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
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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!
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