Algebraic cycles in generalized cohomology theories Mathematisches - - PowerPoint PPT Presentation

Algebraic cycles in generalized cohomology theories

Mathematisches Forschungsinstitut Oberwolfach April 20, 2018 Gereon Quick NTNU

Lefschetz’ s theorem: X projective complex surface

Lefschetz’ s theorem:

2-dim. topological cycle

Given 𝜅: 𝛥 ⊂ X. X projective complex surface

Lefschetz’ s theorem:

2-dim. topological cycle

Given 𝜅: 𝛥 ⊂ X. If 𝛥∼C for an alg. curve C⊂X, then

homologous

X projective complex surface

Lefschetz’ s theorem:

2-dim. topological cycle

Given 𝜅: 𝛥 ⊂ X. If 𝛥∼C for an alg. curve C⊂X, then

homologous

unless 𝛽 is of Hodge type (1,1).

form on X

∫ 𝜅*𝛽 = 0

𝛥

(N) X projective complex surface

Lefschetz’ s theorem:

2-dim. topological cycle

Given 𝜅: 𝛥 ⊂ X. If 𝛥∼C for an alg. curve C⊂X, then

homologous

unless 𝛽 is of Hodge type (1,1).

form on X

∫ 𝜅*𝛽 = 0

𝛥

(N) Lefschetz: If (N) holds for 𝛥, then [𝛥] is “algebraic”.

there is an algebraic curve C～𝛥 class in homology

X projective complex surface

Higher (co-)dimensions: X smooth projective complex algebraic variety

Higher (co-)dimensions: X smooth projective complex algebraic variety

(smooth) subvariety

• f dim. n

𝜅:Z⊂

Higher (co-)dimensions: X smooth projective complex algebraic variety

(smooth) subvariety

• f dim. n

𝜅:Z⊂ ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

forms of type (n,n)

Then

Higher (co-)dimensions: Z⊂X CHp(X)

free abelian group on alg. subvarieties

• f codim. p

modulo

• rat. equiv.

X smooth projective complex algebraic variety

(smooth) subvariety

• f dim. n

𝜅:Z⊂ ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

forms of type (n,n)

Then

Higher (co-)dimensions: Z⊂X CHp(X)

free abelian group on alg. subvarieties

• f codim. p

modulo

• rat. equiv.

⟼ [Zsm]

dual of fund. class (of desingularization) clH

H2p(X;Z) X smooth projective complex algebraic variety

(smooth) subvariety

• f dim. n

𝜅:Z⊂ ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

forms of type (n,n)

Then

Higher (co-)dimensions: Z⊂X CHp(X)

free abelian group on alg. subvarieties

• f codim. p

modulo

• rat. equiv.

⟼ [Zsm]

dual of fund. class (of desingularization) clH

H2p(X;Z) X smooth projective complex algebraic variety

(smooth) subvariety

• f dim. n

𝜅:Z⊂ ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

forms of type (n,n)

Then Hdg2p(X) ⊂

integral Hodge classes

• f type (p,p)

Higher (co-)dimensions: Z⊂X CHp(X)

free abelian group on alg. subvarieties

• f codim. p

modulo

• rat. equiv.
• Hodge’

s question: Is this map surjective? ⟼ [Zsm]

dual of fund. class (of desingularization) clH

H2p(X;Z) X smooth projective complex algebraic variety

(smooth) subvariety

• f dim. n

𝜅:Z⊂ ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

forms of type (n,n)

Then Hdg2p(X) ⊂

integral Hodge classes

• f type (p,p)

Higher (co-)dimensions: Z⊂X CHp(X)

free abelian group on alg. subvarieties

• f codim. p

modulo

• rat. equiv.
• Hodge’

s question: Is this map surjective?

• Question: What is the kernel of this map?

⟼ [Zsm]

dual of fund. class (of desingularization) clH

H2p(X;Z) X smooth projective complex algebraic variety

(smooth) subvariety

• f dim. n

𝜅:Z⊂ ∫ 𝜅*𝛽 = 0 unless 𝛽 ∈ An,n(X).

Z

forms of type (n,n)

Then Hdg2p(X) ⊂

integral Hodge classes

• f type (p,p)

How to do homotopy on Man?

category of complex manifolds

How to do homotopy on Man?

category of complex manifolds

Man Pre

presheaves of sets, i.e., functors: Manop ⟶ Set

How to do homotopy on Man?

category of complex manifolds

M FM

FM: X ⟼ HomMan(X,M)

Man Pre

presheaves of sets, i.e., functors: Manop ⟶ Set

How to do homotopy on Man?

category of complex manifolds

M FM

FM: X ⟼ HomMan(X,M) Presheaves “remember”

HomPre(FM,FM’) = HomMan(M,M’) Man Pre

presheaves of sets, i.e., functors: Manop ⟶ Set

How to do homotopy on Man?

category of complex manifolds

M FM

FM: X ⟼ HomMan(X,M) Presheaves “remember”

HomPre(FM,FM’) = HomMan(M,M’)

• Sets

“rigid” presheaves

• f

Man Pre

presheaves of sets, i.e., functors: Manop ⟶ Set

How to do homotopy on Man?

category of complex manifolds

M FM

FM: X ⟼ HomMan(X,M) Presheaves “remember”

HomPre(FM,FM’) = HomMan(M,M’)

• Sets

“rigid” presheaves

• f

“allow homotopy” switch to

Sets∆

presheaves

• f

Man Pre

presheaves of sets, i.e., functors: Manop ⟶ Set

M FM

FM: X ⟼ discrete simplicial set FM(X)

Man Pre∆

presheaves of simplicial sets functors: Manop ⟶ Set∆

• How to do homotopy on Man?

M FM

FM: X ⟼ discrete simplicial set FM(X)

Man Pre∆

presheaves of simplicial sets functors: Manop ⟶ Set∆

• How to do homotopy on Man?
• Given n≥0, the n-dimensional stalk of F•

= colim F•(Bn(r))

r→0

in Set∆

ball of radius r in n-dim. complex affine space

F•

(n)

M FM

FM: X ⟼ discrete simplicial set FM(X)

Man Pre∆

presheaves of simplicial sets functors: Manop ⟶ Set∆

• How to do homotopy on Man?
• Given n≥0, the n-dimensional stalk of F•

= colim F•(Bn(r))

r→0

in Set∆

ball of radius r in n-dim. complex affine space

F•

(n)

• A map F• → G• is a weak equivalence in Pre∆

if F• → G• is a weak equivalence in Set∆ for all n≥0.

(n) (n)

Homotopy category of Man: Man Pre∆

Homotopy category of Man: Man Pre∆

• homotopy category of

simplicial presheaves on Man

ho =Pre∆[w.e.-1]

Homotopy category of Man: Man Pre∆

• Given M with an open cover {U𝛽}:

FU → FM is a weak equivalence.

• ∐U𝛽 ⇉ ∐U𝛽×XU𝛾 ⇶ …

homotopy category of simplicial presheaves on Man

ho =Pre∆[w.e.-1]

Homotopy category of Man: Man Pre∆

• Given M with an open cover {U𝛽}:

FU → FM is a weak equivalence.

• ∐U𝛽 ⇉ ∐U𝛽×XU𝛾 ⇶ …

homotopy category of simplicial presheaves on Man

ho =Pre∆[w.e.-1]

• Can replace Set∆ with Spectra and get a

stable homotopy category hoPreSpectra of Man.

• S1∧- with S1 viewed as a simplicial (constant) presheaf

is made invertible. sequence of spaces …,En,En+1,… with maps S1∧En →En+1

Homotopy category of SmC: SmC

• smooth complex varieties

Homotopy category of SmC: SmC

• smooth complex varieties

Pre∆

simplicial presheaves on SmC Morel Voevodsky Jardine Joyal Isaksen Dugger …

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

SmC

• smooth complex varieties

Pre∆

simplicial presheaves on SmC Morel Voevodsky Jardine Joyal Isaksen Dugger …

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

SmC

• smooth complex varieties

Pre∆

simplicial presheaves on SmC Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

FU → FX for any X and any (hyper)cover U→X

• SmC
• smooth complex varieties

Pre∆

simplicial presheaves on SmC Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

FU → FX for any X and any (hyper)cover U→X

• SmC
• smooth complex varieties

ho

homotopy category of

Pre∆

simplicial presheaves on SmC Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

FU → FX for any X and any (hyper)cover U→X

• SmC
• smooth complex varieties

ho

homotopy category of

Pre∆

simplicial presheaves on SmC Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

A1xX → X for any X

motivic affine line over C

Homotopy category of SmC:

• Nisnevich covers (replacing open covers)

FU → FX for any X and any (hyper)cover U→X

• SmC
• smooth complex varieties

ho

homotopy category of

• stable motivic homotopy category of SmC
• P1∧- the projective line
• S1∧- the “simplicial circle” and (A1-0)∧- the “Tate circle”

Pre∆

simplicial presheaves on SmC Morel Voevodsky Jardine Joyal Isaksen Dugger …

• Localize with respect to

A1xX → X for any X

motivic affine line over C

Topological realization:

X(C) X SmC ManC 𝜍

P1(C)=CP1≃S2 P1 e.g.

”complex manifold of solutions in C”

Topological realization:

X(C) X SmC ManC 𝜍

P1(C)=CP1≃S2 P1 e.g.

”complex manifold of solutions in C”

Ea (X(C)) Ea,b (X)

mot top

𝜍E

motivic spectrum “topological” “algebraic” induced map

• top. real.

Topological realization:

X(C) X

Questions:

SmC ManC 𝜍

P1(C)=CP1≃S2 P1 e.g.

”complex manifold of solutions in C”

Ea (X(C)) Ea,b (X)

mot top

𝜍E

motivic spectrum “topological” “algebraic” induced map

• top. real.

Topological realization:

X(C) X

Questions: How can we detect whether classes

SmC ManC 𝜍

P1(C)=CP1≃S2 P1 e.g.

”complex manifold of solutions in C”

Ea (X(C)) Ea,b (X)

mot top

𝜍E

motivic spectrum “topological” “algebraic” induced map

• top. real.

Topological realization:

X(C) X

Questions:

• in

E* (X(C))?

top

are topologically trivial, i.e., become 0 in E*,*(X)

mot

How can we detect whether classes

SmC ManC 𝜍

P1(C)=CP1≃S2 P1 e.g.

”complex manifold of solutions in C”

Ea (X(C)) Ea,b (X)

mot top

𝜍E

motivic spectrum “topological” “algebraic” induced map

• top. real.

Topological realization:

X(C) X

• in

E* (X(C))

top

i.e., are in the image of 𝜍E? are algebraic, Questions:

• in

E* (X(C))?

top

are topologically trivial, i.e., become 0 in E*,*(X)

mot

How can we detect whether classes

SmC ManC 𝜍

P1(C)=CP1≃S2 P1 e.g.

”complex manifold of solutions in C”

Ea (X(C)) Ea,b (X)

mot top

𝜍E

motivic spectrum “topological” “algebraic” induced map

• top. real.

Atiyah-Hirzebruch, Totaro, Levine-Morel:

H2*(X;Z)

𝜍H=clH

H

Alg2*(X)⊆ CH*(X)=H2*,*(X;Z)

mot

Atiyah-Hirzebruch, Totaro, Levine-Morel:

H2*(X;Z)

𝜍H=clH

H

Alg2*(X)⊆ CH*(X)=H2*,*(X;Z)

mot

MU2*(X) MGL2*,*(X)

𝜍MGL

universal

• riented

theories

Atiyah-Hirzebruch, Totaro, Levine-Morel:

H2*(X;Z)

𝜍H=clH

H

Alg2*(X)⊆ CH*(X)=H2*,*(X;Z)

mot

MU2*(X) MGL2*,*(X)

𝜍MGL

universal

• riented

theories

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

coeff. ring = Lazard ring

Atiyah-Hirzebruch, Totaro, Levine-Morel:

H2*(X;Z)

𝜍H=clH

H

Alg2*(X)⊆ CH*(X)=H2*,*(X;Z)

mot

⟳

Totaro

MU2*(X) MGL2*,*(X)

𝜍MGL

universal

• riented

theories

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

coeff. ring = Lazard ring

Atiyah-Hirzebruch, Totaro, Levine-Morel:

H2*(X;Z)

𝜍H=clH

H

Alg2*(X)⊆ CH*(X)=H2*,*(X;Z)

mot

⟳

Totaro Levine + Levine-Morel ≈

MU2*(X) MGL2*,*(X)

𝜍MGL

universal

• riented

theories

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

coeff. ring = Lazard ring

Atiyah-Hirzebruch, Totaro, Levine-Morel:

H2*(X;Z)

𝜍H=clH

H

Alg2*(X)⊆ CH*(X)=H2*,*(X;Z)

mot

≉ in general

⟳

Totaro Levine + Levine-Morel ≈

MU2*(X) MGL2*,*(X)

𝜍MGL

universal

• riented

theories

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

coeff. ring = Lazard ring

Atiyah-Hirzebruch, Totaro, Levine-Morel:

H2*(X;Z)

𝜍H=clH

H

Alg2*(X)⊆ CH*(X)=H2*,*(X;Z)

mot

≉ in general

⟳

Totaro

• Atiyah-Hirzebruch: clH is not surjective
• nto integral Hodge classes.

Levine + Levine-Morel ≈

MU2*(X) MGL2*,*(X)

𝜍MGL

universal

• riented

theories

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

coeff. ring = Lazard ring

Atiyah-Hirzebruch, Totaro, Levine-Morel:

H2*(X;Z)

𝜍H=clH

H

Alg2*(X)⊆ CH*(X)=H2*,*(X;Z)

mot

≉ in general

⟳

Totaro

• Atiyah-Hirzebruch: clH is not surjective
• nto integral Hodge classes.
• Totaro: new classes in kernel of clH.

Levine + Levine-Morel ≈

MU2*(X) MGL2*,*(X)

𝜍MGL

universal

• riented

theories

MGL2*,*(X)⊗L*Z MU2*(X)⊗L*Z

coeff. ring = Lazard ring

• I. Kernel:

X smooth projective

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

• I. Kernel:

X smooth projective

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

• I. Kernel:

Hodge classes

X smooth projective

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

Deligne cohomology

• I. Kernel:

Hodge classes

X smooth projective

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

Deligne cohomology Griffiths’ Jacobian

• I. Kernel:

Hodge classes

X smooth projective

CHp(X) clH clHD

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

Deligne cohomology Griffiths’ Jacobian

• I. Kernel:

Hodge classes

X smooth projective

CHp(X) clH clHD

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

Kernel of clH ⊂

Abel- Jacobi map

µH

Deligne cohomology Griffiths’ Jacobian

• I. Kernel:

Hodge classes

X smooth projective

Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins):

CHp(X) clH clHD

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

Kernel of clH ⊂

Abel- Jacobi map

µH

Deligne cohomology Griffiths’ Jacobian

• I. Kernel:

Hodge classes

X smooth projective

Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins):

0 → J2p-1(X) → MU2p(X;Z(p)) → Hdg2p (X) → 0

MU MU D

CHp(X) clH clHD

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

Kernel of clH ⊂

Abel- Jacobi map

µH

Deligne cohomology Griffiths’ Jacobian

• I. Kernel:

Hodge classes

X smooth projective

Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins):

0 → J2p-1(X) → MU2p(X;Z(p)) → Hdg2p (X) → 0

MU MU D

CHp(X) clH clHD

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

Kernel of clH ⊂

Abel- Jacobi map

µH

Deligne cohomology Griffiths’ Jacobian

• I. Kernel:

Hodge classes MU-Hodge classes

X smooth projective

Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins):

0 → J2p-1(X) → MU2p(X;Z(p)) → Hdg2p (X) → 0

MU MU D

CHp(X) clH clHD

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

”Hodge filtered complex cobordism” Kernel of clH ⊂

Abel- Jacobi map

µH

Deligne cohomology Griffiths’ Jacobian

• I. Kernel:

Hodge classes MU-Hodge classes

X smooth projective

Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins):

0 → J2p-1(X) → MU2p(X;Z(p)) → Hdg2p (X) → 0

MU MU D

CHp(X) clH clHD

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

MU-“Jacobian” ”Hodge filtered complex cobordism” Kernel of clH ⊂

Abel- Jacobi map

µH

Deligne cohomology Griffiths’ Jacobian

• I. Kernel:

Hodge classes MU-Hodge classes

X smooth projective

Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins):

MGL2p,p(X) 𝜍MU 𝜍MUD

0 → J2p-1(X) → MU2p(X;Z(p)) → Hdg2p (X) → 0

MU MU D

CHp(X) clH clHD

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

MU-“Jacobian” ”Hodge filtered complex cobordism” Kernel of clH ⊂

Abel- Jacobi map

µH

Deligne cohomology Griffiths’ Jacobian

• I. Kernel:

Hodge classes MU-Hodge classes

X smooth projective

Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins):

MGL2p,p(X) 𝜍MU 𝜍MUD

0 → J2p-1(X) → MU2p(X;Z(p)) → Hdg2p (X) → 0

MU MU D

CHp(X) clH clHD

Recall Deligne’ s diagram

0 → J2p-1(X) → H2p(X;Z(p)) → Hdg2p(X) → 0

D

MU-“Jacobian” ”Hodge filtered complex cobordism” Kernel of 𝜍MU ⊂ µMU ”Abel- Jacobi map” Kernel of clH ⊂

Abel- Jacobi map

µH

Deligne cohomology Griffiths’ Jacobian

• I. Kernel:

Hodge classes MU-Hodge classes

X smooth projective

Examples: The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes: MGL2p,p(X) MU2p(X(C)) JMU (X)

2p-1

Hmot (X;Z)

2p,p

• top. realization

motivic Thom map Abel-Jacobi map for MU

Examples: The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes: MGL2p,p(X) MU2p(X(C))

there is an α ∊

JMU (X)

2p-1

Hmot (X;Z)

2p,p

• top. realization

motivic Thom map Abel-Jacobi map for MU

Examples: The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes: MGL2p,p(X) MU2p(X(C))

there is an α ∊

JMU (X)

2p-1

Hmot (X;Z)

2p,p

• top. realization

motivic Thom map Abel-Jacobi map for MU hence 0 in J2p-1(X)

Examples: The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes: MGL2p,p(X) MU2p(X(C))

there is an α ∊

JMU (X)

2p-1

Hmot (X;Z)

2p,p

• top. realization

motivic Thom map Abel-Jacobi map for MU hence 0 in J2p-1(X)

Examples: The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes: MGL2p,p(X) MU2p(X(C))

there is an α ∊

≠0 JMU (X)

2p-1

Hmot (X;Z)

2p,p

• top. realization

motivic Thom map Abel-Jacobi map for MU hence 0 in J2p-1(X)

In concrete terms:

Given p and X smooth projective.

Elements in MUD (p)(X) consist of triples (f, h, ω):

2p

In concrete terms:

Given p and X smooth projective.

Elements in MUD (p)(X) consist of triples (f, h, ω):

2p

∋ f : Y → X MU2p(X)

(almost) complex manifold proper

In concrete terms:

Given p and X smooth projective.

Elements in MUD (p)(X) consist of triples (f, h, ω):

2p

∋ f : Y → X MU2p(X)

(almost) complex manifold proper V*=MU*⊗C

FpΩ*(X;V*)2p ω ∈

cl closed forms of total degree 2p

In concrete terms:

C*(X;V*)2p-1 ∋ h

Given p and X smooth projective.

Elements in MUD (p)(X) consist of triples (f, h, ω):

2p

∋ f : Y → X MU2p(X)

(almost) complex manifold proper V*=MU*⊗C

FpΩ*(X;V*)2p ω ∈

cl closed forms of total degree 2p

In concrete terms:

C*(X;V*)2p-1 ∋ h

Given p and X smooth projective.

Elements in MUD (p)(X) consist of triples (f, h, ω):

2p

Z*(X;V*)2p

cocycles of total degree 2p

∋ f : Y → X MU2p(X)

(almost) complex manifold proper V*=MU*⊗C

FpΩ*(X;V*)2p ω ∈

cl closed forms of total degree 2p

In concrete terms:

C*(X;V*)2p-1 ∋ h

Given p and X smooth projective.

Elements in MUD (p)(X) consist of triples (f, h, ω):

2p

f Z*(X;V*)2p

cocycles of total degree 2p

∋ f : Y → X MU2p(X)

(almost) complex manifold proper V*=MU*⊗C

FpΩ*(X;V*)2p ω ∈

cl closed forms of total degree 2p

In concrete terms:

C*(X;V*)2p-1 ∋ h

Given p and X smooth projective.

ω

Elements in MUD (p)(X) consist of triples (f, h, ω):

2p

f Z*(X;V*)2p

cocycles of total degree 2p

∋ f : Y → X MU2p(X)

(almost) complex manifold proper V*=MU*⊗C

FpΩ*(X;V*)2p ω ∈

cl closed forms of total degree 2p

In concrete terms:

C*(X;V*)2p-1 ∋ h

Given p and X smooth projective.

ω

Elements in MUD (p)(X) consist of triples (f, h, ω):

2p

= ∂h

• f

Z*(X;V*)2p

cocycles of total degree 2p

∋ f : Y → X MU2p(X)

(almost) complex manifold proper V*=MU*⊗C

FpΩ*(X;V*)2p ω ∈

cl closed forms of total degree 2p

In concrete terms:

C*(X;V*)2p-1 ∋ h

Given p and X smooth projective.

ω

Elements in MUD (p)(X) consist of triples (f, h, ω):

2p

= ∂h

• f

Z*(X;V*)2p

cocycles of total degree 2p

∋ f : Y → X MU2p(X)

(almost) complex manifold proper V*=MU*⊗C

FpΩ*(X;V*)2p ω ∈

cl closed forms of total degree 2p “f∗ of universal genus of curvature form” of normal bundle of Y if Y is a smooth projective variety

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: MGLS → 𝜃 (MUD) * Let S be a scheme of finite type over Z, and let 𝜃 be the generic point.

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

Arakelov algebraic cobordism: MGLS → 𝜃 (MUD) * Let S be a scheme of finite type over Z, and let 𝜃 be the generic point. MGLArakelov → homotopy fibre represents “Arakelov algebraic cobordism” Question: What is the arithmetic-geometric information encoded in the Chern classes in Arakelov algebraic cobordism?

X(C) X

manifold of solutions in C

Ea (X(C)) Ea,b (X)

mot top

𝜍E

motivic spectrum

Sm Man 𝜍

topological algebraic induced map

• II. Image:

Recall:

X(C) X

manifold of solutions in C

Ea (X(C)) Ea,b (X)

mot top

𝜍E

motivic spectrum

Sm Man 𝜍

topological algebraic induced map

• II. Image:

E* (X(C))

top

are in the image of 𝜍E? are algebraic, i.e., Question:

• How can we detect whether classes in

Recall:

X(C) X

manifold of solutions in C

Ea (X(C)) Ea,b (X)

mot top

𝜍E

motivic spectrum

Sm Man 𝜍

topological algebraic induced map

• II. Image:

E* (X(C))

top

are in the image of 𝜍E? are algebraic, i.e., Question:

• How can we detect whether classes in

Recall:

not not

X(C) X

manifold of solutions in C

Ea (X(C)) Ea,b (X)

mot top

𝜍E

motivic spectrum

Sm Man 𝜍

topological algebraic induced map

• II. Image:

E* (X(C))

top

are in the image of 𝜍E? are algebraic, i.e., Question:

• How can we detect whether classes in

Recall:

not not

• How can we construct such classes?

Fix a prime p. A different perspective:

Fix a prime p.

Brown-Peterson, Quillen

MU(p) splits as a wedge of suspensions of spectra BP with BP = Z(p)[v1,v2,…]. *

|vi|=2(pi-1)

A different perspective:

Fix a prime p.

Brown-Peterson, Quillen

MU(p) splits as a wedge of suspensions of spectra BP with BP = Z(p)[v1,v2,…]. *

|vi|=2(pi-1) quotient map

BP BP/(vn+1,…) =: BP⟨n⟩ with BP⟨n⟩ = Z(p)[v1,…,vn] *

For every n:

A different perspective:

Fix a prime p.

Brown-Peterson, Quillen

MU(p) splits as a wedge of suspensions of spectra BP with BP = Z(p)[v1,v2,…]. *

|vi|=2(pi-1) quotient map

BP BP/(vn+1,…) =: BP⟨n⟩ with BP⟨n⟩ = Z(p)[v1,…,vn] *

For every n:

A different perspective: BP⟨n⟩ BP … …

HZ(p) HFp

BP⟨0⟩ BP⟨-1⟩ The Brown-Peterson tower (Wilson): BP⟨1⟩

p=2: 2-local connective K-theory

Milnor operations:

Milnor operations:

For every n:

BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Milnor operations:

with an induced exact sequence (for any space X)

BP⟨n⟩* (X)

+|vn|

BP⟨n⟩*(X) BP⟨n-1⟩*(X) BP⟨n⟩*+|vn|+1 (X)

qn For every n:

BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Milnor operations:

with an induced exact sequence (for any space X)

BP⟨n⟩* (X)

+|vn|

BP⟨n⟩*(X) BP⟨n-1⟩*(X) BP⟨n⟩*+|vn|+1 (X)

qn

H*(X;Fp) H*

+|vn|+1(X;Fp) Qn BP⟨n-1⟩ HFp Thom map For every n:

BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Milnor operations:

with an induced exact sequence (for any space X)

BP⟨n⟩* (X)

+|vn|

BP⟨n⟩*(X) BP⟨n-1⟩*(X) BP⟨n⟩*+|vn|+1 (X)

qn nth Milnor

• peration:

Q0=Bockstein Qn=P Qn-1-Qn-1P

pn-1 pn-1

H*(X;Fp) H*

+|vn|+1(X;Fp) Qn BP⟨n-1⟩ HFp Thom map For every n:

BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 Question: Is 𝝱 algebraic?

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 Question: Is 𝝱 algebraic?

CH*(X)

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 Question: Is 𝝱 algebraic?

CH*(X)

LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 𝝱n-1 qn𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 𝝱n-1 qn𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

= 0

if LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 𝝱n-1 qn𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

= 0

if

𝝱n

then LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 Qn𝝱 𝝱n-1 qn𝝱n-1 Question: Is 𝝱 algebraic?

CH*(X)

= 0

if

𝝱n

then LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X)

⟲

𝝱 Qn𝝱 𝝱n-1 qn𝝱n-1

CH*(X)

= 0

if

𝝱n

then LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 Qn𝝱 𝝱n-1 qn𝝱n-1

CH*(X)

= 0

if

𝝱n

then LMT

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 Qn𝝱 𝝱n-1 qn𝝱n-1

CH*(X)

= 0

if

𝝱n

then LMT

If Qn𝝱 ≠ 0, ≠ 0

if

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 Qn𝝱 𝝱n-1 qn𝝱n-1

CH*(X)

𝝱n

then LMT

If Qn𝝱 ≠ 0, ≠ 0

if

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 Qn𝝱 𝝱n-1 ≠ 0 qn𝝱n-1

CH*(X)

𝝱n

then LMT

If Qn𝝱 ≠ 0, ≠ 0

if

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 Qn𝝱 𝝱n-1 ≠ 0 qn𝝱n-1

CH*(X)

𝝱n

then LMT

✘ If Qn𝝱 ≠ 0, ≠ 0

if

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 ✘ Qn𝝱 𝝱n-1 ≠ 0 qn𝝱n-1

CH*(X)

𝝱n

then LMT

✘ If Qn𝝱 ≠ 0, ≠ 0

if

The LMT obstruction in action: BP⟨n⟩2*(X) BP⟨n-1⟩2*(X) BP⟨n⟩2*+|vn|+1(X) H2*(X;Fp) H2*

+|vn|+1(X;Fp) Qn qn

↺

BP2*(X) Levine-Morel-Totaro obstruction:

⟲

𝝱 ✘ Qn𝝱 𝝱n-1 ≠ 0 qn𝝱n-1 then 𝝱 is not algebraic.

CH*(X)

𝝱n

then LMT

✘ If Qn𝝱 ≠ 0, ≠ 0

if

Voevodsky’ s motivic Milnor operations:

Voevodsky’ s motivic Milnor operations: There are motivic operations Qn ∈ 𝓑 2pn-1,pn-1

mod p-motivic Steenrod algebra

mot

Voevodsky’ s motivic Milnor operations: There are motivic operations Qn ∈ 𝓑 2pn-1,pn-1

mod p-motivic Steenrod algebra

mot

H H

i+2pn-1,j+pn-1 (X;Fp)

(X;Fp)

i,j mot mot mod p-motivic cohomology

For a smooth complex variety X: Qn

mot

Voevodsky’ s motivic Milnor operations: H (X;Fp)

2i,i mot

= CHi(X;Z/p) H (X;Fp)

i,j mot

= 0 if i>2j. and Recall: There are motivic operations Qn ∈ 𝓑 2pn-1,pn-1

mod p-motivic Steenrod algebra

mot

H H

i+2pn-1,j+pn-1 (X;Fp)

(X;Fp)

i,j mot mot mod p-motivic cohomology

For a smooth complex variety X: Qn

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

topological realization

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

= 0

topological realization

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

= 0

topological realization

mot

Qn

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

= 0 𝝱

topological realization

mot

Qn

mot

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

= 0 𝝱

topological realization

mot

Qn

mot

Qn𝝱 ≠ 0

if

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

= 0 𝝱 ✘

topological realization

mot

Qn

mot

Qn𝝱 ≠ 0

if

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

= 0 𝝱 ✘ Observation: The LMT-obstruction is particular to smooth varieties and bidegrees (2i,i).

topological realization

mot

Qn

mot

Qn𝝱 ≠ 0

if

Obstructions revisited: H H

2i+2pn-1,i+pn-1(X;Fp) Qn

(X;Fp)

2i,i mot mot

H H2i(X;Fp)

2i+2pn-1

X smooth complex variety

(X;Fp)

↺

Qn

= 0 𝝱 ✘ Observation: The LMT-obstruction is particular to smooth varieties and bidegrees (2i,i).

topological realization

mot

Qn

mot

Example: Qn𝛋≠0 for 𝛋 the fundamental class of a suitable Eilenberg-MacLane space, though 𝛋 is algebraic.

Qn𝝱 ≠ 0

if

Back to our task:

Back to our task: Study Alg2*(X) and its complement in E2*(X).

E top

Back to our task: Study Alg2*(X) and its complement in E2*(X).

E top

For example: E=BP⟨n⟩?

Back to our task: Study Alg2*(X) and its complement in E2*(X).

E top

For example: E=BP⟨n⟩? Recall: BP and BP⟨n⟩ exist in the motivic world (e.g. Vezzosi, Hopkins, Hu-Kriz, Ormsby, Hoyois, Ormsby-Østvær).

Back to our task: Study Alg2*(X) and its complement in E2*(X).

E top

For example: E=BP⟨n⟩? Recall: BP and BP⟨n⟩ exist in the motivic world (e.g. Vezzosi, Hopkins, Hu-Kriz, Ormsby, Hoyois, Ormsby-Østvær).

will drop the “top” again

Question: How can we produce non-algebraic elements in BP⟨n⟩2*(X)?

top

Back to the cofibre sequence:

Back to the cofibre sequence: BP⟨n-1⟩*(X) BP⟨n⟩* +|vn|+1(X).

qn and the induced map

Recall: BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Back to the cofibre sequence: For example: BP⟨n-1⟩*(X) BP⟨n⟩* +|vn|+1(X).

qn and the induced map

Recall: BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Back to the cofibre sequence: n=0: H*(X;Fp) H*+1 (X;Z(p)),

q0 Bockstein homomorphism

For example: BP⟨n-1⟩*(X) BP⟨n⟩* +|vn|+1(X).

qn and the induced map

Recall: BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

Back to the cofibre sequence: n=0: H*(X;Fp) H*+1 (X;Z(p)),

q0 Bockstein homomorphism

For example: n=1: H*(X;Z(p))

+2p-1(X), q1

BP⟨1⟩* ⋮ BP⟨n-1⟩*(X) BP⟨n⟩* +|vn|+1(X).

qn and the induced map

Recall: BP⟨n⟩ BP⟨n⟩ BP⟨n-1⟩ BP⟨n⟩

vn |vn|

∑

|vn|+1

∑

stable cofibre sequence

A diagram chase: Hk(X;Fp)

A diagram chase: Hk(X;Fp) Hk+1(X;Z(p))

q0

A diagram chase: Hk(X;Fp) Hk+1(X;Z(p))

q0 q1

BP⟨1⟩k+1+2p-1(X)

A diagram chase: Hk(X;Fp) Hk+1(X;Z(p))

q0 q1

BP⟨1⟩k+1+2p-1(X) (X)

q2

⋮ BP⟨n⟩k

qn +|v0|+…+|vn|

A diagram chase: Hk(X;Fp) Hk+1(X;Z(p))

q0 q1

BP⟨1⟩k+1+2p-1(X) (X)

q2

⋮ BP⟨n⟩k

qn +|v0|+…+|vn|

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

A diagram chase: Hk(X;Fp) Hk+1(X;Z(p))

q0 q1

BP⟨1⟩k+1+2p-1(X) (X)

q2

⋮ BP⟨n⟩k

qn +|v0|+…+|vn|

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1 BP⟨n+1⟩

Hk (X;Fp)

+|v0|+…+|vn+1| HFp Thom map

A diagram chase: Hk(X;Fp) Hk+1(X;Z(p))

q0 q1

BP⟨1⟩k+1+2p-1(X) (X)

q2

⋮ BP⟨n⟩k

qn +|v0|+…+|vn|

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1 BP⟨n+1⟩

Hk (X;Fp)

+|v0|+…+|vn+1| HFp Thom map Qn+1Qn…Q0

Lifting classes: We get

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

𝝱 𝜒(𝝱)

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

𝝱 𝜒(𝝱) (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

𝝱 𝜒(𝝱) (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

≠0 ≠0

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

𝝱 𝜒(𝝱) ✘ (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

≠0 ≠0

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

𝝱 𝜒(𝝱) ✘ (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

• If Qn+1..Q0(𝝱)≠0, then 𝜒(𝝱) is not algebraic.

≠0 ≠0

Lifting classes: We get Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒:=qn…q0

• a map 𝜒

𝝱 𝜒(𝝱) ✘ (X) BPk+|v0|+…+|vn|

Qn+1Qn…Q0

Hk (X;Fp)

+|v0|+…+|vn+1|

• an obstruction

(X) BP⟨n+1⟩k+|v0|+…+|vn+1|

qn+1

But we also pay a price…

the degree increases

• If Qn+1..Q0(𝝱)≠0, then 𝜒(𝝱) is not algebraic.

≠0 ≠0

Wilson’ s unstable splitting: The price is as little as possible.

Wilson’ s unstable splitting: The price is as little as possible. Theorem (Wilson): For any finite complex X, the map BPi(X) BP⟨n⟩i(X) is surjective if i≤2(pn+…+p+1).

Wilson’ s unstable splitting: The price is as little as possible. Theorem (Wilson): For any finite complex X, the map BPi(X) BP⟨n⟩i(X) is surjective if i≤2(pn+…+p+1). Recall |vn|=2pn-1, hence |v0|+…+|vn|=2(pn+…+1)-n-1.

Wilson’ s unstable splitting: The price is as little as possible. Theorem (Wilson): For any finite complex X, the map BPi(X) BP⟨n⟩i(X) is surjective if i≤2(pn+…+p+1). Recall |vn|=2pn-1, hence |v0|+…+|vn|=2(pn+…+1)-n-1. Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒=qn…q0

(X) BPk+|v0|+…+|vn|

Wilson’ s unstable splitting: The price is as little as possible. Theorem (Wilson): For any finite complex X, the map BPi(X) BP⟨n⟩i(X) is surjective if i≤2(pn+…+p+1). Recall |vn|=2pn-1, hence |v0|+…+|vn|=2(pn+…+1)-n-1. Hk(X;Fp) (X) BP⟨n⟩k+|v0|+…+|vn|

𝜒=qn…q0

(X) BPk+|v0|+…+|vn|

need to pick k≥n+3

✘

Examples of non-algebraic classes:

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof:

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof: Let Gk:=∏k(Z/p) (following Atiyah-Hirzebruch).

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof: Let Gk:=∏k(Z/p) (following Atiyah-Hirzebruch).

We know:

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof: Let Gk:=∏k(Z/p) (following Atiyah-Hirzebruch).

We know: • H*(BGk;Fp) = Fp[y1,…,yk]⊗𝚳(x1,…,xk);

|yi|=2 |xi|=1

Examples of non-algebraic classes: Theorem (Q.): For every n, there is a smooth proj. complex variety X and a class in BP⟨n⟩2*(X). BP⟨n⟩2*,*

mot(X)

which is not in the image of the map

2(pn+…+1)+2 (X)

BP⟨n⟩

• top. realization

Proof: Let Gk:=∏k(Z/p) (following Atiyah-Hirzebruch).

We know:

• Qj(xi)=yi , Qj(yi)=0.

pj

• H*(BGk;Fp) = Fp[y1,…,yk]⊗𝚳(x1,…,xk);

|yi|=2 |xi|=1

Proof continued: H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk);

Qj(xi)=yi , Qj(yi)=0.

pj

Proof continued: Choose: k=n+3 and 𝝱:=x1…xn+3 in Hn+3(BGk;Fp).

H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk); Qj(xi)=yi , Qj(yi)=0.

pj

Proof continued: Check: Qn+1…Q0(𝝱)≠0. Choose: k=n+3 and 𝝱:=x1…xn+3 in Hn+3(BGk;Fp).

H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk); Qj(xi)=yi , Qj(yi)=0.

pj

Proof continued: Check: Qn+1…Q0(𝝱)≠0. Choose: k=n+3 and 𝝱:=x1…xn+3 in Hn+3(BGk;Fp).

H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk); Qj(xi)=yi , Qj(yi)=0.

pj

is not in the image of the map

2(pn+…+1)+2 (BGn+3)

BP⟨n⟩ Hence x:=qn…q0(𝝱) in BP⟨n⟩ BP (BGn+3)

2(pn+…+1)+2

(BGn+3).

2(pn+…+1)+2

Proof continued: Check: Qn+1…Q0(𝝱)≠0. Choose: k=n+3 and 𝝱:=x1…xn+3 in Hn+3(BGk;Fp).

H*(BGk;Fp)= Fp[y1,…,yk]⊗𝚳(x1,…,xk); Qj(xi)=yi , Qj(yi)=0.

pj

is not in the image of the map

2(pn+…+1)+2 (BGn+3)

BP⟨n⟩ Hence x:=qn…q0(𝝱) in BP⟨n⟩ BP (BGn+3)

2(pn+…+1)+2

(BGn+3).

2(pn+…+1)+2

Finally, set X = Godeaux-Serre variety associated to the group Gn+3 and pullback x via X BGn+3 × CP∞.

a 2(pn+1+…+1)+1- connected map

□

Let’ s check the numbers:

Let’ s check the numbers: The minimal complex dimension of X is 2n+3-1 for p=2.

Let’ s check the numbers: The minimal complex dimension of X is 2n+3-1 for p=2. BP⟨1⟩ is 2-local connective complex K-theory ku. For n=1 and p=2:

Let’ s check the numbers: There is a smooth proj. variety X of dimension 15

• ver C with a non-algebraic class in BP⟨1⟩8(X).

The minimal complex dimension of X is 2n+3-1 for p=2. BP⟨1⟩ is 2-local connective complex K-theory ku. For n=1 and p=2:

Some final remarks:

Some final remarks:

• For n=0: the example of Atiyah and Hirzebruch.

Some final remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Other types of non-alg. classes in H*(X;Z):

Some final remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Other types of non-alg. classes in H*(X;Z):

･ Kollar: non-torsion classes on hypersurface in P4.

Some final remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Other types of non-alg. classes in H*(X;Z):

･ Kollar: non-torsion classes on hypersurface in P4. ･ Voisin: torsion classes based on Kollar’ s example.

Some final remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Other types of non-alg. classes in H*(X;Z):

･ Kollar: non-torsion classes on hypersurface in P4. ･ Voisin: torsion classes based on Kollar’ s example.

“not topological”

Some final remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Other types of non-alg. classes in H*(X;Z):

･ Kollar: non-torsion classes on hypersurface in P4. ･ Voisin: torsion classes based on Kollar’ s example.

“not topological”

･ Yagita, Pirutka-Yagita, Kameko, and others:

non-torsion class in H*(BG;Z) for alg. group G with (Z/p)3⊂G

Some final remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Other types of non-alg. classes in H*(X;Z):

･ Kollar: non-torsion classes on hypersurface in P4. ･ Voisin: torsion classes based on Kollar’ s example.

“not topological”

･ Yagita, Pirutka-Yagita, Kameko, and others:

non-torsion class in H*(BG;Z) for alg. group G with (Z/p)3⊂G

･ Antieau:

class in H*(BG;Z) for alg. group G, represent. theory

• n which the Qi’

s vanish, but a higher differential in the AH-spectral sequence is nontrivial.

Some final remarks:

• For n=0: the example of Atiyah and Hirzebruch.
• Other types of non-alg. classes in H*(X;Z):

･ Kollar: non-torsion classes on hypersurface in P4. ･ Voisin: torsion classes based on Kollar’ s example.

“not topological”

･ Yagita, Pirutka-Yagita, Kameko, and others:

non-torsion class in H*(BG;Z) for alg. group G with (Z/p)3⊂G

･ Antieau:

class in H*(BG;Z) for alg. group G, represent. theory

• n which the Qi’

s vanish, but a higher differential in the AH-spectral sequence is nontrivial. non-torsion classes in BP⟨n⟩*(BG)

Thank you!