mixed hodge structures with modulusmap of this talk our goal. to generalize deligne’s category mhs...
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Mixed Hodge structures with modulus
Joint work with Florian Ivorra
Takao Yamazaki (Tohoku University)
June 28, 2018, Singapore
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Map of this talk
Our goal. To generalize Deligne’s category MHS of mixedHodge structures to that of MHS with modulus MHSM;
“level ≤ 1 parts” (indicated by 1) admit geometric description:
(semi-abel.) ⊂ (Deligne 1-motives) MHS1
⊃ ⊃ ⊃
(comm. alg. gp.) ⊂ (Laumon 1-motives) MHSM1
2/3 of this talk is devoted to “level ≤ 1 parts”, previouslyknown by Barbieri-Viale, Kato, Russell in different languages.
We then explain the whole category MHSM, and itsapplication to generalize Kato-Russell’s construction ofAlbanese varieties with modulus to 1-motives.
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Commutative algebraic groupsA commutative algebraic group G over C is an extension
0 → Gmul × Gadd → G → Gab → 0,
Gab : abel. var., Gmul Gsm , Gadd Gt
a (s , t ∈ Z≥0).
G is called semi-abelian if Gadd = 0.
AV = ab. var. ⊂ SA = semi-ab. ⊂ AG = alg. gp..
X : cpt. Riemann surface, Y : effective divisor on X(Y = a1P1 + · · · + ar Pr ; Pi ∈ X distinct, ai ∈ Z>0).Generalized Jacobian Jac(X , Y ) ∈ AG classifies pairs(L , σ) of a degee zero line b’dl. L on X and σ : L |Y OY .
• Y = ∅ ⇒ Jac(X , ∅) = Jac(X ) ∈ AV;
• Y : reduced (i.e. a1 = · · · = ar = 1)⇒ Jac(X , Y ) ∈ SA.
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DualityA ∈ AV ⇒ ∃A ∨ ∈ AV : dual of A ; Jac(X )∨ Jac(X ).
To extend ∨ to AG, we need Laumon 1-motives:Def. A Laumon 1-motive M is a two-term cpx. of the form
(∗) M = [Zs × Gta → G] (s , t ∈ Z≥0, G ∈ AG).
Regard G ∈ AG as a Laumon 1-motive by [0 → G].M in (∗) is called a Deligne 1-motive if t = 0 and G ∈ SA.M D
1= Deligne 1-motives ⊂M L
1= Laumon 1-motives
Generalized Jacobian can be generalized to 1-motives:Attach Jac(X , Y , Z) ∈M L
1to a triple (X , Y , Z) of X : cpt.
Riemann surface and Y , Z : eff. divisors s.t. |Y | ∩ |Z | = ∅;
Jac(X , Y , ∅) = Jac(X , Y ), Jac(X , Y , Z)∨ Jac(X , Z , Y ).
This is best explained from the viewpoint of Hodge theory.
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AV and HS 1
Recall. ∃ equiv. of cat. AV HS1 : Hodge str. of level ≤ 1.
Def. A Hodge structure of level ≤ 1 is a pair H = (HZ, F0) of
a) HZ : free Z-module of finite rank,
b) F0 ⊂ HC := HZ ⊗ C : C-subspace;
subject to conditions (HC = F0 ⊕ F0 and “polarizable”).
• Jac(X ) ↔ H1(X ,Z) for X : cpt. Riemann surf.
• ∨ : AV → AV corresponds to an easy linear algebraoperation Hom(−,Z(1)) on HS1.
• J (X )∨ J (X ) explained by Poincaré duality.
The equivalence AV HS1 is extended to M D1
by Deligne.
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M D1
and MHS1
Thm. (Deligne). ∃ equiv. of cat. M D1 MHS1.
Def. A mixed HS of level ≤ 1 H = (HZ, F0,W−1,W−2) is:
a) HZ : free Z-module of finite rank,
b) F0 ⊂ HC := HZ ⊗ C : C-subspace,
c) W−2 ⊂ W−1 ⊂ HZ : Z-submodules,
s.t. HZ/W−1 Zs ,W−1/W−2 ∈ HS1,W−2 Z(1)t (s , t ∈ Z≥0)
• (X , Y , Z) as above, with Y , Z reduced⇒Jac(X , Y , Z) ↔ H1(X \ Y , Z ,Z) : relative homology.
• ∨ : M D1→M D
1↔ an easy lin. alg. operation on MHS1.
• Poincaré duality explains Jac(X , Y , Z)∨ Jac(X , Z , Y ).
The equivalence M D1 MHS1 is extended to M L
1.
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M L1
and MHSM1
At least 3 known categories: M L1 FHS1 H1 MHSM1.
• FHS1: formal HS (Barbieri-Viale, 2007)• H1 : MHS with additive part (Kato and Russell, 2012)• MHSM1: MHS with modulus (Ivorra and Y.).
Def. A MHSM of level ≤ 1 H = (H,U, V ,F ) consists of:
a) H = (HZ, F0,W−1,W−2) ∈ MHS1.
b) U, V : finite dim. C-vector sp.
c) F ⊂ HC ⊕ U ⊕ V : C-subspace.
satisfying certain conditions (details come later).
If [Zs × Gta → G] ↔ (H,U, V ,F ) by M L
1 MHSM1,
[Zs → G/Gadd] ↔ H by M D1 MHS1,
U Lie(Gadd), V Lie(Gta).
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Jacobian 1-motives
For (X , Y , Z) as above, Y , Z not. nec. reduced,Jac(X , Y , Z) ↔ (H,U, V ,F ) under M L
1 MHSM1 with
• H = H1(X \ Yred , Zred ,Z) ∈ MHS1 (Deligne),
• U = H0(X ,OX (−Yred )/OX (−Y )).
• V = H0(X ,OX (Z − Zred )/OX ).
• F := Im(H0(X ,Ω1X
(Z)) → H1(X , [OX (−Y ) → Ω1X
(Z)])) HC ⊕ U ⊕ V . (This map tuns out to be injective.)
N.B. Y , Z : reduced⇒ U = V = 0, i.e. Jac(X , Y , Z) ∈M D1
.
∨ : M L1→M L
1↔ an easy lin. alg. operation on MHSM1.
Poincaré and Serre dualities imply
Jac(X , Y , Z)∨ Jac(X , Z , Y ).
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MHS of arbitrary level
MHS : Deligne’s cat. of mixed Hodge structures.
Formal property. MHS is abelian; it contains MHS1.
Geometry. X : smooth proper variety of dimension d ,Y , Z ⊂ X : effective reduced divisors, |Y | ∩ |Z | = ∅.∃Hn(X , Y , Z) ∈ MHS s.t. HZ = Hn(X \ Z , Y ,Z) (n ∈ Z).
Duality. Hn(X , Y , Z)∨ H2d−n(X , Z , Y )(d )/(tor).
Albanese. One has H2d−1(X , Y , Z)(d )/(tor)∈ MHS1, and itcorresponds to the Albanese 1-motive Alb (X , Y , Z) ∈M D
1via MHS1 M D
1. [d = 1 ⇒ Alb (X , Y , Z) = Jac(X , Y , Z).]
NB. Deligne constructed Hn(S) ∈ MHS for any variety S .One has H2d−1(S)(d )/(torsion)∈ MHS1 if d = dim S , and itcorresponds to the Albanese 1-motive Alb (S) ∈M D
1.
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MHSM of arbitrary level
Def. A MHS with modulus is H = (H,U•, V•, F p p∈Z),a) H ∈ MHS,
b) · · · → Up up
→ Up−1 up−1
→ · · · : chain of C-linear maps,
c) · · · → Vp v p
→ Vp−1 v p−1
→ · · · : chain of C-linear maps,
d) F p ⊂ HC ⊕ Up ⊕ Vp : C-subspaces (∀p ∈ Z),
subject to the following conditions:
• dim C[⊕p (Up ⊕ Vp )] < ∞.
• (idHC ⊕ up ⊕ v p )(F p ) ⊂ F p−1.
• For x ∈ HC: x ∈ Fp HC ⇔ ∃u ∈ Up s.t. x + u ∈ F p .
• F p → HC ⊕ Up ⊕ Vp ↠ Vp : surj.
• Up → HC ⊕ Up ⊕ Vp ↠ HC ⊕ Up ⊕ Vp/F p : inj.
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Main results
Formal property. MHSM is abelian; it contains MHSM1.
Geometry. X : smooth proper variety of dimension d .Y , Z ⊂ X : eff. divisors, not nec. reduced, |Y | ∩ |Z | = ∅,(Y + Z)red : strict normal crossing. n ∈ Z.∃H n(X , Y , Z) ∈ MHSM s.t. H = Hn(X , Yred , Zred ) ∈ MHS.
Duality. H n(X , Y , Z)∨ H 2d−n(X , Z , Y )(d )/(tor).
Albanese. H 2d−1(X , Y , Z)(d )/(tor)∈ MHSM1, correspondingto Albanese 1-motive Alb (X , Y , Z) ∈M L
1via MHSM1 M L
1.
Jacobian. When d = 1, we have Alb (X , Y , Z) = Jac(X , Y , Z),and Jac(X , Y , Z)∨ Jac(X , Z , Y ) by Duality (above).
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Relation with previous works
Kato and Russell (2012) constructed Alb (X , Y ) ∈ AG forsmooth proper X , eff. divisor Y (generalizing Jac(X , Y ) forcurves). It agrees with our Alb (X , Y , ∅).Bloch and Srinivas (2000) constructed enriched HS andHn(S) ∈ EHS for proper (but possibly singular) var. S .Barbieri-Viale (2007) and Mazzari (2011) generalized EHS toformal HS. This is (almost) same as MHSM with V• = 0.
Deligne constructed Hn(S) ∈ MHS for arbitrary (possiblysingular/non-proper) S , which is basis of all constructions.
Problem. Can one define Hn(S) ∈ MHSM for such S?