On topological invariants for algebraic cobordism 27th Nordic - - PowerPoint PPT Presentation

On topological invariants for algebraic cobordism

27th Nordic Congress of Mathematicians, Celebrating the 100th anniversary of Institut Mittag-Leffler Gereon Quick NTNU 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…

𝛽 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.

• For a differential form 𝛽 write 𝛽 ∈ 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 𝛽 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.

• For a differential form 𝛽 write 𝛽 ∈ 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

• 𝛽 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.
• For a differential form 𝛽 write 𝛽 ∈ 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

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

Let 𝜅: 𝛥 ⊂ 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

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

Let 𝜅: 𝛥 ⊂ X be a topological cycle on X of dimension k. We can integrate 𝛽 over 𝛥: ∫ 𝜅*𝛽.

𝛥

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

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

Let 𝜅: 𝛥 ⊂ X be a topological cycle on X of dimension k. We can integrate 𝛽 over 𝛥: ∫ 𝜅*𝛽.

𝛥

If 𝛥 = Z happens to be an algebraic subvariety of X, say of complex dimension n, then ∫ 𝜅*𝛽 = 0 unless 𝛽 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 𝜅: 𝛥 ⊂ 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 𝜅: 𝛥 ⊂ X to be “algebraic” (homologous to an algebraic subvariety Z of dimension n): 𝛥∼Z ⇒ ∫ 𝜅*𝛽 = 0 if 𝛽 ∉ An,n(X).

𝛥

Hodge’ s question: This imposes a necessary condition on a topological cycle 𝜅: 𝛥 ⊂ X to be “algebraic” (homologous to an algebraic subvariety Z of dimension n): 𝛥∼Z ⇒ ∫ 𝜅*𝛽 = 0 if 𝛽 ∉ An,n(X).

𝛥

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

Hodge’ s question: This imposes a necessary condition on a topological cycle 𝜅: 𝛥 ⊂ X to be “algebraic” (homologous to an algebraic subvariety Z of dimension n): 𝛥∼Z ⇒ ∫ 𝜅*𝛽 = 0 if 𝛽 ∉ An,n(X).

𝛥

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 𝜅: 𝛥 ⊂ X to be “algebraic” (homologous to an algebraic subvariety Z of dimension n): 𝛥∼Z ⇒ ∫ 𝜅*𝛽 = 0 if 𝛽 ∉ An,n(X).

𝛥

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 𝜅: 𝛥 ⊂ X to be “algebraic” (homologous to an algebraic subvariety Z of dimension n): 𝛥∼Z ⇒ ∫ 𝜅*𝛽 = 0 if 𝛽 ∉ An,n(X).

𝛥

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 𝜅: 𝛥 ⊂ X to be “algebraic” (homologous to an algebraic subvariety Z of dimension n): 𝛥∼Z ⇒ ∫ 𝜅*𝛽 = 0 if 𝛽 ∉ An,n(X).

𝛥

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 curve Let C be a smooth proj. complex curve of genus g.

A short digression: the Jacobian of a curve Let 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 curve Let C be a smooth proj. complex curve of genus g. Let ω1, …, ωg be a basis of (holom.) 1-forms on C. ⌠ ⌡q ω1

p

⌠ ⌡q ωg

p

⎛ ⎝ ⎞ ⎠ , ..., Then every pair of points p,q∈C defines a g-tuple of complex numbers

A short digression: the Jacobian of a curve Let 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 ⌠ ⌡q ω1

p

⌠ ⌡q ωg

p

⎛ ⎝ ⎞ ⎠ , ..., Then every pair of points p,q∈C defines a g-tuple of complex numbers

A short digression: the Jacobian of a curve Let 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 ⌠ ⌡q ω1

p

⌠ ⌡q ωg

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 curve Let 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 ⌠ ⌡q ω1

p

⌠ ⌡q ωg

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

• f ωj’

s over loops

/Λ

A short digression: the Jacobian of a curve Let 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 ⌠ ⌡q ω1

p

⌠ ⌡q ωg

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

• f ωj’

s over loops

/Λ Jacobian variety of C =: J(C)

A short digression: the Jacobian of a curve Let 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 ⌠ ⌡q ω1

p

⌠ ⌡q ωg

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

• f 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” 𝜉 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 𝜉D(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 𝜉D: t ↦ 𝜉D(t) ∈ J. Choose a point p0 on all Ct. Then ∑i pi(t) - dp0 is a divisor of degree 0 and defines a point 𝜉D(t) ∈ J(Ct).

Poincaré’ s Existence Theorem:

Every normal function 𝜉 arises as the normal function 𝜉D associated to an algebraic curve D. Poincaré’ s Existence Theorem:

Every normal function 𝜉 arises as the normal function 𝜉D associated to an algebraic curve D. Poincaré’ s Existence Theorem: Then Lefschetz proved:

Every normal function 𝜉 arises as the normal function 𝜉D associated to an algebraic curve D. Poincaré’ s Existence Theorem:

• Every normal function 𝜉 defines a class

𝜃(𝜉)∈H2(X;Z) of Hodge type (1,1) such that 𝜃(𝜉D) = clH(D). Then Lefschetz proved:

Every normal function 𝜉 arises as the normal function 𝜉D associated to an algebraic curve D. Poincaré’ s Existence Theorem:

• Every normal function 𝜉 defines a class

𝜃(𝜉)∈H2(X;Z) of Hodge type (1,1) such that 𝜃(𝜉D) = clH(D).

• Every class in H2(X;Z) of Hodge type (1,1) arises

as 𝜃(𝜉) for some normal function 𝜉. 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 ⌠ ⌡Γ ω

|

→ ∈ 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 ⌠ ⌡Γ ω

|

→ ∈ 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⊂X Let X be a smooth projective complex variety.

An interesting diagram: Zp(X) clH Z⊂X Let X be a smooth projective complex variety.

An interesting diagram: Hdg2p(X) Zp(X) clH Z⊂X Let 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-Jacobi map 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-Jacobi map 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-Jacobi map 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-Jacobi map 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-Jacobi map 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-Jacobi map 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) clH clHD Zp(X)h=Kernel of clH ⊂ Abel-Jacobi map 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 cobordism

• f Levine and Morel

Φ: Ω*(X) → MU *(X)

2

Another interesting map for smooth complex varieties: algebraic cobordism

• f Levine and Morel

complex cobordism of the top. space X(C)

Φ: Ω*(X) → MU *(X)

2

Another interesting map for smooth complex varieties: algebraic cobordism

• f Levine and Morel

complex cobordism of the 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 cobordism

• f Levine and Morel

complex cobordism of the 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=1

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=1

X

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=1

X

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=1

X

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=1

X

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).

X X

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) Φ ΦD [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) Φ ΦD 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) Φ ΦD Ωp(X)top:=Kernel of Φ ⊂ “Abel-Jacobi map” 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(Xx∆•;V*) K(FpA *(X;V*),n) Given n, p

The Abel-Jacobi map: Sing•MUn(X) Zn(Xx∆•;V*) K(FpA *(X;V*),n) simpl.map. space Given n, p

The Abel-Jacobi map: Sing•MUn(X) Zn(Xx∆•;V*) K(FpA *(X;V*),n) cocycles simpl.map. space Given n, p

The Abel-Jacobi map: Sing•MUn(X) Zn(Xx∆•;V*) K(FpA *(X;V*),n) V*:=MU*⊗C cocycles simpl.map. space Given n, p

The Abel-Jacobi map: Sing•MUn(X) Zn(Xx∆•;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(Xx∆•;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(Xx∆•;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(Xx∆•;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(Xx∆•;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(Xx∆•;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(Xx∆•;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(Xx∆•;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(Xx∆•;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(Xx∆•;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) Φ µMU 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 ∊ Φ µMU 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 ∊ Φ µMU 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 ∊ Φ µMU 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 Φ µMU JMU (X)

2p-1

Thank you!