On some invariants for algebraic cobordism cycles International Workshop on... · On some...

On some invariants for algebraic cobordism

cycles

International Workshop on Motives, Tokyo February 15, 2016

Gereon QuickNTNU

Point of departure: Poincaré, Lefschetz, Hodge…

Let X⊂PN be a smooth projective complex variety.



Let zj be a local coordinate on X.



Let zj be a local coordinate on X. Write zj = xj + iyj with real coordinates xj and yj.




There are associated differential forms

dzj = dxj + idyj and dzj = dxj - idyj.-




There are associated differential forms

dzj = dxj + idyj and dzj = dxj - idyj.-

Every k-form 𝛼 on X can then be written as

𝛼 = ∑j fj dzj1∧…∧dzjp∧dzjp+1∧…∧dzjk.- -

Integrals over cycles:

𝛼 = ∑j fj dzj1∧…∧dzjp∧dzj1∧…∧dzjq.- -

Let’s write 𝛼 ∈ Ap,q(X) if 𝛼 is of the form


exactly p dzj’s for all j




exactly p dzj’s for all j exactly q dzj’s for all j-







Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k.





Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k. We can integrate 𝛼 over 𝛤: ∫ 𝜄*𝛼. 𝛤





Let 𝜄: 𝛤 ⊂ X be a topological cycle on X of dimension k. We can integrate 𝛼 over 𝛤: ∫ 𝜄*𝛼. 𝛤

If 𝛤 = Z happens to be an algebraic subvariety of X, say of complex dimension n, then ∫ 𝜄*𝛼 = 0 unless 𝛼 lies in An,n(X). Z

Hodge’s question:

Hodge’s question:This imposes a necessary condition on a topological cycle 𝜄: 𝛤 ⊂ X to be algebraic (homologous to an algebraic subvariety Z dimension n):


𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤


𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤

Hodge wondered: Is this condition also sufficient? ? ⇐ ?


𝛤∼Z ⇒ ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X). 𝛤

Hodge wondered: Is this condition also sufficient? ? ⇐ ?

Usually, we ask this question in cohomological terms…

Hodge’s question in terms of cohomology:

Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Z

Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Zits Poincaré dual complex cohomology class is represented by a form in Ap,p(X) with p = dimX - n.

Hodge’s question in terms of cohomology:Since an algebraic subvariety 𝜄: Z ⊂ X of dimension n satisfies ∫ 𝜄*𝛼 = 0 if 𝛼 ∉ An,n(X), Zits Poincaré dual complex cohomology class is represented by a form in Ap,p(X) with p = dimX - n.Such classes are said to be “Hodge-type (p,p)”. The subgroup of such classes is denoted by

Hp,p(X) ⊂ H2p(X;C).


Hp,p(X) ⊂ H2p(X;C).The Hodge Conjecture: The map

clH: Zp(X) → Hp,p(X) ∩ H2p(X;Z) is surjective.


Hp,p(X) ⊂ H2p(X;C).The Hodge Conjecture: The map

clH: Zp(X) → Hp,p(X) ∩ H2p(X;Z) is surjective.⊗Q

switch to Q-coefficients

×

A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g.

A short digression: the Jacobian of a curveLet C be a smooth proj. complex curve of genus g. Let ω1, …, ωg be a basis of (holom.) 1-forms on C.


⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,

Then every pair of points p,q∈C defines a g-tuple of complex numbers


µ: Div0(C) → ℂg

⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,




⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,


group of formal sums ∑i(pi-qi)



⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,


group of formal sums ∑i(pi-qi) lattice of integrals

of ωj’s over loops

/Λ



⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,


group of formal sums ∑i(pi-qi) lattice of integrals

of ωj’s over loops

/Λ

Jacobian variety of C

=: J(C)



⌠⌡qω1

p⌠⌡qωg

p⎛⎝

⎞⎠, ...,


group of formal sums ∑i(pi-qi)

Jacobi Inversion Theorem: The map (Abel-Jacobi map)

is surjective.

lattice of integrals of ωj’s over loops

/Λ

Jacobian variety of C

=: J(C)

More canonically:

More canonically:

µ: Div0(C) → J(C) = H0(C; Ω1hol)∨/H1(C;Z)

⌠⌡qiω

pi

∑(pi-qi) ⟼ mod H1(C;Z).∑i⎝⎛ ⟼ ⎞

⎠ω

More canonically:

Abel’s Theorem: The kernel of this map is the subgroup of principal divisors.

µ: Div0(C) → J(C) = H0(C; Ω1hol)∨/H1(C;Z)

⌠⌡qiω

pi

∑(pi-qi) ⟼ mod H1(C;Z).∑i⎝⎛ ⟼ ⎞

⎠ω

Lefschetz’s proof for (1,1)-classes:

For simplicity, let X⊂PN be a surface.


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 J(Ct)





J := ⋃t J(Ct)

and a fibre space π: J → P1 (of complex Lie groups).





J := ⋃t J(Ct)

and a fibre space π: J → P1 (of complex Lie groups).

A “normal function” 𝜈 is a holomorphic section of π.


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).




Choose a point p0 on all Ct. Then ∑i pi(t) - dp0 is a divisor of degree 0 and defines a point

𝜈D(t) ∈ J(Ct).



Hence D defines a normal function

𝜈D: t ↦ 𝜈D(t) ∈ J.

Choose a point p0 on all Ct. Then ∑i pi(t) - dp0 is a divisor of degree 0 and defines a point

𝜈D(t) ∈ J(Ct).

Poincaré’s Existence Theorem:

Every normal function 𝜈 arises as the normal function 𝜈D associated to an algebraic curve D.




Then Lefschetz proved:



• Every normal function 𝜈 defines a class 𝜂(𝜈)∈H2(X;Z) of Hodge type (1,1) such that 𝜂(𝜈D) = clH(D).




• Every normal function 𝜈 defines a class 𝜂(𝜈)∈H2(X;Z) of Hodge type (1,1) such that 𝜂(𝜈D) = clH(D).

• Every class in H2(X;Z) of Hodge type (1,1) arises as 𝜂(𝜈) for some normal function 𝜈.


X a smooth projective complex variety with dimX=n.

Griffiths: in higher dimensions…


Z⊂X a closed subvariety of codimension p.





We assume that Z is the boundary of a differentiable chain Γ ∈ C2n-2p+1(X;Z).




Let ω be a smooth form of degree 2n-2p+1.




Then ⌠⌡Γω⟼ defines a map A2n-2p+1(X)→C.ω⎛⎝⎞⎠


Let ω be a smooth form of degree 2n-2p+1.


We would like this map to

Deligne, Griffiths, …:



• descend to a map on cohomology,




• and to become independent of the choice of Γ.





Let ω’ be a form such that dω’= ω.





Let ω’ be a form such that dω’= ω.

Then Stokes’ formula tells us

⌠⌡Γω ⌠

⌡Zω’=

⌠⌡Γω ⌠

⌡Zω’=

The intermediate Jacobian of Griffiths and the Abel-Jacobi map:

⌠⌡Γω ⌠

⌡Zω’=

If ω is a form of degree (p’,2n-2p+1-p’) with p’ ≥ n-p+1, this integral vanishes.


⌠⌡Γω ⌠

⌡Zω’=


We obtain a well-defined map⌠⌡Γ

⟼Z

µ: Zp(X)h → J2p-1(X) = Fn-p+1H2n-2p+1(X;C)∨/H2n-2p+1(X;Z)

for some Γ with Z=∂Γ


⌠⌡Γω ⌠

⌡Zω’=


We obtain a well-defined map⌠⌡Γ

⟼Z

µ: Zp(X)h → J2p-1(X) = Fn-p+1H2n-2p+1(X;C)∨/H2n-2p+1(X;Z)

for some Γ with Z=∂Γ

J2p-1(X) is a complex torus and is called Griffiths’ intermediate Jacobian.


The Jacobian and Griffiths’ theorem:


J2p-1(X) is, in general, not an abelian variety.



But it varies homomorphically in families.




Have an induced a map:

Griffp(X):= Zp(X)h/Zp(X)alg → J2p-1(X)/J2p-1(X)alg




Griffith’s theorem: Let X⊂P4 be a general quintic hypersurface. There are lines L and L’ on X such that µ(L-L’) is a non torsion element in J3(X).

Have an induced a map:

Griffp(X):= Zp(X)h/Zp(X)alg → J2p-1(X)/J2p-1(X)alg

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

An interesting diagram:

Zp(X)Let X be a smooth projective complex variety.


Zp(X) Z⊂XLet X be a smooth projective complex variety.


Zp(X)

clH

Z⊂XLet X be a smooth projective complex variety.


Hdg2p(X)

Zp(X)

clH

Z⊂XLet X be a smooth projective complex variety.


Hdg2p(X)

Zp(X)

clH

Z⊂X

[Zsm]

−

Let X be a smooth projective complex variety.


Hdg2p(X)

Zp(X)

clH

Zp(X)h=Kernel of clH ⊂ Z⊂X

[Zsm]

−



Hdg2p(X)

Zp(X)

clH

Zp(X)h=Kernel of clH ⊂

Abel-Jacobimap µ

Z⊂X

[Zsm]

−



Hdg2p(X)

Zp(X)

clH


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−



Hdg2p(X)

Zp(X)

clH


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−


→ HD (X;Z(p)) →2p


Hdg2p(X)

Zp(X)

clH


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−


→ HD (X;Z(p)) →2p

Deligne cohomology combines topological with Hodge theoretic information


Hdg2p(X)

Zp(X)

clH


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−


→ 0→ HD (X;Z(p)) →2p



Hdg2p(X)

Zp(X)

clH


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−


0 → → 0→ HD (X;Z(p)) →2p



Hdg2p(X)

Zp(X)

clHclHD


Abel-Jacobimap µ

J2p-1(X)

Z⊂X

[Zsm]

−


0 → → 0→ HD (X;Z(p)) →2p


Another interesting map for smooth complex varieties:

Φ: Ω*(X) → MU *(X)2


Φ: Ω*(X) → MU *(X)2


algebraic cobordismof Levine and Morel

Φ: Ω*(X) → MU *(X)2



complex cobordism ofthe top. space X(C)

Φ: Ω*(X) → MU *(X)2



complex cobordism ofthe top. space X(C)

Recall: MUi(X(C)) is (roughly speaking) generated by maps f: M→X(C) of codimension i modulo the relation g-1(0) ∼ g-1(1) for maps g: M’→X(C)xR such that g is transversal to e0: X(C)→X(C)xR and e1: X(C)→X(C)xR.

Algebraic cobordism in a nutshell:


Ωp(X) is generated (roughly speaking) by proj. maps f:Y→X of codimension p with Y smooth modulo …



Levine’s and Pandharipande’s “double point relation”:



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.

What can we say about the map Φ?

Ω*(X) MU2*(X)Φ


Ω*(X) MU2*(X)Φ[Y→X]


Ω*(X) MU2*(X)Φ[Y→X] [Y(C)→X(C)]⟼


• The image: Ω*(X) MU2*(X)Φ[Y→X] [Y(C)→X(C)]⟼


• The image:

CH*(X)

Ω*(X) MU2*(X)Φ[Y→X] [Y(C)→X(C)]⟼


• The image:

CH*(X)

Ω*(X) MU2*(X)Φ

Hdg2*(X) ⊆ H2*(X;Z)clH

[Y→X] [Y(C)→X(C)]⟼


• The image:

CH*(X)There is a “Hodge-theoretic” restriction for ImΦ.

Ω*(X) MU2*(X)Φ


[Y→X] [Y(C)→X(C)]⟼


• The image:

• The kernel:


Ω*(X) MU2*(X)Φ


[Y→X] [Y(C)→X(C)]⟼


• The image:

• The kernel:


Griffiths’ theorem suggests that Φ is not injective.

Ω*(X) MU2*(X)Φ


[Y→X] [Y(C)→X(C)]⟼


• The image:

• 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 KerΦ?

Ω*(X) MU2*(X)Φ


[Y→X] [Y(C)→X(C)]⟼

What can we say about ImΦ?

Ω*(X) → MU2*(X)


Ω*(X) → MU2*(X)HdgMU2*(X) ⊂


Ω*(X) → MU2*(X)HdgMU2*(X) ⊂

This map is not surjective, …


Ω*(X) → MU2*(X)

… but the classical topological arguments do not work!

HdgMU2*(X) ⊂

This map is not surjective, …

Ω*(X) MU2*(X)Φ

Atiyah-Hirzebruch and Totaro:

Ω*(X)⊗L*Z MU2*(X)⊗L*Z

Ω*(X) MU2*(X)Φ



Ω*(X) MU2*(X)Φ


CH*(X)


Ω*(X) MU2*(X)Φ


CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH


Ω*(X) MU2*(X)Φ


CH*(X) Hdg2*(X) ⊆ H2*(X;Z)clH

Totaro


Ω*(X) MU2*(X)Φ


CH*(X)

Levine-Morel ≈


Totaro


Ω*(X) MU2*(X)Φ


CH*(X)

Levine-Morel ≈ ≉ in general


Totaro


Ω*(X) MU2*(X)Φ


CH*(X)


Atiyah-Hirzebruch: clH is not surjective.


Totaro


Ω*(X) MU2*(X)Φ


CH*(X)


Atiyah-Hirzebruch: clH is not surjective.


This argument does not work for Φ.

Totaro

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


Let X⊂P4 a very general hypersurface of degree d.



Lefschetz’s hyperplane theorem:




• H2(X;Z) is torsion-free and generated by the class h=c1(𝓞X(1)) of a hyperplane section on X,




• H2(X;Z) is torsion-free and generated by the class h=c1(𝓞X(1)) of a hyperplane section on X,

• H4(X;Z) is torsion-free and generated by a class α such that ∫ α∙h=1, and all classes are Hodge classes.

X

Kollar’s examples: H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X

Kollar’s examples:

Assume now d=p3 for a prime p≥5.

H2(X;Z)=Zh, H4(X;Z)=Zα, ∫ α∙h=1X


Kollar: Then any algebraic curve C on X has degree divisible by p.







Consequence:





Consequence: Let C be a curve on X and [C]∈H4(X;Z) be its cohomology class.





Consequence: Let C be a curve on X and [C]∈H4(X;Z) be its cohomology class.

Then [C]= kα for some k and ∫ [C]∙h=k.X


Kollar: p divides the degree of any curve on X.





k = ∫ [C]∙hX




k = ∫ [C]∙hX

= number of intersection points of C with h




k = ∫ [C]∙hX


= degree of C




k = ∫ [C]∙hX


= degree of C ⟹ p divides k




k = ∫ [C]∙hX



In particular, α is not algebraic (for k cannot be 1).




k = ∫ [C]∙hX



In particular, α is not algebraic (for k cannot be 1).

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

Consequences 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).



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.

What about the kernel?

Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼

Φ(Y)=0 creates a new equivalence relation.


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼


top. cobordismrelation


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼


double point relation



Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼




⊆


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼




⊆ algebraic equivalence


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼





of A. Krishna and J. Park


Φ: Ω*(X) → MU *(X)2

[Y→X] [Y(C)→X(C)]⟼





⊆

of A. Krishna and J. Park

A new diagram:


A new diagram:

Ωp(X)


A new diagram:

Ωp(X) [Y→X]


A new diagram:

Ωp(X) [Y→X]


HdgMU(X)2p

A new diagram:

Ωp(X) [Y→X]

[Y(C)→X(C)]

−


HdgMU(X)2p

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


HdgMU(X)2p

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


MUD (p)(X) →2p HdgMU(X)2p

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


MUD (p)(X) →2p HdgMU(X)2p

combines topol. cobordism with Hodge theoretic information (jt. with M. Hopkins)

A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


MUD (p)(X) →2p HdgMU(X)2p → 0


A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


0 →JMU (X) →2p-1 MUD (p)(X) →2p HdgMU(X)2p → 0


A new diagram:

Ωp(X)

Φ

[Y→X]

[Y(C)→X(C)]

−


0 →JMU (X) →2p-1 MUD (p)(X) →2p HdgMU(X)2p

complex torus ≈ MU2p-1(X)⊗R/Z

→ 0


A new diagram:

Ωp(X)

Φ

Kernel of Φ ⊂ [Y→X]

[Y(C)→X(C)]

−




→ 0


A new diagram:

Ωp(X)

Φ

Kernel of Φ ⊂

“Abel-Jacobimap” µMU

[Y→X]

[Y(C)→X(C)]

−




→ 0


A new diagram:

Ωp(X)

ΦΦD

Kernel of Φ ⊂

“Abel-Jacobimap” µMU

[Y→X]

[Y(C)→X(C)]

−




→ 0


Examples:

The new Abel-Jacobi map is able to detect interesting algebraic cobordism classes:

Ωp(X)

MU2p(X)CHp(X)

ΦµMU

JMU (X)2p-1

Examples:


Ωp(X)

MU2p(X)CHp(X)

∃ α ∊

ΦµMU

JMU (X)2p-1

Examples:


Ωp(X)

MU2p(X)CHp(X)

∃ α ∊

0ΦµMU

JMU (X)2p-1

Examples:


Ωp(X)

MU2p(X)CHp(X)

∃ α ∊

0 0ΦµMU

JMU (X)2p-1

Examples:


Ωp(X)

MU2p(X)CHp(X)

∃ α ∊

0 0≠0ΦµMU

JMU (X)2p-1

The Abel-Jacobi map:


Given α ∈ Ωp(X), by Rost’s degree formula (due to Levine-Morel)

with Zi⊂X of codim ci≥0, Zi a res. of singularities and 𝛾i ∈ π2p+2 MU and

α = ∑i=1𝛾i[Zi→X] k

~~

ci





~~

ci

Assume Φ(α)=0. Then there are chains (modulo torsion)Γi ∈ C2n-2p+1-2ci(X;Z) such that





~~

ci

Assume Φ(α)=0. Then there are chains (modulo torsion)Γi ∈ C2n-2p+1-2ci(X;Z) such that

“Image of α = ∂(∑i=1𝛾iΓi) ∈ C2n-2p-2∗(X;Z)⊗π2∗MU”k


Taking integrals modulo MU2n-2p+1(X) yields a map

n = dim X

Fn-p+1H2n-2p+1(X;π2∗MU⊗C)∨ MU2n-2p+1(X)/Ker(Φ) →

α ⌠⌡Γi

⟼ mod MU2n-2p+1(X).

≈ JMU2p-1(X)

∑i=1 𝛾i k

Thank you!

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