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

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

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

Poincaré’ s Existence Theorem: Every normal function 𝜉 arises as the normal function 𝜉 D associated to an algebraic curve D. Then Lefschetz proved: • Every normal function 𝜉 defines a class 𝜃 ( 𝜉 ) ∈ H 2 (X;Z) of Hodge type (1,1) such that 𝜃 ( 𝜉 D ) = cl H (D).

Poincaré’ s Existence Theorem: Every normal function 𝜉 arises as the normal function 𝜉 D associated to an algebraic curve D. Then Lefschetz proved: • Every normal function 𝜉 defines a class 𝜃 ( 𝜉 ) ∈ H 2 (X;Z) of Hodge type (1,1) such that 𝜃 ( 𝜉 D ) = cl H (D). • Every class in H 2 (X;Z) of Hodge type (1,1) arises as 𝜃 ( 𝜉 ) for some normal function 𝜉 .

Griffiths: Higher dimensions X a smooth projective complex variety with dimX=n.

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 . ⌠ → ω ω Then ∈ F n-p+1 H 2n-2p+1 (X;C) ∨ . | ⌡ 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 . ⌠ → ω ω Then ∈ F n-p+1 H 2n-2p+1 (X;C) ∨ . | ⌡ Γ But the value depends on the choice of Γ .

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 ⌠ Z for some Γ with Z= ∂ Γ ⟼ ⌡ Γ µ: Z p (X) h → F n-p+1 H 2n-2p+1 (X;C) ∨ /H 2n-2p+1 (X;Z)

The intermediate Jacobian of Griffiths and the Abel-Jacobi map: We obtain a well-defined map ⌠ Z for some Γ with Z= ∂ Γ ⟼ ⌡ Γ µ: Z p (X) h → F n-p+1 H 2n-2p+1 (X;C) ∨ /H 2n-2p+1 (X;Z) ≈ H 2p-1 (X;Z) ⊗ R/Z

The intermediate Jacobian of Griffiths and the Abel-Jacobi map: We obtain a well-defined map ⌠ Z for some Γ with Z= ∂ Γ ⟼ ⌡ Γ µ: Z p (X) h → F n-p+1 H 2n-2p+1 (X;C) ∨ /H 2n-2p+1 (X;Z) ≈ H 2p-1 (X;Z) ⊗ R/Z = J 2p-1 (X)

The intermediate Jacobian of Griffiths and the Abel-Jacobi map: We obtain a well-defined map ⌠ Z for some Γ with Z= ∂ Γ ⟼ ⌡ Γ µ: Z p (X) h → F n-p+1 H 2n-2p+1 (X;C) ∨ /H 2n-2p+1 (X;Z) ≈ H 2p-1 (X;Z) ⊗ R/Z = J 2p-1 (X) J 2p-1 (X) is a complex torus and is called Griffiths’ intermediate Jacobian.

The Jacobian and Griffiths’ theorem:

The Jacobian and Griffiths’ theorem: J 2p-1 (X) is, in general, not an abelian variety.

The Jacobian and Griffiths’ theorem: J 2p-1 (X) is, in general, not an abelian variety. But it varies homomorphically in families.

The Jacobian and Griffiths’ theorem: J 2p-1 (X) is, in general, not an abelian variety. But it varies homomorphically in families. Have an induced a map: Griff p (X):= Z p (X) h /Z p (X) alg → J 2p-1 (X)/J 2p-1 (X) alg

The Jacobian and Griffiths’ theorem: J 2p-1 (X) is, in general, not an abelian variety. But it varies homomorphically in families. Have an induced a map: Griff p (X):= Z p (X) h /Z p (X) alg → J 2p-1 (X)/J 2p-1 (X) alg Griffith’ s theorem: Let X ⊂ P 4 be a general quintic hypersurface. There are lines L and L ’ on X such that µ(L-L ’) is a non torsion element in J 3 (X).

An interesting diagram: Let X be a smooth projective complex variety.

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

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

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) Z ⊂ X cl H

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) Z ⊂ X cl H Hdg 2p (X)

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) Z ⊂ X − cl H [Z sm ] Hdg 2p (X)

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) h =Kernel of cl H ⊂ Z p (X) Z ⊂ X − cl H [Z sm ] Hdg 2p (X)

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) h =Kernel of cl H ⊂ Z p (X) Z ⊂ X − Abel-Jacobi cl H map µ [Z sm ] Hdg 2p (X)

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) h =Kernel of cl H ⊂ Z p (X) Z ⊂ X − Abel-Jacobi cl H map µ [Z sm ] J 2p-1 (X) Hdg 2p (X)

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) h =Kernel of cl H ⊂ Z p (X) Z ⊂ X − Abel-Jacobi cl H map µ [Z sm ] 2p J 2p-1 (X) → H D (X;Z(p)) → Hdg 2p (X)

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) h =Kernel of cl H ⊂ Z p (X) Z ⊂ X − Abel-Jacobi cl H map µ [Z sm ] 2p J 2p-1 (X) → H D (X;Z(p)) → Hdg 2p (X) Deligne cohomology combines topological with Hodge theoretic information

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) h =Kernel of cl H ⊂ Z p (X) Z ⊂ X − Abel-Jacobi cl H map µ [Z sm ] 2p J 2p-1 (X) → H D (X;Z(p)) → Hdg 2p (X) → 0 Deligne cohomology combines topological with Hodge theoretic information

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) h =Kernel of cl H ⊂ Z p (X) Z ⊂ X − Abel-Jacobi cl H map µ [Z sm ] 2p J 2p-1 (X) → H D (X;Z(p)) → Hdg 2p (X) → 0 0 → Deligne cohomology combines topological with Hodge theoretic information

An interesting diagram: Let X be a smooth projective complex variety. Z p (X) h =Kernel of cl H ⊂ Z p (X) Z ⊂ X − Abel-Jacobi cl HD cl H map µ [Z sm ] 2p J 2p-1 (X) → H D (X;Z(p)) → Hdg 2p (X) → 0 0 → Deligne cohomology combines topological with Hodge theoretic information

Another interesting map for smooth complex varieties:

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 algebraic cobordism of Levine and Morel

Another interesting map for smooth complex varieties: Φ : Ω *(X) → MU *(X) 2 algebraic cobordism complex cobordism of of Levine and Morel the top. space X(C)

Another interesting map for smooth complex varieties: Φ : Ω *(X) → MU *(X) 2 algebraic cobordism complex cobordism of of Levine and Morel the top. space X(C) Ω p (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”:

Another interesting map for smooth complex varieties: Φ : Ω *(X) → MU *(X) 2 algebraic cobordism complex cobordism of of Levine and Morel the top. space X(C) Ω p (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”: π -1 (0) ∼ π -1 ( ∞ ) for projective morphisms π : Y’ → XxP 1 such that π -1 (0) is smooth and π -1 ( ∞ )=A ∪ D B w here A and B are smooth and meet transversally in D.

What can we say about the map Φ ? Φ MU 2 *(X) Ω *(X)

What can we say about the map Φ ? [Y → X] Φ MU 2 *(X) Ω *(X)

What can we say about the map Φ ? [Y → X] [Y(C) → X(C)] ⟼ Φ MU 2 *(X) Ω *(X)

What can we say about the map Φ ? [Y → X] [Y(C) → X(C)] ⟼ Φ MU 2 *(X) • The image: Ω *(X)

What can we say about the map Φ ? [Y → X] [Y(C) → X(C)] ⟼ Φ MU 2 *(X) • The image: Ω *(X) Z*(X)/ rat.eq = CH*(X)

What can we say about the map Φ ? [Y → X] [Y(C) → X(C)] ⟼ Φ MU 2 *(X) • The image: Ω *(X) Z*(X)/ rat.eq = CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

What can we say about the map Φ ? [Y → X] [Y(C) → X(C)] ⟼ Φ MU 2 *(X) • The image: Ω *(X) Z*(X)/ rat.eq = CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

What can we say about the map Φ ? [Y → X] [Y(C) → X(C)] ⟼ Φ MU 2 *(X) • The image: Ω *(X) Z*(X)/ rat.eq = CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

What can we say about the map Φ ? [Y → X] [Y(C) → X(C)] ⟼ Φ MU 2 *(X) • The image: Ω *(X) Z*(X)/ rat.eq = CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H There is a “Hodge-theoretic” restriction for Im Φ .

What can we say about the map Φ ? [Y → X] [Y(C) → X(C)] ⟼ Φ MU 2 *(X) • The image: Ω *(X) Z*(X)/ rat.eq = CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H There is a “Hodge-theoretic” restriction for Im Φ . • The kernel:

What can we say about the map Φ ? [Y → X] [Y(C) → X(C)] ⟼ Φ MU 2 *(X) • The image: Ω *(X) Z*(X)/ rat.eq = CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H There is a “Hodge-theoretic” restriction for Im Φ . • The kernel: Griffiths’ theorem suggests that Φ is not injective.

What can we say about the map Φ ? [Y → X] [Y(C) → X(C)] ⟼ Φ MU 2 *(X) • The image: Ω *(X) Z*(X)/ rat.eq = CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H There is a “Hodge-theoretic” restriction for Im Φ . • The kernel: Griffiths’ theorem suggests that Φ is not injective. Question: Is there is an “Abel-Jacobi-invariant” which is able to detect elements in Ker Φ ?

The image: Φ MU 2 *(X) Ω *(X)

The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X)

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X)

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z CH*(X)

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z Totaro CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z Totaro Levine-Morel ≈ CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z ≉ in general Totaro Levine-Morel ≈ CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z ≉ in general Totaro Levine-Morel ≈ CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H Atiyah-Hirzebruch: cl H is not surjective.

not surjective, but … The image: Hdg MU2 *(X) ∩ Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z ≉ in general Totaro Levine-Morel ≈ CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H Atiyah-Hirzebruch: cl H is not surjective. This argument does not work for Φ .

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

Kollar’ s examples: (see also Soulé-Voisin et. al.) Let X ⊂ P 4 a very general hypersurface of degree d=p 3 for a prime p ≥ 5.