Generalized Deligne- Beilinson cohomology and regulators Regulators - PowerPoint PPT Presentation

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X)

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X) H 2 *(X;Z)

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X) CH*(X) H 2 *(X;Z)

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X) CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ MU* Z CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ MU* Z CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ MU* Z CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ MU* Z Totaro CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ MU* Z Totaro Levine-Morel ≈ CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ MU* Z ≉ in general Totaro Levine-Morel ≈ CH*(X) Hdg 2 *(X) ⊆ H 2 *(X;Z) cl H

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

Atiyah-Hirzebruch and Totaro: Φ MU 2 *(X) Ω *(X) Ω *(X) ⊗ L* Z MU 2 *(X) ⊗ MU* 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:

Kollar’ s examples: Let X ⊂ P 4 a very general hypersurface of degree d=p 3 for a prime p ≥ 5.

Kollar’ s examples: H 2 (X;Z)=Zh, Let X ⊂ P 4 a very general hypersurface of degree d=p 3 for a prime p ≥ 5.

H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: H 2 (X;Z)=Zh, X Let X ⊂ P 4 a very general hypersurface of degree d=p 3 for a prime p ≥ 5.

H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: H 2 (X;Z)=Zh, X and all classes are Hodge classes. Let X ⊂ P 4 a very general hypersurface of degree d=p 3 for a prime p ≥ 5.

H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: H 2 (X;Z)=Zh, X and all classes are Hodge classes. Let X ⊂ P 4 a very general hypersurface of degree d=p 3 for a prime p ≥ 5. Kollar: Then any algebraic curve C on X has degree divisible by p.

H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: H 2 (X;Z)=Zh, X and all classes are Hodge classes. Let X ⊂ P 4 a very general hypersurface of degree d=p 3 for a prime p ≥ 5. Kollar: Then any algebraic curve C on X has degree divisible by p. Let C be a curve on X and [C] ∈ H 4 (X;Z) be its cohomology class.

H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: H 2 (X;Z)=Zh, X and all classes are Hodge classes. Let X ⊂ P 4 a very general hypersurface of degree d=p 3 for a prime p ≥ 5. Kollar: Then any algebraic curve C on X has degree divisible by p. Let C be a curve on X and [C] ∈ H 4 (X;Z) be its cohomology class. Then [C]= n α for some n and ∫ [C] ∙ h=n. X

H 2 (X;Z)=Zh, H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: X Kollar: p divides the degree of any curve on X. n = ∫ [C] ∙ h X

H 2 (X;Z)=Zh, H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: X Kollar: p divides the degree of any curve on X. n = ∫ [C] ∙ h X = number of intersection points of C with h

H 2 (X;Z)=Zh, H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: X Kollar: p divides the degree of any curve on X. n = ∫ [C] ∙ h X = number of intersection points of C with h = degree of C

H 2 (X;Z)=Zh, H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: X Kollar: p divides the degree of any curve on X. n = ∫ [C] ∙ h X = number of intersection points of C with h = degree of C ⇒ p divides n

H 2 (X;Z)=Zh, H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: X Kollar: p divides the degree of any curve on X. n = ∫ [C] ∙ h X = number of intersection points of C with h = degree of C ⇒ p divides n In particular, α is not algebraic (for n cannot be 1).

H 2 (X;Z)=Zh, H 4 (X;Z)=Z α , ∫ α∙ h=1 Kollar’ s examples: X Kollar: p divides the degree of any curve on X. n = ∫ [C] ∙ h X = number of intersection points of C with h = degree of C ⇒ p divides n In particular, α is not algebraic (for n cannot be 1). But d α is algebraic (for ∫ d α∙ h = d = ∫ h 2 ∙ h ⇒ d α =h 2 ). X X

Consequence for Φ : Ω *(X) → MU 2 *(X):

Consequence for Φ : Ω *(X) → MU 2 *(X): Let X ⊂ P 4 be a very general hypersurface as above.

Consequence for Φ : Ω *(X) → MU 2 *(X): Let X ⊂ P 4 be a very general hypersurface as above. Then MU 4 (X) ↠ H 4 (X;Z) is surjective, s argument implies that Φ is not and thus Kollar’ surjective (on Hodge classes).

Consequence for Φ : Ω *(X) → MU 2 *(X): Let X ⊂ P 4 be a very general hypersurface as above. Then MU 4 (X) ↠ H 4 (X;Z) is surjective, s argument implies that Φ is not and thus Kollar’ surjective (on Hodge classes). These examples are “not topological”: there is a dense subset of hypersurfaces Y ⊂ P 4 such that the generator in H 4 (Y;Z) is algebraic.

Generalized Hodge filtered cohomology theories (joint work with Michael J. Hopkins): Recall the diagram for Chow groups Kernel of cl H ⊂ CH p (X) cl HD cl H 0 → J 2p-1 (X) → H D2p (X;Z(p)) → Hdg 2p (X) → 0

Generalized Hodge filtered cohomology theories (joint work with Michael J. Hopkins): Recall the diagram for Chow groups Kernel of cl H ⊂ CH p (X) cl HD cl H 0 → J 2p-1 (X) → H D2p (X;Z(p)) → Hdg 2p (X) → 0 Our goal: define new invariants for algebraic cobordism which combine

Generalized Hodge filtered cohomology theories (joint work with Michael J. Hopkins): Recall the diagram for Chow groups Kernel of cl H ⊂ CH p (X) cl HD cl H 0 → J 2p-1 (X) → H D2p (X;Z(p)) → Hdg 2p (X) → 0 Our goal: define new invariants for algebraic cobordism which combine • Hodge theoretical information and

Generalized Hodge filtered cohomology theories (joint work with Michael J. Hopkins): Recall the diagram for Chow groups Kernel of cl H ⊂ CH p (X) cl HD cl H 0 → J 2p-1 (X) → H D2p (X;Z(p)) → Hdg 2p (X) → 0 Our goal: define new invariants for algebraic cobordism which combine • Hodge theoretical information and • topological information of complex cobordism.

Deligne cohomology:

Deligne cohomology: Given an integer p ≥ 0.

Deligne cohomology: Given an integer p ≥ 0. The Deligne complex of sheaves Z D (p) on the complex manifold X is defined by

Deligne cohomology: Given an integer p ≥ 0. The Deligne complex of sheaves Z D (p) on the complex manifold X is defined by a homotopy cartesian square of sheaves of complexes.

Deligne cohomology: Given an integer p ≥ 0. The Deligne complex of sheaves Z D (p) on the complex manifold X is defined by a homotopy cartesian square of sheaves of complexes. A * sheaf of hol. forms on X

Deligne cohomology: Given an integer p ≥ 0. The Deligne complex of sheaves Z D (p) on the complex manifold X is defined by a homotopy cartesian square of sheaves of complexes. A * F p A * sheaf of hol. forms on X

Deligne cohomology: Given an integer p ≥ 0. The Deligne complex of sheaves Z D (p) on the complex manifold X is defined by a homotopy cartesian Z square of sheaves of complexes. A * F p A * sheaf of hol. forms on X

Deligne cohomology: Given an integer p ≥ 0. The Deligne complex of sheaves Z D (p) on the complex manifold X is defined by a homotopy cartesian Z Z D (p) square of sheaves of complexes. htpy cart. A * F p A * sheaf of hol. forms on X

Deligne cohomology: Given an integer p ≥ 0. The Deligne complex of sheaves Z D (p) on the complex manifold X is defined by a homotopy cartesian Z Z D (p) square of sheaves of complexes. htpy cart. A * F p A * sheaf of hol. forms on X Deligne cohomology is the hypercohomology of this n complex: H D (X;Z(p)) = H n (X;Z D (p)).

A homotopy cartesian square of sheaves of complexes Z Z D (p) A * F p A *

A homotopy cartesian square of pre sheaves of complexes Z Z D (p) A * F p A *

A homotopy cartesian square of pre sheaves of Z Z D (p) A * F p A *

A homotopy cartesian square of pre sheaves of spectra on the site of complex manifolds Z Z D (p) A * F p A *

A homotopy cartesian square of pre sheaves of spectra on the site of complex manifolds Z A * F p A *

A homotopy cartesian square of pre sheaves of spectra on the site of complex manifolds H=Eilenberg-MacLane Z spectrum functor for complexes A * F p A *

A homotopy cartesian square of pre sheaves of spectra on the site of complex manifolds H=Eilenberg-MacLane H Z spectrum functor for complexes A * F p A *

A homotopy cartesian square of pre sheaves of spectra on the site of complex manifolds H=Eilenberg-MacLane H Z spectrum functor for complexes H A * F p A *

A homotopy cartesian square of pre sheaves of spectra on the site of complex manifolds H=Eilenberg-MacLane H Z spectrum functor for complexes H A * H F p A *

A homotopy cartesian square of pre sheaves of spectra on the site of complex manifolds H=Eilenberg-MacLane H Z HZ D (p) spectrum functor for complexes H A * H F p A *

A homotopy cartesian square of pre sheaves of spectra on the site of complex manifolds H=Eilenberg-MacLane H Z HZ D (p) spectrum functor for complexes H A * H F p A * HZ D (p) represents Deligne cohomology in the homotopy category of presheaves of spectra.

A homotopy cartesian square of pre sheaves of spectra on the site of complex manifolds H=Eilenberg-MacLane H Z spectrum functor for complexes H A * H F p A * HZ D (p) represents Deligne cohomology in the homotopy category of presheaves of spectra.

A homotopy cartesian square of pre sheaves of spectra on the site of complex manifolds H=Eilenberg-MacLane H spectrum functor for complexes H A * H F p A * HZ D (p) represents Deligne cohomology in the homotopy category of presheaves of spectra.