On algebraic cycles with modulus RIKEN iTHEMS Hiroyasu MIYAZAKI BU-KEIO Workshop 2019 June 25 2019, Boston University
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)
<latexit sha1_base64="dzNY+pxlkj/RBqRMZBOJWL05NVg=">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</latexit>
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)
X(Q) = {trivial solutions}<latexit sha1_base64="fXMoJmgon2JBbbSaFNvHz9xDNI0=">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</latexit>
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⇤⌦•
<latexit sha1_base64="wvJ7mb9u25CHRSd9C78Nl8spBCY=">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</latexit>
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!