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

Generalized Deligne- Beilinson cohomology and regulators

Regulators IV in Paris May 23, 2016 Gereon Quick NTNU

Euler, Abel and Jacobi: Let f(x) be a polynomial of degree 3 with simple roots.

Euler, Abel and Jacobi: Let f(x) be a polynomial of degree 3 with simple roots. ⌡ ⌠ I(t):= 1/√f(x)dx.

t

We would like to evaluate the integral

In modern terms:

In modern terms: Set E:y2 = f(x) and consider the space E(C) of complex solutions. Set ω = dx/y.

In modern terms: Set E:y2 = f(x) and consider the space E(C) of complex solutions. Set ω = dx/y. Then I(t) = ω for some point P on E(C). ⌠ ⌡0

P

In modern terms: Set E:y2 = f(x) and consider the space E(C) of complex solutions. Set ω = dx/y. Then I(t) = ω for some point P on E(C). ⌠ ⌡0

P

Calculating the integral depends on the choice of a homotopy class of paths from 0 to P.

In modern terms: Set E:y2 = f(x) and consider the space E(C) of complex solutions. Set ω = dx/y. Then I(t) = ω for some point P on E(C). ⌠ ⌡0

P

Calculating the integral depends on the choice of a homotopy class of paths from 0 to P. E(C) C E(C) ~ Euler: this is a group homomorphism. Hence P ⟼ I(P) is really a function on the universal cover E(C) of E(C): ~

The Jacobian of E and the Abel-Jacobi map:

Choose two closed loops γ1 and γ2 which form a basis of H1(E(C);Z)≈H1(S1xS1;Z)≈ZxZ. The Jacobian of E and the Abel-Jacobi map:

Choose two closed loops γ1 and γ2 which form a basis of H1(E(C);Z)≈H1(S1xS1;Z)≈ZxZ. Let and be the periods of E. λ1= ⌠ ⌡γ1 ω λ2= ⌠ ⌡

γ2

ω The Jacobian of E and the Abel-Jacobi map:

Choose two closed loops γ1 and γ2 which form a basis of H1(E(C);Z)≈H1(S1xS1;Z)≈ZxZ. Let and be the periods of E. λ1= ⌠ ⌡γ1 ω λ2= ⌠ ⌡

γ2

ω The Jacobian of E and the Abel-Jacobi map: The map P ⟼ ⌠ ⌡0 ω

P

defines an isomorphism E(C) → C/(Zλ1⊕Zλ2) ≈ Jac(E).

Lefschetz’ s theorem for (1,1)-classes: For simplicity, let X⊂PN be a surface.

Lefschetz’ s theorem 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 theorem 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 Jac(Ct) For simplicity, let X⊂PN be a surface.

Lefschetz’ s theorem 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 Jac(Ct) and a fibre space π: J → P1 (of complex Lie groups). For simplicity, let X⊂PN be a surface.

Lefschetz’ s theorem 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 Jac(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) ∈ Jac(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) ∈ Jac(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) (=cycle class of the curve 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) (=cycle class of the curve D).

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

as 𝜃(𝜉) for some normal function 𝜉. Then Lefschetz proved:

Deligne’ s diagram: Let X be a smooth projective complex variety.

Deligne’ s diagram: CHp(X) Let X be a smooth projective complex variety.

Deligne’ s diagram: CHp(X) Z⊂X Let X be a smooth projective complex variety.

Deligne’ s diagram: CHp(X) Z⊂X Hdg2p(X) clH [Zsm]

−

Let X be a smooth projective complex variety.

Deligne’ s diagram: CHp(X) Z⊂X Hdg2p(X) clH [Zsm]

−

Let X be a smooth projective complex variety. clHD HD (X;Z(p)) →

2p

Deligne cohomology combines topological with Hodge theoretic information

Deligne’ s diagram: CHp(X) Z⊂X Hdg2p(X) clH [Zsm]

−

Let X be a smooth projective complex variety.

→ 0

clHD HD (X;Z(p)) →

2p

Deligne cohomology combines topological with Hodge theoretic information

Deligne’ s diagram: CHp(X) Z⊂X Hdg2p(X) clH [Zsm]

−

Let X be a smooth projective complex variety. J2p-1(X) → 0 →

→ 0

clHD HD (X;Z(p)) →

2p

Deligne cohomology combines topological with Hodge theoretic information

Deligne’ s diagram: CHp(X) Z⊂X Hdg2p(X) clH [Zsm]

−

Let X be a smooth projective complex variety. J2p-1(X) → 0 →

→ 0

clHD HD (X;Z(p)) →

2p

Deligne cohomology combines topological with Hodge theoretic information Griffiths’ intermediate Jacobian (a complex torus)

Deligne’ s diagram: CHp(X) CHp(X)h=Kernel of clH ⊂ Z⊂X Hdg2p(X) clH [Zsm]

−

Let X be a smooth projective complex variety. J2p-1(X) → 0 →

→ 0

clHD HD (X;Z(p)) →

2p

Deligne cohomology combines topological with Hodge theoretic information Griffiths’ intermediate Jacobian (a complex torus)

Deligne’ s diagram: CHp(X) CHp(X)h=Kernel of clH ⊂ Abel-Jacobi map µ Z⊂X Hdg2p(X) clH [Zsm]

−

Let X be a smooth projective complex variety. J2p-1(X) → 0 →

→ 0

clHD HD (X;Z(p)) →

2p

Deligne cohomology combines topological with Hodge theoretic information Griffiths’ intermediate Jacobian (a complex torus)

Another map for smooth complex varieties:

Another map for smooth complex varieties: Ω*(X)

Another map for smooth complex varieties: Ω*(X) algebraic cobordism

• f Levine and Morel

Another map for smooth complex varieties: Ω*(X) algebraic cobordism

• f Levine and Morel
• gen. by proj. maps f:Y→X, Y smooth, modulo Levine’

s and Pandharipande’ s “double point relation”:

Another map for smooth complex varieties: Ω*(X) algebraic cobordism

• f Levine and Morel
• gen. by proj. maps f:Y→X, Y smooth, modulo Levine’

s and Pandharipande’ s “double point relation”:

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.

Another map for smooth complex varieties: Ω*(X) algebraic cobordism

• f Levine and Morel
• gen. by proj. maps f:Y→X, Y smooth, modulo Levine’

s and Pandharipande’ s “double point relation”:

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.

2

→ MU *(X)

complex cobordism of the top. space X(C)

Another map for smooth complex varieties: Ω*(X) algebraic cobordism

• f Levine and Morel
• gen. by proj. maps f:Y→X, Y smooth, modulo Levine’

s and Pandharipande’ s “double point relation”:

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.

[Y→X]

2

→ MU *(X)

complex cobordism of the top. space X(C)

Another map for smooth complex varieties: Ω*(X) algebraic cobordism

• f Levine and Morel
• gen. by proj. maps f:Y→X, Y smooth, modulo Levine’

s and Pandharipande’ s “double point relation”:

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.

[Y→X] [Y(C)→X(C)] ⟼

2

→ MU *(X)

complex cobordism of the top. space X(C)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

H2*(X;Z)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

CH*(X) H2*(X;Z)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

Hdg2*(X) ⊆

clH

CH*(X) H2*(X;Z)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

Ω*(X)⊗L*Z MU2*(X)⊗MU*Z Hdg2*(X) ⊆

clH

CH*(X) H2*(X;Z)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

Ω*(X)⊗L*Z MU2*(X)⊗MU*Z Hdg2*(X) ⊆

clH

CH*(X) H2*(X;Z)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

Ω*(X)⊗L*Z MU2*(X)⊗MU*Z Hdg2*(X) ⊆

clH

CH*(X) H2*(X;Z)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

Ω*(X)⊗L*Z MU2*(X)⊗MU*Z Hdg2*(X) ⊆

clH

Totaro

CH*(X) H2*(X;Z)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

Levine-Morel ≈

Ω*(X)⊗L*Z MU2*(X)⊗MU*Z Hdg2*(X) ⊆

clH

Totaro

CH*(X) H2*(X;Z)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

Levine-Morel ≈

Ω*(X)⊗L*Z MU2*(X)⊗MU*Z

≉ in general

Hdg2*(X) ⊆

clH

Totaro

CH*(X) H2*(X;Z)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

Levine-Morel ≈

Ω*(X)⊗L*Z MU2*(X)⊗MU*Z

≉ in general

Atiyah-Hirzebruch: clH is not surjective.

Hdg2*(X) ⊆

clH

Totaro

CH*(X) H2*(X;Z)

Ω*(X)

MU2*(X)

Φ Atiyah-Hirzebruch and Totaro:

Levine-Morel ≈

Ω*(X)⊗L*Z MU2*(X)⊗MU*Z

≉ in general

Atiyah-Hirzebruch: clH is not surjective.

Hdg2*(X) ⊆

clH This argument does not work for Φ.

Totaro

CH*(X) H2*(X;Z)

Kollar’ s examples:

Kollar’ s examples: Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5.

Kollar’ s examples: Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Zh,

Kollar’ s examples: Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Zh,

X

H4(X;Z)=Zα, ∫ α∙h=1

Kollar’ s examples: Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Zh,

X

H4(X;Z)=Zα, ∫ α∙h=1

and all classes are Hodge classes.

Kollar’ s examples: Kollar: Then any algebraic curve C on X has degree divisible by p. Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Zh,

X

H4(X;Z)=Zα, ∫ α∙h=1

and all classes are Hodge classes.

Kollar’ s examples: Kollar: Then any algebraic curve C on X has degree divisible by p. Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Zh, Let C be a curve on X and [C]∈H4(X;Z) be its cohomology class.

X

H4(X;Z)=Zα, ∫ α∙h=1

and all classes are Hodge classes.

Kollar’ s examples: Kollar: Then any algebraic curve C on X has degree divisible by p. Let X⊂P4 a very general hypersurface of degree d=p3 for a prime p≥5. H2(X;Z)=Zh, Let C be a curve on X and [C]∈H4(X;Z) be its cohomology class. Then [C]= nα for some n and ∫ [C]∙h=n.

X X

H4(X;Z)=Zα, ∫ α∙h=1

and all classes are Hodge classes.

Kollar’ s examples: Kollar: p divides the degree of any curve on X. H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1

X

n = ∫ [C]∙h

X

Kollar’ s examples: Kollar: p divides the degree of any curve on X. H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1

X

n = ∫ [C]∙h

X

= number of intersection points of C with h

Kollar’ s examples: Kollar: p divides the degree of any curve on X. H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1

X

n = ∫ [C]∙h

X

= number of intersection points of C with h = degree of C

Kollar’ s examples: Kollar: p divides the degree of any curve on X. H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1

X

n = ∫ [C]∙h

X

= number of intersection points of C with h = degree of C ⇒ p divides n

Kollar’ s examples: Kollar: p divides the degree of any curve on X. H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1

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).

Kollar’ s examples: Kollar: p divides the degree of any curve on X. H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1

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

X X

Consequence for Φ: Ω*(X) → MU2*(X):

Consequence for Φ: Ω*(X) → MU2*(X): Let X⊂P4 be a very general hypersurface as above.

Consequence 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).

Consequence 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.

Generalized Hodge filtered cohomology theories (joint work with Michael J. Hopkins): 0 → J2p-1(X) → HD2p(X;Z(p)) → Hdg2p(X) → 0

CHp(X) clH clHD Kernel of clH ⊂

Recall the diagram for Chow groups

Generalized Hodge filtered cohomology theories (joint work with Michael J. Hopkins): Our goal: define new invariants for algebraic cobordism which combine 0 → J2p-1(X) → HD2p(X;Z(p)) → Hdg2p(X) → 0

CHp(X) clH clHD Kernel of clH ⊂

Recall the diagram for Chow groups

Generalized Hodge filtered cohomology theories (joint work with Michael J. Hopkins): Our goal: define new invariants for algebraic cobordism which combine

• Hodge theoretical information and

0 → J2p-1(X) → HD2p(X;Z(p)) → Hdg2p(X) → 0

CHp(X) clH clHD Kernel of clH ⊂

Recall the diagram for Chow groups

Generalized Hodge filtered cohomology theories (joint work with Michael J. Hopkins): Our goal: define new invariants for algebraic cobordism which combine

• Hodge theoretical information and

0 → J2p-1(X) → HD2p(X;Z(p)) → Hdg2p(X) → 0

CHp(X) clH clHD Kernel of clH ⊂

Recall the diagram for Chow groups

• topological information of complex cobordism.

Deligne cohomology:

Deligne cohomology: Given an integer p≥0.

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

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

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

sheaf of hol. forms on X

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

sheaf of hol. forms on X

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

sheaf of hol. forms on X

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

sheaf of hol. forms on X

ZD(p)

htpy cart.

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

sheaf of hol. forms on X

ZD(p)

htpy cart.

Deligne cohomology is the hypercohomology of this complex: HD(X;Z(p)) = Hn(X;ZD(p)).

n

Z A * FpA * ZD(p) sheaves of A homotopy cartesian square of complexes

Z A * FpA * ZD(p) sheaves of pre A homotopy cartesian square of complexes

Z A * FpA * ZD(p) sheaves of pre A homotopy cartesian square of

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

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

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

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

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

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

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

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

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

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

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

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

A * FpA * sheaves of pre spectra on the site of complex manifolds MU A homotopy cartesian square of H H H=Eilenberg-MacLane spectrum functor for complexes HZD(p) represents Deligne cohomology in the homotopy category of presheaves of spectra. V*:=MU*⊗C

A * FpA * sheaves of pre spectra on the site of complex manifolds MU (V*) A homotopy cartesian square of H H H=Eilenberg-MacLane spectrum functor for complexes HZD(p) represents Deligne cohomology in the homotopy category of presheaves of spectra. V*:=MU*⊗C

A * FpA * sheaves of pre spectra on the site of complex manifolds MU (V*) (V*) A homotopy cartesian square of H H H=Eilenberg-MacLane spectrum functor for complexes HZD(p) represents Deligne cohomology in the homotopy category of presheaves of spectra. V*:=MU*⊗C

A * FpA * sheaves of pre spectra on the site of complex manifolds MU (V*) (V*) A homotopy cartesian square of H H H=Eilenberg-MacLane spectrum functor for complexes HZD(p) represents Deligne cohomology in the homotopy category of presheaves of spectra. V*:=MU*⊗C MUD(p)

htpy cart.

Hodge filtered complex bordism:

MU HA *(V*) HFpA *(V*) MUD(p)

Hodge filtered complex bordism:

MU HA *(V*) HFpA *(V*) MUD(p)

X a complex manifold, and n, p integers

Hodge filtered complex bordism:

MU HA *(V*) HFpA *(V*) MUD(p)

X a complex manifold, and n, p integers MUD(p)(X):=HomHoPre(∑∞(X+),∑nMUD(p)) We define:

n

Hodge filtered complex bordism:

MU HA *(V*) HFpA *(V*) MUD(p)

X a complex manifold, and n, p integers “HFC bordism” groups sit in long exact sequences. MUD(p)(X):=HomHoPre(∑∞(X+),∑nMUD(p)) We define:

n

Let’ s get concrete:

Let’ s get concrete: Given n, p and a smooth projective variety X.

Let’ s get concrete: Given n, p and a smooth projective variety X. Elements in MUD(p)(X) consist of triples (f, h, ω):

n

Let’ s get concrete: ∋ f : X → MUn Map(X, MUn) Given n, p and a smooth projective variety X. Elements in MUD(p)(X) consist of triples (f, h, ω):

n

Let’ s get concrete: ∋ f : X → MUn Map(X, MUn) Given n, p and a smooth projective variety X. Elements in MUD(p)(X) consist of triples (f, h, ω):

n

V*=MU*⊗C

FpA *(X;V*)n ω ∈

cl

Let’ s get concrete: ∋ f : X → MUn Map(X, MUn) Given n, p and a smooth projective variety X. Elements in MUD(p)(X) consist of triples (f, h, ω):

n

V*=MU*⊗C

FpA *(X;V*)n ω ∈

cl closed forms of total degree n

Let’ s get concrete: C*(X;V*)n-1 ∋ h ∋ f : X → MUn Map(X, MUn) Given n, p and a smooth projective variety X. Elements in MUD(p)(X) consist of triples (f, h, ω):

n

V*=MU*⊗C

FpA *(X;V*)n ω ∈

cl closed forms of total degree n

Let’ s get concrete: C*(X;V*)n-1 ∋ h ∋ f : X → MUn Map(X, MUn) Given n, p and a smooth projective variety X. Elements in MUD(p)(X) consist of triples (f, h, ω):

n

V*=MU*⊗C

FpA *(X;V*)n ω ∈

cl

Z*(X;V*)n

cocycles of total degree n closed forms of total degree n

Let’ s get concrete: C*(X;V*)n-1 ∋ h ∋ f : X → MUn Map(X, MUn) Given n, p and a smooth projective variety X. Elements in MUD(p)(X) consist of triples (f, h, ω):

n

f

V*=MU*⊗C

FpA *(X;V*)n ω ∈

cl

Z*(X;V*)n

cocycles of total degree n closed forms of total degree n

Let’ s get concrete: C*(X;V*)n-1 ∋ h ∋ f : X → MUn Map(X, MUn) Given n, p and a smooth projective variety X. ω Elements in MUD(p)(X) consist of triples (f, h, ω):

n

f

V*=MU*⊗C

FpA *(X;V*)n ω ∈

cl

Z*(X;V*)n

cocycles of total degree n closed forms of total degree n

Let’ s get concrete: C*(X;V*)n-1 ∋ h ∋ f : X → MUn Map(X, MUn) Given n, p and a smooth projective variety X. ω Elements in MUD(p)(X) consist of triples (f, h, ω):

n

= ∂h

• f

V*=MU*⊗C

FpA *(X;V*)n ω ∈

cl

Z*(X;V*)n

cocycles of total degree n closed forms of total degree n

For smooth (not projective) complex varieties, we have to take colimits over compactifications.

For smooth (not projective) complex varieties, we have to take colimits over compactifications.

• Motivic ring spectrum MUD := ∨pMUD(p)

For smooth (not projective) complex varieties, we have to take colimits over compactifications.

• Motivic ring spectrum MUD := ∨pMUD(p)
• Projective bundle formula

For smooth (not projective) complex varieties, we have to take colimits over compactifications.

• Motivic ring spectrum MUD := ∨pMUD(p)
• Projective bundle formula
• Transfers: a projective morphism induces a push-

forward map.

For smooth (not projective) complex varieties, we have to take colimits over compactifications.

• Motivic ring spectrum MUD := ∨pMUD(p)
• Projective bundle formula
• Transfers: a projective morphism induces a push-

forward map. ΦD : Ω*(X) → MUD*(∗)(X) =⊕pMUD (p)(X)

2p 2

• This induces a natural ring homomorphism

A new diagram: Let X be a smooth projective complex variety.

A new diagram: Ωp(X) Let X be a smooth projective complex variety.

A new diagram: Ωp(X) [Y→X] Let X be a smooth projective complex variety.

A new diagram: Ωp(X) [Y→X] Let X be a smooth projective complex variety. Φ [Y(C)→X(C)]

−

HdgMU(X)

2p

A new diagram: Ωp(X) [Y→X] Let X be a smooth projective complex variety. ΦD

MUD (p)(X)

2p

Φ [Y(C)→X(C)]

−

HdgMU(X)

2p

A new diagram: Ωp(X) [Y→X] Let X be a smooth projective complex variety. ΦD

MUD (p)(X)

2p

Φ [Y(C)→X(C)]

−

HdgMU(X)

2p

→

A new diagram: Ωp(X) [Y→X] Let X be a smooth projective complex variety. ΦD

MUD (p)(X)

2p

Φ [Y(C)→X(C)]

−

HdgMU(X)

2p

→ 0 →

A new diagram: Ωp(X) [Y→X] Let X be a smooth projective complex variety. 0 →JMU (X) →

2p-1

ΦD

MUD (p)(X)

2p

Φ [Y(C)→X(C)]

−

HdgMU(X)

2p

→ 0 →

A new diagram: Ωp(X) [Y→X] Let X be a smooth projective complex variety. 0 →JMU (X) →

2p-1

ΦD

MUD (p)(X)

2p

Φ [Y(C)→X(C)]

−

HdgMU(X)

2p

“new Jacobian”: a complex torus ≈ MU2p-1(X)⊗R/Z

→ 0 →

A new diagram: Ωp(X) Ωp(X)top:=Kernel of Φ ⊂ [Y→X] Let X be a smooth projective complex variety. 0 →JMU (X) →

2p-1

ΦD

MUD (p)(X)

2p

Φ [Y(C)→X(C)]

−

HdgMU(X)

2p

“new Jacobian”: a complex torus ≈ MU2p-1(X)⊗R/Z

→ 0 →

A new diagram: Ωp(X) Ωp(X)top:=Kernel of Φ ⊂ “Abel-Jacobi map” µMU [Y→X] Let X be a smooth projective complex variety. 0 →JMU (X) →

2p-1

ΦD

MUD (p)(X)

2p

Φ [Y(C)→X(C)]

−

HdgMU(X)

2p

“new Jacobian”: a complex torus ≈ MU2p-1(X)⊗R/Z

→ 0 →

The Abel-Jacobi map:

The Abel-Jacobi map: Given [Y→X]∈Ωp(X) (with X smooth projective).

The Abel-Jacobi map: Assume [Y(C)→X(C)]=0 in MU2p(X), Given [Y→X]∈Ωp(X) (with X smooth projective).

The Abel-Jacobi map: Assume [Y(C)→X(C)]=0 in MU2p(X), Given [Y→X]∈Ωp(X) (with X smooth projective). and let (f, h, ω) be the image in MUD (p)(X).

2p

The Abel-Jacobi map: Assume [Y(C)→X(C)]=0 in MU2p(X), Given [Y→X]∈Ωp(X) (with X smooth projective).

• H : XxI → MU2p

and let (f, h, ω) be the image in MUD (p)(X).

2p

The Abel-Jacobi map: Assume [Y(C)→X(C)]=0 in MU2p(X), Given [Y→X]∈Ωp(X) (with X smooth projective).

• H : XxI → MU2p
• 𝜃 ∈ FpA

*(X;V*)2p-1

and let (f, h, ω) be the image in MUD (p)(X).

2p

The Abel-Jacobi map: Assume [Y(C)→X(C)]=0 in MU2p(X), Given [Y→X]∈Ωp(X) (with X smooth projective).

• H : XxI → MU2p
• 𝜃 ∈ FpA

*(X;V*)2p-1

and let (f, h, ω) be the image in MUD (p)(X).

2p

Then (0,H-𝜃,0) = (f,h,ω) in MUD (p)(X),

2p

The Abel-Jacobi map: and H-𝜃 ∈ C*(X;V*)2p-1 defines an element in MU2p-1(X)⊗R, unique modulo MU2p-1(X). Assume [Y(C)→X(C)]=0 in MU2p(X), Given [Y→X]∈Ωp(X) (with X smooth projective).

• H : XxI → MU2p
• 𝜃 ∈ FpA

*(X;V*)2p-1

and let (f, h, ω) be the image in MUD (p)(X).

2p

Then (0,H-𝜃,0) = (f,h,ω) in MUD (p)(X),

2p

The Abel-Jacobi map: and H-𝜃 ∈ C*(X;V*)2p-1 defines an element in MU2p-1(X)⊗R, unique modulo MU2p-1(X). Abel-Jacobi map: Ωp(X)top→ MU2p-1(X)⊗R/Z Assume [Y(C)→X(C)]=0 in MU2p(X), Given [Y→X]∈Ωp(X) (with X smooth projective).

• H : XxI → MU2p
• 𝜃 ∈ FpA

*(X;V*)2p-1

and let (f, h, ω) be the image in MUD (p)(X).

2p

Then (0,H-𝜃,0) = (f,h,ω) in MUD (p)(X),

2p

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

New “regulator maps”:

New “regulator maps”: Let E be an oriented motivic (ring) spectrum and E(C) be its associated topological spectrum.

New “regulator maps”: Let E be an oriented motivic (ring) spectrum and E(C) be its associated topological spectrum. E → E(C)D := ∨pE(C)D(p)

New “regulator maps”: Let E be an oriented motivic (ring) spectrum and E(C) be its associated topological spectrum. E → E(C)D := ∨pE(C)D(p) We can consider such a map as an (integral) “regulator map”.

New “regulator maps”: Let E be an oriented motivic (ring) spectrum and E(C) be its associated topological spectrum. E → E(C)D := ∨pE(C)D(p) We can consider such a map as an (integral) “regulator map”. For example, if E represents algebraic K-theory: Kn(X) → ⊕pKD (p)(X)

2p-n

Arakelov algebraic cobordism:

Arakelov algebraic cobordism: Let S be a scheme of finite type over Z, and let 𝜃 be the generic point.

Arakelov algebraic cobordism: Let MGL denote Voevodsky’ s Thom spectrum representing algebraic cobordism. Let S be a scheme of finite type over Z, and let 𝜃 be the generic point.

Arakelov algebraic cobordism: Let MGL denote Voevodsky’ s Thom spectrum representing algebraic cobordism. MGLS → 𝜃 (MUD) * Let S be a scheme of finite type over Z, and let 𝜃 be the generic point.

Arakelov algebraic cobordism: Let MGL denote Voevodsky’ s Thom spectrum representing algebraic cobordism. MGLArakelov → homotopy fibre represents “Arakelov algebraic cobordism” MGLS → 𝜃 (MUD) * Let S be a scheme of finite type over Z, and let 𝜃 be the generic point.

Thank you!