On algebraic cycles with modulusmath.bu.edu/keio2019/talks/Miyazaki.pdf · 2019-07-05 · On...

On algebraic cycles with modulus

RIKEN iTHEMS

Hiroyasu MIYAZAKI

BU-KEIO Workshop 2019 June 25 2019, Boston University

Key words

Motives with modulus

Motives

Higher Chow group with modulus

Invariants

Geometric Objects Invariants

Invariant = essential aspect of shape

Number of Holes （genus）

１= =

Geometric Objects Invariants

Invariant = essential aspect of shape

Number of Holes （genus）

１= =

https://ja.wikipedia.org/wiki/位相幾何学

Geometric Objects Invariants

Invariant = essential aspect of shape

Number of Holes （genus）

2= =

https://en.wikipedia.org/wiki/Genus-two_surface

Geometric Objects Invariants

Invariant = essential aspect of shape

Number of Holes （genus）

3= =

Geometric Objects Invariants

Invariant = essential aspect of shape

Number of Holes （genus）

3= =

https://en.wikipedia.org/wiki/Genus_g_surface#Genus_3

Arithmetic Geometry

Zeros of polynomials are the main subjects

Real solution

Zeros of polynomials are the main subjects

Rational solution

Zeros of polynomials are the main subjects

Integer solution

Zeros of polynomials are the main subjects

Complex solution

1 point compactification

dimℝ = 2

∞

Polynomial

Complex points Real points Rational points

A space independent of the area of solutions

Polynomial

Algebraic Variety

Complex points Real points Rational points

A space independent of the area of solutions

Polynomial

Algebraic Variety

Complex points # of Rational points

?Genus #X(Q)

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Faltings’s theorem

（smooth projective）algebraic curve

（Also called Modell’s conjecture）

If genus (number of holes) is greater than 1 then there are only finitely many rational solutions

/ℚ

topological data

arithmetic data

mysterious relation between

Invariants

?

（Fermat’s conjecture）

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Motif

topological data

（genus etc）

analytic data

（zeta etc）

arithmetic data

（rat. points etc）

　Mysteriously related invariants.

Difficult to compare because they look too different

Alg var

de Rham cohomology （group）

crystalline cohomology （group）

étale cohomology （group）

topological data

（genus etc）

analytic data

（zeta etc）

arithmetic data

（rat. points etc）

Can compare Groups

Motive

A. Grothendieck

There should be a universal cohomology

de Rham cohomology （group）

crystalline cohomology （group）

étale cohomology （group）

https://en.wikipedia.org/wiki/Alexander_Grothendieck

Mixed Motive

A. Grothendieck

There should be a universal cohomology

de Rham cohomology （group）

crystalline cohomology （group）

étale cohomology （group）

V. Voevodsky

We can construct it

ℳ(X)https://en.wikipedia.org/wiki/Alexander_Grothendieck https://en.wikipedia.org/wiki/Vladimir_Voevodsky

étale cohomology

V. Voevodsky

New connection through motives

Milnor’s K-group

J. W. Milnor

Milnor’s conjecture (l=2) Bloch-Kato conjecture

Mixed Motiveℳ(X)

https://en.wikipedia.org/wiki/Vladimir_Voevodskyhttps://en.wikipedia.org/wiki/John_Milnor

Homotopy invariance

• Voevodsky’s construction of motives is abstract.

• For concrete applications, we must compute them.

The most important property of motives is

ℳ(X) ≅ ℳ(X × 𝔸1)

ℳ(X)X

𝔸1 is a replacement of [0,1]

Homotopy invariance (HI)

HI is strong!• It enables us to catch geometric information well.

• It makes motives computable.

HomDM(ℳ(X)[n], ℳ(Y)) ≃ CHdim Y(X × Y, n)

Th (Voevodsky)For any smooth X and Y,  we have

Higher Chow group (concrete group)

CHr(X, n) ≅ CHr(X × 𝔸1, n)

Higher Chow group satisfies HI, too (Bloch).

This is the most fundamental property.

What is higher Chow?

Singular (co)homology?

XContinuous maps

Topological Space

[0,1]

[0,1]2

[0,1]3

[0,1]0

Singular (co)homology?

XContinuous maps[0,1]n

:= ℤ {continuous maps from 𝔸n to X}Cn(X)

C2(X)

C3(X)

C1(X)

complex homology

Hn(X)

Higher Chow group?

Xmaps of varieties

Algebraic Variety

𝔸1

𝔸2

𝔸3

𝔸0 Much fewer than Continuous maps…

Doesn’t work.

Higher Chow group?

XAlgebraic

Algebraic Variety

𝔸1

𝔸2

𝔸3

Cycles

𝔸0

Higher Chow group?

XAlgebraic𝔸n

Cycles

:= ℤ {closed subvarieties of X × 𝔸n of codim r at good position}zr(X, n)

𝔸n

X

𝔸n

X

r = dim X

Graph

Not Graph

Higher Chow group?

XAlgebraic𝔸n

Cycles

:= ℤ {closed subvarieties of X × 𝔸n of codim r at good position}zr(X, n)

zr(X,2)

zr(X,3)

zr(X,1)

complex homology

CHr(X, n)

Back to the story

HI is strong!• It enables us to catch geometric information well.

• It makes motives computable.

HomDM(ℳ(X)[n], ℳ(Y)) ≃ CHdim Y(X × Y, n)

Th (Voevodsky)For any smooth X and Y,  we have

Higher Chow group is connected to many other invariants.

HI is too strong!• It disables us from catching arithmetic information.

πab1 (X)X

Arithmetic fundamental group (knows ramifications)

πab1 (X) does not satisfy homotopy invariance.

E.g.

It cannot be captured by motives!

We have to generalize motives.

How?

We have to generalize motives.

HomDM(ℳ(X)[n], ℳ(Y)) ≃ CHdim Y(X × Y, n)Higher Chow group (’86, Bloch)Motives (’00 Voevodsky)

Abstract side Concrete side

We have to generalize motives.

HomDM(ℳ(X)[n], ℳ(Y)) ≃ CHdim Y(X × Y, n)Binda-Saito

(2014)generalized

Higher Chow group with modulus

Abstract side Concrete side

Higher Chow group (’86, Bloch)Motives (’00 Voevodsky)

We have to generalize motives.

HomDM(ℳ(X)[n], ℳ(Y)) ≃ CHdim Y(X × Y, n)Higher Chow group

Kahn-M. -Saito-Yamazaki

(upcoming)generalized generalized

Motives

Higher Chow group with modulus

Motives with modulus

Abstract side Concrete side

Binda-Saito (2014)

We have to generalize motives.

HomDM(ℳ(X)[n], ℳ(Y)) ≃ CHdim Y(X × Y, n)Higher Chow group

Higher Chow group with modulus

Kahn-M. -Saito-Yamazaki

(upcoming)

Motives with modulus

generalized generalized

Abstract side Concrete side

Motives

First thing to study!

Binda-Saito (2014)

Higher Chow group with modulus

Basic idea is to replace X with

“Pair of spaces”

Precisely, D is a Cartier divisor on X .

dim X = 1

dim X = 2

𝒳 = (X, D)

Basic idea is to replace X with

𝒳 = (X, D) “Pair of spaces”

Precisely, D is a Cartier divisor on X .

CHr(𝒳, n) = CHr(X, D, n)Higher Chow group with modulus

CHr(𝒳, n) = CHr(X, D, n)

E.g.

CHr(X, ∅, n) = CHr(X, n)

CHr(X × 𝔸1, m(X × {0}), n) = TCHr(X, n; m)Additive higher Chow group (Bloch-Esnault, Park)

Computes de Rham-Witt complex W⇤⌦•

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non-HI

πab1 (X)geo ≅ lim←

m≥1

CHdim X(X, mD,0)deg=0

Th (Kerz-Saito)For any X smooth over a finite field, we have

where X ⊂ X is a compactification s.t. D = X∖X is Cartier.

Chow with modulus captures

ramifications!

• Higher Chow group with modulus (CHM) is good.

• But it does not satisfy homotopy invariance.

•

• Higher Chow group with modulus (CHM) is good.

• But it does not satisfy homotopy invariance.

• Q1. How far is CHM from HI?

• Q2. How does CHM depend on multiplicities of D?

• Q3. Is there a generalization of HI for CHM?

Main Results

(J. Algebraic Geometry 28 (2019) 339-390)

“Cube invariance of higher Chow groups with modulus”

Cor

CHr(𝒳, n) ⊗ ℤ[1/p] ≅ CHr(𝒳 × 𝔸1, n) ⊗ ℤ[1/p]

How far is CHM from HI ?

CHr(𝒳 × 𝔸1, n) ≅ CHr(𝒳, n) ⊕ NCHr(𝒳, n)

Th (M.)1) There exists a canonical splitting

2) NCHr(𝒳, n) is a p-group when ch(k) = p > 0

Obstruction to HI

“Non-HI part” is p-primary torsion

𝒳 × 𝔸1 := (X × 𝔸1, D × 𝔸1)

(Known by Binda-Cao-Kai-Sugiyama for r = dim X, n = 0, X proper) .

How does CHM depend on D ?

If ch(k) = p > 0

Th (M.)

CHr(X, D, n) ⊗ ℤ[1/p] ≅ CHr(X, mD, n) ⊗ ℤ[1/p] ∀m ≥ 1

Only p-primary torsion part depends on multiplicity of D

Remark: ∃ similar results in charactetristic 0.

(Known by Binda-Cao-Kai-Sugiyama for r = dim X, n = 0, X proper) .

Generalization of HI ?Th (M.)

CHr(𝒳, n) ≅ CHr(𝒳 × □∨ , n)

Remark:  Motives with modulus satisfy the same property.

For any 𝒳 = (X, D),  we have a caonical isomorphsim

where □∨ = (ℙ1, − ∞) .

If D = ∅,  then this coincides with HI of higher Chow group.

This suggest the connection between MwM and CHM.

Future?

HomDM(ℳ(X)[n], ℳ(Y)) ≃ CHdim Y(X × Y, n)Higher Chow group

Higher Chow group with modulus

Kahn-M. -Saito-Yamazaki

(upcoming)

Motives with modulus

generalized generalized

Abstract side Concrete side

Motives

Binda-Saito (2014)

Future?

HomDM(ℳ(X)[n], ℳ(Y)) ≃ CHdim Y(X × Y, n)Higher Chow group

Higher Chow group with modulus

Kahn-M. -Saito-Yamazaki

(upcoming)

Motives with modulus

generalized generalized

Abstract side Concrete side

Motives

Binda-Saito (2014)

≃???

→ Better control of π1, K-group etc.

Thank you very much!