𝜍 Sm C Man C Topological realization: X(C) X ”complex e.g. P 1 P 1 (C)=CP 1 ≃ S 2 manifold of motivic induced map solutions in C” top. real. spectrum 𝜍 E E a,b (X) E a (X(C)) mot top “algebraic” “topological” Questions: How can we detect whether classes
𝜍 Sm C Man C Topological realization: X(C) X ”complex e.g. P 1 P 1 (C)=CP 1 ≃ S 2 manifold of motivic induced map solutions in C” top. real. spectrum 𝜍 E E a,b (X) E a (X(C)) mot top “algebraic” “topological” Questions: How can we detect whether classes • in E* , *(X) are topologically trivial, mot i.e., become 0 in E* (X(C))? top
𝜍 Sm C Man C Topological realization: X(C) X ”complex e.g. P 1 P 1 (C)=CP 1 ≃ S 2 manifold of motivic induced map solutions in C” top. real. spectrum 𝜍 E E a,b (X) E a (X(C)) mot top “algebraic” “topological” Questions: How can we detect whether classes • in E* , *(X) are topologically trivial, mot i.e., become 0 in E* (X(C))? top • in are algebraic, E* (X(C)) top i.e., are in the image of 𝜍 E ?
Atiyah-Hirzebruch, Totaro, Levine-Morel: 𝜍 H =cl H CH*(X)=H 2 * , *(X;Z) Alg 2 *(X) ⊆ H 2 *(X;Z) mot H
Atiyah-Hirzebruch, Totaro, Levine-Morel: universal oriented 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) theories 𝜍 H =cl H CH*(X)=H 2 * , *(X;Z) Alg 2 *(X) ⊆ H 2 *(X;Z) mot H
Atiyah-Hirzebruch, Totaro, Levine-Morel: universal oriented 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) theories coeff. ring = Lazard MGL 2 * , *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z ring 𝜍 H =cl H CH*(X)=H 2 * , *(X;Z) Alg 2 *(X) ⊆ H 2 *(X;Z) mot H
Atiyah-Hirzebruch, Totaro, Levine-Morel: universal oriented 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) theories coeff. ring = Lazard MGL 2 * , *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z ring Totaro ⟳ 𝜍 H =cl H CH*(X)=H 2 * , *(X;Z) Alg 2 *(X) ⊆ H 2 *(X;Z) mot H
Atiyah-Hirzebruch, Totaro, Levine-Morel: universal oriented 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) theories coeff. ring = Lazard MGL 2 * , *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z ring Levine + Totaro Levine-Morel ≈ ⟳ 𝜍 H =cl H CH*(X)=H 2 * , *(X;Z) Alg 2 *(X) ⊆ H 2 *(X;Z) mot H
Atiyah-Hirzebruch, Totaro, Levine-Morel: universal oriented 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) theories coeff. ring = Lazard MGL 2 * , *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z ring Levine + ≉ in general Totaro Levine-Morel ≈ ⟳ 𝜍 H =cl H CH*(X)=H 2 * , *(X;Z) Alg 2 *(X) ⊆ H 2 *(X;Z) mot H
Atiyah-Hirzebruch, Totaro, Levine-Morel: universal oriented 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) theories coeff. ring = Lazard MGL 2 * , *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z ring Levine + ≉ in general Totaro Levine-Morel ≈ ⟳ 𝜍 H =cl H CH*(X)=H 2 * , *(X;Z) Alg 2 *(X) ⊆ H 2 *(X;Z) mot H • Atiyah-Hirzebruch: cl H is not surjective onto integral Hodge classes.
Atiyah-Hirzebruch, Totaro, Levine-Morel: universal oriented 𝜍 MGL MGL 2 * , *(X) MU 2 *(X) theories coeff. ring = Lazard MGL 2 * , *(X) ⊗ L* Z MU 2 *(X) ⊗ L* Z ring Levine + ≉ in general Totaro Levine-Morel ≈ ⟳ 𝜍 H =cl H CH*(X)=H 2 * , *(X;Z) Alg 2 *(X) ⊆ H 2 *(X;Z) mot H • Atiyah-Hirzebruch: cl H is not surjective onto integral Hodge classes. • Totaro: new classes in kernel of cl H .
I. Kernel: X smooth projective
I. Kernel: X smooth projective Recall Deligne’ s diagram 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D
I. Kernel: X smooth projective Recall Deligne’ s diagram 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Hodge classes
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Hodge classes
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Griffiths’ Hodge Jacobian classes
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology CH p (X) cl HD cl H 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Griffiths’ Hodge Jacobian classes
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology Kernel of cl H ⊂ CH p (X) Abel- cl HD µ H cl H Jacobi map 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Griffiths’ Hodge Jacobian classes
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology Kernel of cl H ⊂ CH p (X) Abel- cl HD µ H cl H Jacobi map 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Griffiths’ Hodge Jacobian classes Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins):
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology Kernel of cl H ⊂ CH p (X) Abel- cl HD µ H cl H Jacobi map 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Griffiths’ Hodge Jacobian classes Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins): 0 → J 2p-1 (X) → MU 2p (X;Z(p)) → Hdg 2p (X) → 0 D MU MU
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology Kernel of cl H ⊂ CH p (X) Abel- cl HD µ H cl H Jacobi map 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Griffiths’ Hodge Jacobian classes Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins): MU-Hodge classes 0 → J 2p-1 (X) → MU 2p (X;Z(p)) → Hdg 2p (X) → 0 D MU MU
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology Kernel of cl H ⊂ CH p (X) Abel- cl HD µ H cl H Jacobi map 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Griffiths’ Hodge Jacobian classes Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins): MU-Hodge classes 0 → J 2p-1 (X) → MU 2p (X;Z(p)) → Hdg 2p (X) → 0 D MU MU ”Hodge filtered complex cobordism”
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology Kernel of cl H ⊂ CH p (X) Abel- cl HD µ H cl H Jacobi map 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Griffiths’ Hodge Jacobian classes Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins): MU-Hodge classes 0 → J 2p-1 (X) → MU 2p (X;Z(p)) → Hdg 2p (X) → 0 D MU MU ”Hodge filtered MU-“Jacobian” complex cobordism”
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology Kernel of cl H ⊂ CH p (X) Abel- cl HD µ H cl H Jacobi map 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Griffiths’ Hodge Jacobian classes Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins): MU-Hodge MGL 2p,p (X) classes 𝜍 MU 𝜍 MUD 0 → J 2p-1 (X) → MU 2p (X;Z(p)) → Hdg 2p (X) → 0 D MU MU ”Hodge filtered MU-“Jacobian” complex cobordism”
I. Kernel: X smooth projective Deligne Recall Deligne’ s diagram cohomology Kernel of cl H ⊂ CH p (X) Abel- cl HD µ H cl H Jacobi map 0 → J 2p-1 (X) → H 2p (X;Z(p)) → Hdg 2p (X) → 0 D Griffiths’ Hodge Jacobian classes Generalized Hodge filtered cohomology theories (joint work with Mike Hopkins): MU-Hodge ”Abel- Kernel of 𝜍 MU ⊂ MGL 2p,p (X) classes Jacobi 𝜍 MU 𝜍 MUD µ MU map” 0 → J 2p-1 (X) → MU 2p (X;Z(p)) → Hdg 2p (X) → 0 D MU MU ”Hodge filtered MU-“Jacobian” complex cobordism”
Examples: The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes: MGL 2p,p (X) motivic Thom map top. realization Abel-Jacobi map for MU 2p,p 2p-1 H mot (X;Z) J MU (X) MU 2p (X(C))
Examples: The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes: there is an α ∊ MGL 2p,p (X) motivic Thom map top. realization Abel-Jacobi map for MU 2p,p 2p-1 H mot (X;Z) J MU (X) MU 2p (X(C))
Examples: The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes: there is an α ∊ MGL 2p,p (X) motivic Thom map top. realization Abel-Jacobi map for MU hence 0 0 in J 2p-1 (X) 2p,p 2p-1 H mot (X;Z) J MU (X) MU 2p (X(C))
Examples: The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes: there is an α ∊ MGL 2p,p (X) motivic Thom map top. realization Abel-Jacobi map for MU hence 0 0 0 in J 2p-1 (X) 2p,p 2p-1 H mot (X;Z) J MU (X) MU 2p (X(C))
Examples: The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes: there is an α ∊ MGL 2p,p (X) motivic Thom map top. realization Abel-Jacobi map for MU hence 0 0 ≠ 0 0 in J 2p-1 (X) 2p,p 2p-1 H mot (X;Z) J MU (X) MU 2p (X(C))
In concrete terms: Given p and X smooth projective. 2p Elements in MU D (p)(X) consist of triples (f, h, ω ):
In concrete terms: Given p and X smooth projective. 2p Elements in MU D (p)(X) consist of triples (f, h, ω ): (almost) complex manifold ∋ f : Y → X MU 2p (X) proper
In concrete terms: Given p and X smooth projective. 2p Elements in MU D (p)(X) consist of triples (f, h, ω ): (almost) complex manifold ∋ f : Y → X MU 2p (X) proper ω ∈ F p Ω *(X;V*) 2p cl closed forms of total degree 2p V*=MU* ⊗ C
In concrete terms: Given p and X smooth projective. 2p Elements in MU D (p)(X) consist of triples (f, h, ω ): (almost) complex manifold ∋ f : Y → X MU 2p (X) proper ω ∈ F p Ω *(X;V*) 2p cl closed forms of C*(X;V*) 2p-1 ∋ h total degree 2p V*=MU* ⊗ C
In concrete terms: Given p and X smooth projective. 2p Elements in MU D (p)(X) consist of triples (f, h, ω ): (almost) complex manifold ∋ f : Y → X MU 2p (X) proper ω ∈ F p Ω *(X;V*) 2p Z*(X;V*) 2p cocycles of cl total degree 2p closed forms of C*(X;V*) 2p-1 ∋ h total degree 2p V*=MU* ⊗ C
In concrete terms: Given p and X smooth projective. 2p Elements in MU D (p)(X) consist of triples (f, h, ω ): (almost) complex manifold ∋ f : Y → X MU 2p (X) proper ω ∈ F p Ω *(X;V*) 2p Z*(X;V*) 2p cocycles of cl total degree 2p f closed forms of C*(X;V*) 2p-1 ∋ h total degree 2p V*=MU* ⊗ C
In concrete terms: Given p and X smooth projective. 2p Elements in MU D (p)(X) consist of triples (f, h, ω ): (almost) complex manifold ∋ f : Y → X MU 2p (X) proper ω ∈ F p Ω *(X;V*) 2p Z*(X;V*) 2p cocycles of cl total degree 2p ω f closed forms of C*(X;V*) 2p-1 ∋ h total degree 2p V*=MU* ⊗ C
In concrete terms: Given p and X smooth projective. 2p Elements in MU D (p)(X) consist of triples (f, h, ω ): (almost) complex manifold ∋ f : Y → X MU 2p (X) proper ω ∈ F p Ω *(X;V*) 2p Z*(X;V*) 2p cocycles of cl total degree 2p ω - f = ∂ h closed forms of C*(X;V*) 2p-1 ∋ h total degree 2p V*=MU* ⊗ C
In concrete terms: Given p and X smooth projective. 2p Elements in MU D (p)(X) consist of triples (f, h, ω ): (almost) complex “f ∗ of universal genus of manifold curvature form” of normal ∋ f : Y → X MU 2p (X) bundle of Y if Y is a proper smooth projective variety ω ∈ F p Ω *(X;V*) 2p Z*(X;V*) 2p cocycles of cl total degree 2p ω - f = ∂ h closed forms of C*(X;V*) 2p-1 ∋ h total degree 2p V*=MU* ⊗ C
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 S be a scheme of finite type over Z, and let 𝜃 be the generic point. MGL S → 𝜃 ( MU D ) *
Arakelov algebraic cobordism: Let S be a scheme of finite type over Z, and let 𝜃 be the generic point. MGL Arakelov → MGL S → 𝜃 ( MU D ) * homotopy fibre represents “Arakelov algebraic cobordism”
Arakelov algebraic cobordism: Let S be a scheme of finite type over Z, and let 𝜃 be the generic point. MGL Arakelov → MGL S → 𝜃 ( MU D ) * homotopy fibre represents “Arakelov algebraic cobordism” Question: What is the arithmetic-geometric information encoded in the Chern classes in Arakelov algebraic cobordism?
𝜍 Sm Man Recall: II. Image: X(C) X manifold of solutions in C motivic induced map spectrum 𝜍 E E a,b (X) E a (X(C)) mot top algebraic topological
𝜍 Sm Man Recall: II. Image: X(C) X manifold of solutions in C motivic induced map spectrum 𝜍 E E a,b (X) E a (X(C)) mot top algebraic topological Question: • How can we detect whether classes in E* (X(C)) are algebraic, i.e., top are in the image of 𝜍 E ?
𝜍 Sm Man Recall: II. Image: X(C) X manifold of solutions in C motivic induced map spectrum 𝜍 E E a,b (X) E a (X(C)) mot top algebraic topological Question: • How can we detect whether classes in E* (X(C)) are algebraic, i.e., not top not are in the image of 𝜍 E ?
𝜍 Sm Man Recall: II. Image: X(C) X manifold of solutions in C motivic induced map spectrum 𝜍 E E a,b (X) E a (X(C)) mot top algebraic topological Question: • How can we detect whether classes in E* (X(C)) are algebraic, i.e., not top not are in the image of 𝜍 E ? • How can we construct such classes?
A different perspective: Fix a prime p.
A different perspective: Brown-Peterson, Fix a prime p. Quillen |v i |=2(p i -1) MU (p) splits as a wedge of suspensions of spectra BP with BP = Z (p) [v 1 ,v 2 ,…]. *
A different perspective: Brown-Peterson, Fix a prime p. Quillen |v i |=2(p i -1) MU (p) splits as a wedge of suspensions of spectra BP with BP = Z (p) [v 1 ,v 2 ,…]. * quotient map BP BP/(v n+1 ,…) =: BP ⟨ n ⟩ For every n: with BP ⟨ n ⟩ = Z (p) [v 1 ,…,v n ] *
A different perspective: Brown-Peterson, Fix a prime p. Quillen |v i |=2(p i -1) MU (p) splits as a wedge of suspensions of spectra BP with BP = Z (p) [v 1 ,v 2 ,…]. * quotient map BP BP/(v n+1 ,…) =: BP ⟨ n ⟩ For every n: with BP ⟨ n ⟩ = Z (p) [v 1 ,…,v n ] * The Brown-Peterson tower (Wilson): … … BP BP ⟨ n ⟩ BP ⟨ 1 ⟩ BP ⟨ 0 ⟩ BP ⟨ -1 ⟩ p=2: 2-local HZ (p) HF p connective K-theory
Milnor operations:
Milnor operations: For every n: stable cofibre sequence v n |v n | |v n |+1 ∑ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ ∑
Milnor operations: For every n: stable cofibre sequence v n |v n | |v n |+1 ∑ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ ∑ with an induced exact sequence (for any space X) +|v n | BP ⟨ n ⟩ * (X) BP ⟨ n ⟩ *(X) q n BP ⟨ n ⟩ * +|v n |+1 (X) BP ⟨ n-1 ⟩ *(X)
Milnor operations: For every n: stable cofibre sequence v n |v n | |v n |+1 ∑ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ ∑ with an induced exact sequence (for any space X) +|v n | BP ⟨ n ⟩ * (X) BP ⟨ n ⟩ *(X) q n BP ⟨ n ⟩ * +|v n |+1 (X) BP ⟨ n-1 ⟩ *(X) BP ⟨ n-1 ⟩ Thom map +|v n |+1 (X;F p ) HF p H*(X;F p ) H* Q n
Milnor operations: For every n: stable cofibre sequence v n |v n | |v n |+1 ∑ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n ⟩ BP ⟨ n-1 ⟩ ∑ with an induced exact sequence (for any space X) +|v n | BP ⟨ n ⟩ * (X) BP ⟨ n ⟩ *(X) q n BP ⟨ n ⟩ * +|v n |+1 (X) BP ⟨ n-1 ⟩ *(X) BP ⟨ n-1 ⟩ nth Milnor Thom operation: map Q 0 =Bockstein p n-1 p n-1 +|v n |+1 (X;F p ) Q n =P Q n-1 -Q n-1 P HF p H*(X;F p ) H* Q n
The LMT obstruction in action: BP 2 *(X) ⟲ q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) ↺ +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Q n
The LMT obstruction in action: BP 2 *(X) ⟲ q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) ↺ +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Q n 𝝱
The LMT obstruction in action: BP 2 *(X) ⟲ q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) ↺ +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * Q n 𝝱 Question: Is 𝝱 algebraic?
The LMT obstruction in action: BP 2 *(X) ⟲ q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) ↺ +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * CH*(X) Q n 𝝱 Question: Is 𝝱 algebraic?
The LMT obstruction in action: BP 2 *(X) LMT ⟲ q n BP ⟨ n ⟩ 2 * +|v n |+1 (X) BP ⟨ n ⟩ 2 *(X) BP ⟨ n-1 ⟩ 2 *(X) ↺ +|v n |+1 (X;F p ) H 2 *(X;F p ) H 2 * CH*(X) Q n 𝝱 Question: Is 𝝱 algebraic?
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