Mixed Hodge structures with modulus

Joint work with Florian Ivorra

Takao Yamazaki (Tohoku University)

June 28, 2018, Singapore

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.

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.

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.

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.

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.

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).

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 ).

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.

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.

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).

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?