, H4(X;Z)=Z∙α, ∫ α∙h=1X

Kollar’s examples: (see also Soulé-Voisin et. al.)Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Z∙h, H4(X;Z)=Z∙α, ∫ α∙h=1X

both torsion-free and all classes are Hodge classes

Kollar’s examples: (see also Soulé-Voisin et. al.)Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Z∙h, H4(X;Z)=Z∙α, ∫ α∙h=1X

both torsion-free and all classes are Hodge classes

Kollar: p divides the degree of any curve on X.

Kollar’s examples: (see also Soulé-Voisin et. al.)Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Z∙h, H4(X;Z)=Z∙α, ∫ α∙h=1X

both torsion-free and all classes are Hodge classes

Kollar: p divides the degree of any curve on X.

This implies: α is not algebraic (since we needed a curve of degree 1).

Kollar’s examples: (see also Soulé-Voisin et. al.)Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Z∙h, H4(X;Z)=Z∙α, ∫ α∙h=1X

both torsion-free and all classes are Hodge classes

Kollar: p divides the degree of any curve on X.

But dα is algebraic (for ∫ dα∙h = d = ∫ h2∙h ⇒ dα=h2). XX

This implies: α is not algebraic (since we needed a curve of degree 1).

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.

A new diagram:

Let X be any smooth projective complex variety.

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

Let X be any smooth projective complex variety.

(joint work with Mike Hopkins)

A new diagram:

Ωp(X) [Y→X]

Let X be any smooth projective complex variety.

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

Φ

[Y→X]

Let X be any smooth projective complex variety.

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

Φ

[Y→X]

Let X be any smooth projective complex variety.

HdgMU(X)2p

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−

Let X be any smooth projective complex variety.

HdgMU(X)2p

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−

Let X be any smooth projective complex variety.

MUD (p)(X) →2p HdgMU(X)2p

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−

Let X be any smooth projective complex variety.

MUD (p)(X) →2p HdgMU(X)2p

combines topol. cobordism with Hodge theoretic information

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−

Let X be any smooth projective complex variety.

MUD (p)(X) →2p HdgMU(X)2p → 0

combines topol. cobordism with Hodge theoretic information

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−

Let X be any smooth projective complex variety.

0 →JMU (X) →2p-1 MUD (p)(X) →2p HdgMU(X)2p → 0

combines topol. cobordism with Hodge theoretic information

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−

Let X be any 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

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

ΦΦD

[Y→X]

[Y(C)→X(C)]

−

Let X be any 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

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

ΦΦD

Ωp(X)top:=Kernel of Φ ⊂ [Y→X]

[Y(C)→X(C)]

−

Let X be any 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

(joint work with Mike Hopkins)

A new diagram:

Ωp(X)

ΦΦD

Ωp(X)top:=Kernel of Φ ⊂

“Abel-Jacobimap” µMU

[Y→X]

[Y(C)→X(C)]

−

Let X be any 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

(joint work with Mike Hopkins)

The Abel-Jacobi map:

The Abel-Jacobi map: Given n, p

The Abel-Jacobi map:

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

Given n, p

The Abel-Jacobi map:

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

simpl.map. space

Given n, p

The Abel-Jacobi map:

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n) cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

V*:=MU*⊗C

cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

V*:=MU*⊗C

EM-space cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

MUn(p)(X)htpy. cart.

V*:=MU*⊗C

EM-space cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

MUn(p)(X)htpy. cart.

π0

V*:=MU*⊗C

EM-space cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

MUn(p)(X)htpy. cart.

π0

V*:=MU*⊗C

Elements in MUDn(p)(X) consist of (f, h, ω):

EM-space cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

• f : X → MUn Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

MUn(p)(X)htpy. cart.

π0

V*:=MU*⊗C

Elements in MUDn(p)(X) consist of (f, h, ω):

EM-space cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

• f : X → MUn • ω ∈ FpAn(X;V*)

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

MUn(p)(X)htpy. cart.

π0

V*:=MU*⊗C

Elements in MUDn(p)(X) consist of (f, h, ω):

EM-space cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

• f : X → MUn • ω ∈ FpAn(X;V*) • h ∈ Cn-1(X;V*) such that “∂h = f-ω”

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

MUn(p)(X)htpy. cart.

π0

V*:=MU*⊗C

Elements in MUDn(p)(X) consist of (f, h, ω):

EM-space cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

• f : X → MUn • ω ∈ FpAn(X;V*) • h ∈ Cn-1(X;V*) such that “∂h = f-ω”

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

MUn(p)(X)htpy. cart.

π0

If n=2p, [f]=0 and [ω]=0, then (f, h, ω) defines an element in MU2p-1(X)⊗R, uniquely modulo MU2p-1(X).

V*:=MU*⊗C

Elements in MUDn(p)(X) consist of (f, h, ω):

EM-space cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

• f : X → MUn • ω ∈ FpAn(X;V*) • h ∈ Cn-1(X;V*) such that “∂h = f-ω”

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

MUn(p)(X)htpy. cart.

π0

If n=2p, [f]=0 and [ω]=0, then (f, h, ω) defines an element in MU2p-1(X)⊗R, uniquely modulo MU2p-1(X).This gives the Abel-Jacobi map Ωp(X)top→ MU2p-1(X)⊗R/Z

V*:=MU*⊗C

Elements in MUDn(p)(X) consist of (f, h, ω):

EM-space cocycles

simpl.map. space

Given n, p

The Abel-Jacobi map:

• f : X → MUn • ω ∈ FpAn(X;V*) • h ∈ Cn-1(X;V*) such that “∂h = f-ω”

Sing•MUn(X)

Zn(Xxƥ;V*)K(FpA*(X;V*),n)

MUn(p)(X)htpy. cart.

π0

If n=2p, [f]=0 and [ω]=0, then (f, h, ω) defines an element in MU2p-1(X)⊗R, uniquely modulo MU2p-1(X).This gives the Abel-Jacobi map Ωp(X)top→ MU2p-1(X)⊗R/Z

(get functional via Kronecker pairing)

V*:=MU*⊗C

Elements in MUDn(p)(X) consist of (f, h, ω):

EM-space cocycles

simpl.map. space

Given n, p

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

Thank you!