Irregular Hodge theory: Applications to arithmetic and mirror - - PowerPoint PPT Presentation

Irregular Hodge theory: Applications to arithmetic and mirror symmetry

Claude Sabbah

Centre de Mathématiques Laurent Schwartz CNRS, École polytechnique, Institut Polytechnique de Paris Palaiseau, France

Origins and motivations of irreg. Hodge theory

Deligne, 1984.

Origins and motivations of irreg. Hodge theory

Deligne, 1984. ∙ Griffiths’ regularity theorem: ∙ (푉 , ∇): alg. vect. bdle with connect. on a quasi-proj. curve. ∙ (푉 , ∇) underlies a PVHS ⟹ ∇ has reg. sing. at ∞. ∙ E.g., regularity of the Gauss-Manin connection. ∙ Complex analogues of exponential sums over finite fields: (푉 , ∇) with irreg. sing. at ∞. ∙ Is there a Hodge realization for such objects? ∙ Typical example: “푒푥” on 픸

1  푗

⟶ ℙ1, i.e., (푗∗풪픸

1, d + d푥).

∙ Deligne defines a ↘ filtration 퐹∙(푗∗푉 ) in many examples. ∙ ⟿ Filtration of the de Rham complex 퐹 푝 DR(푗∗푉 , ∇) ∶= {0 → 퐹 푝(푗∗푉 ) ∇ ← ← ← ← ← ← ← ← ← ← ← ← ← → Ω1

ℙ1⊗퐹 푝−1(푗∗푉 ) → 0}

∙ In these examples, degeneration at 퐸1, i.e., 푯1(ℙ1, 퐹 푝 DR(푗∗푉 , ∇))  ⟶ 푯1(ℙ1, DR(푗∗푉 , ∇)). ∙ Filtration indexed by 푝 ∈ 퐴 + ℕ, 퐴 ⊂ [0, 1) finite.

∙ What could be the use of a “Hodge filtration” which does not lead to Hodge theory? A hope it that it imposes bounds to 푝-adic valuations of eigenvalues of Frobenius. Adolphson-Sperber, 1987–89. ∙ Lower bound of the 푝-adic Newton polygon of the 퐿-function attached to a nondeg. Laurent pol. 푓 ∈ ℤ[푥±1

1 , … , 푥±1 푛 ] given

by a Newton polygon attached to 푓. ∙ ⟿ Answers Deligne’s hope, but no Hodge filtration. ∙ (Would like to interpret this as “Newton above Hodge”.) Simpson, 1990. ∙ Non abelian Hodge theory on curves. Correspondence be- tween (푉 , ∇) with reg. sing. (tame) at ∞ and stable tame par- abolic Higgs bdles. ∙ Simpson suggests it would be possible to extend this corre- spondence to (푉 , ∇) wild (i.e., with irreg. sing.). ∙ ⟿ Positive answer on curves by CS and Biquard-Boalch (2000 ±휀). ∙ Positive answer (any dimension) by T. Mochizuki (2011). ∙ Drawback: no Hodge filtration. Mirror symmetry for Fano’s. ∙ Need to consider a pair (푋, 푓), 푓 ∶ 푋 → 픸

1, 푋 smooth

quasi-proj., as possible mirror of a Fano mfld. ∙ ⟿ Various cohomologies 퐻∙(푋, 푓) attached to (푋, 푓), e.g. ∙ dual of Betti homology (Lefschetz thimbles), ∙ de Rham cohomology: hypercohom of (Ω∙

푋, d + d푓),

∙ Periodic cyclic homology, ∙ Exponential motives. Questions on the Hodge theory of Landau-Ginzburg models. ∙ If (푋, 푓) is mirror of a Fano mfld 푌 , what is the Hodge filtra- tion on 퐻∙(푋, 푓) corresponding to that of 퐻∙(푌 )? ∙ If 푌 is a Fano orbifold (e.g. toric, like ℙ(푤0, … , 푤푛)), 퐻∙

• rb(푌 )

(Chen-Ruan) has rational exponents (corresponding to “twisted sectors”). Natural to expect that 퐹∙ for (푋, 푓) is indexed by 퐴 + ℕ, 퐴 ⊂ [0, 1) ∩ ℚ. ∙ If 푌 is a Fano mfld, how to translate to 퐹∙퐻푛(푋, 푓) Hard Lefschetz for 푐1(푇 푌 )?

퐸1-degeneration

Hodge realization for a pair (푋, 푓). ∙ 푋 smooth quasi-proj. ∙ Choose a compact. 푓 ∶ 푋 → ℙ1 of 푓 s.t. 퐷 = 푋 ∖ 푋 ncd. ∙ 푃 ∶= 푓 ∗(∞), |푃 | ⊂ 퐷. 퐻푘

dR(푋, 푓) ≃

{푯푘(푋, (Ω∙

푋(∗퐷), d + d푓)),

푯푘(푋, (Ω∙

푋(log 퐷, 푓), d + d푓))

Ω푘

푋(log 퐷, 푓) ∶ =

{ 휔 ∈ Ω푘

푋(log 퐷) ∣ d푓 ∧ 휔 ∈ Ω푘+1 푋 (log 퐷)

} = { 휔 ∈ Ω푘

푋(log 퐷) ∣ (d + d푓∧)휔 ∈ Ω푘+1 푋 (log 퐷)

} ∙ Quasi-isomorphic filtered complexes: ∙ Yu: 퐹∙(Ω∙

푋(∗퐷), d + d푓),

∙ K-K-P: 퐹∙(Ω∙

푋(log 퐷, 푓), d + d푓)).

퐹 푝(Ω∙

푋(log 퐷, 푓), d) ∶= {0 → Ω푝(log 퐷, 푓) 푝

→ ⋯ → Ω푛(log 퐷, 푓) → 0} ∙ Recall: for 푋 quasi-projective (and 푓 ≡ 0) Theorem (Degeneration at 퐸1, Deligne (Hodge II, 1972)). 푯∙(푋, 퐹 푝(Ω∙

푋(log 퐷), d)) 

⟶ 푯∙(푋, (Ω∙

푋(log 퐷), d)) ≃ 퐻∙(푋, ℂ).

Theorem (Esnault-S.-Yu, Katzarkov-Kontsevich-Pantev, M. Saito,

• T. Mochizuki).

∙ The spectral seq. for 퐹∙(Ω∙

푋(∗퐷), d + d푓), equivalently for

퐹∙(Ω∙

푋(log 퐷, 푓), d + d푓)), degenerates at 퐸1.

∙ ⟿ Irreg. Hodge filtr. 퐹∙퐻푘

dR(푋, 푓).

∙ Four different proofs: ∙ M. Saito uses a comparison with nearby cycles of 푓 along 푓 ∗(∞) and Steenbrink/Schmid limit theorems. ∙ K-K-P use reduction to char. 푝 à la Deligne-Illusie. But need assumption that 푓 ∗(∞) is reduced. ∙ E-S-Y use reduction to 푋 = 픸

1 by pushing forward by 푓

and previous results on CS extending the original construc- tion of Deligne on curves by means of twistor D-modules. ∙ T. Mochizuki uses the full strength of twistor D-modules in arbitrary dimensions. ∙ Can take into account multiplicities of 푓 ∗(∞) to refine 퐹∙ and index it by 퐴 + ℕ, 퐴 = { 퓁∕푚푖 ∣ 0 ⩽ 퓁 < 푚푖, 푚푖 = mult. of a component of 푓 ∗(∞) } .

Computation of Hodge numbers by means of irregular Hodge theory

∙ Standard course of calculus: often easier to compute convolu- tion 푓 ⋆ 푔 by applying Fourier transformation. ∙ Same idea for Hodge nbrs. ∙ Arithmetic motivation: Functional equation for the 퐿-function attached to symmetric power moments of Kloosterman sums. ∙ Complex analogue of the Kloosterman sums: modified Bessel differential equation on 픾m. ∙ Kl2 ∶ (풪2

픾m, ∇),

∇(푣0, 푣1) = (푣0, 푣1) ⋅ ( 0 푧 1 0 ) ⋅ d푧 푧 . ∙ For 푘 ⩾ 1, want to consider Sym푘 Kl2: ∙ free ℂ[푧, 푧−1]-mod. rk 푘 + 1 with connection, and its de Rham cohomology 퐻1

dR(픾m, Sym푘 Kl2) = coker

[ ∇ ∶ Sym푘 Kl2 ⟶ Sym푘 Kl2⊗d푧 푧 ] Theorem (Fresán-S-Yu). Assume 푘 odd for simplicity. ∙ 퐻1

dR(픾m, Sym푘 Kl2) canonically endowed with a MHS of weights

푘 + 1 & 2푘 + 2. ∙ dim 퐻1

dR(픾m, Sym푘 Kl2)푝,푞 = 1 if 푝 + 푞 = 푘 + 1 and 푝 =

2, … , 푘 − 1 or 푝 = 푞 = 푘 + 1, and 0 otherwise. Synopsis. ∙ Motivations. Series of papers by Broadhurst-Roberts: some Feynman integrals expressed as period integrals ∫

∞

퐼0(푡)푎퐾0(푡)푏푡푐d푡 (퐼0, 퐾0 ∶ “modified Bessel functions”). ⟿ various conjectures on 퐿 fns of Kloosterman moments. ∙ On Sym푘 Kl2, ∇ has a regular sing. at 푧 = 0, but an irregular

• ne at ∞, hence does not underlie a PVHS (Griffiths th.).

∙ 퐻1

dR(픾m, Sym푘 Kl2) has a motivic interpretation: this explains

the MHS. ∙ Sym푘 Kl2 underlies a variation of irregular Hodge structure (i.e., an irregular mixed Hodge module on ℙ1 ⊃ 픾m). ∙ ⟹ 퐻1

dR(픾m, Sym푘 Kl2) endowed with an irregular Hodge

filtration. ∙ We prove that this irreg. Hodge filtr. coincides with the Hodge

• filtr. of the MHS.

∙ We compute this irreg. Hodge filtration by toric methods of Adolphson-Sperber & Yu. (Irreg. analogue of Danilov-Khovanski computation for toric hypersurfaces).

Computation of Hodge numbers by means of irregular Hodge theory

∙ Standard course of calculus: often easier to compute convolu- tion 푓 ⋆ 푔 by applying Fourier transformation. ∙ Same idea for Hodge nbrs. ∙ Arithmetic motivation: Functional equation for the 퐿-function attached to symmetric power moments of Kloosterman sums. ∙ Complex analogue of the Kloosterman sums: modified Bessel differential equation on 픾m. ∙ Kl2 ∶ (풪2

픾m, ∇),

∇(푣0, 푣1) = (푣0, 푣1) ⋅ ( 0 푧 1 0 ) ⋅ d푧 푧 . ∙ For 푘 ⩾ 1, want to consider Sym푘 Kl2: ∙ free ℂ[푧, 푧−1]-mod. rk 푘 + 1 with connection, and its de Rham cohomology 퐻1

dR(픾m, Sym푘 Kl2) = coker

[ ∇ ∶ Sym푘 Kl2 ⟶ Sym푘 Kl2⊗d푧 푧 ] Theorem (Fresán-S-Yu). Assume 푘 odd for simplicity. ∙ 퐻1

dR(픾m, Sym푘 Kl2) canonically endowed with a MHS of weights

푘 + 1 & 2푘 + 2. ∙ dim 퐻1

dR(픾m, Sym푘 Kl2)푝,푞 = 1 if 푝 + 푞 = 푘 + 1 and 푝 =

2, … , 푘 − 1 or 푝 = 푞 = 푘 + 1, and 0 otherwise. Motivic interpretation. ∙ (Kl2, ∇) is the Gauss-Manin conn. of (풪픾2

m, d + d(푥 + 푧∕푥))

by the proj. 픾m × 픾m → 픾m (푥, 푧) ↦ 푧. ∙ (⨂푘 Kl2, ∇): G-M conn. of (풪픾m×픾푘

m, d + d(푓푘))

푓푘(푥1, … , 푥푘, 푧) = ∑

푖(푥푖 + 푧∕푥푖)

∙ Set ̃ Kl2 = [2]∗Kl2, [2] ∶ 푡 ↦ 푡2. Set 푦푖 = 푥푖∕푡. ∙ Then (⨂푘 ̃ Kl2, ∇): G-M conn. of 퐸푡⋅푔푘 ∶=(풪픾m×픾푘

m, d + d(푡 ⋅ 푔푘))

푔푘(푦1, … , 푦푘) = ∑

푖(푦푖 + 1∕푦푖) ∶ 픾푘 m → 픸 1.

∙ 퐻1

dR(픾m, Sym푘 Kl2) ≃ 퐻1 dR(픾m, ⨂푘 ̃

Kl2)픖푘×휇2 ≃ 퐻푘+1

dR (픾m × 픾푘 m, 푡 ⋅ 푔푘)픖푘×휇2

∙ General fact (Fresán-Jossen, F-S-Y): 푈 smooth quasi-proj., 푔 ∶ 푈 → 픸

1 regular, 퐻푛 dR(픾m × 푈, 푡 ⋅ 푔) underlies a Nori

motive, hence endowed with a canonical MHS. ∙ Analogue of Fourier inversion formula for ℎ ∶ ℝ → ℝ: ℎ(0) = ⋆ ∫ℝ ̂ ℎ(푡) d푡 = ⋆ ∫ℝ2 푒2휋푖 푡⋅ℎ(푥)d푡 d푥. ∙ Set 풦 = 푔−1

푘 (0) ⊂ 픾푘

• m. Variant of what we want:

퐻푘+1(픸

1 × 픾푘 m, 푡 ⋅ 푔푘) ≃ 퐻푘−1 c

(풦)∨(−푘).

Irregular mixed Hodge structures

There exist various generalizations of a MHS on a 푘-vect. space (푘 = ℚ, ℝ, ℂ). ∙ Mixed twistor structure (Simpson, 1997). ∙ ⟿ mixed twistor 퐷-module (T. Mochizuki, 2011). ∙ Semi-infinite pure Hodge structure (Barannikov, 2001). ∙ ⟿ Construction of Frobenius mfld structures. ∙ Pure TERP structure (Hertling, 2002). ∙ ⟿ tt∗ geometry on Frobenius manifolds. ∙ Non-commutative Hodge structure (Katzarkov-Kontsevich- Pantev, 2008). ∙ ⟿ Hodge theory for periodic cyclic homology of some dg-algebras. ∙ Exponential mixed Hodge structure (Kontsevich-Soibelman, 2011). ∙ ⟿ Hodge theory for cohomological Hall algebras. ∙ Irregular Hodge structure (S-Yu, 2018). ∙ ⟿ General framework for the irregular Hodge filtration. ∙ Example. 퐻푘

dR(푋, 푓) “underlies” an exponential MHS, hence

an irreg. MHS, 퐹∙퐻푘

dR(푋, 푓) is the irreg. Hodge filtration.

Integrable mixed twistor structure. ∙ Object ((풯, ∇), 푊∙ ) : ∙ 풯: hol. vect. bdle on ℙ1 = 픸

1 푢 ∪ 픸 1 푣 (twistor structure),

∙ ∇: merom. connection on 풯, pole of order ⩽ 2 at 0 & ∞, no other pole (integrable twistor structure), ∙ 푊∙: ↗ filtr. of (풯, ∇) such that each gr푊

퓁 (풯, ∇) is pure of

weight 퓁, i.e., gr푊

퓁 풯 ≃ 풪푟퓁 ℙ1(퓁) (integr. mixed twistor str.).

∙ Can add: polarization (in the pure case), real or rational struc- ture (on the local system ker ∇ on ℂ∗ + Stokes struct. at 0, ∞). ∙ Associated vector space 퐻: 풯

1 = fibre at 1

Irregular Hodge filtration. ∙ ((풯, ∇), 푊∙ ) ⟿ ↘ filtration on 퐻: ∙ (ℳ, ∇) = (풯, ∇)|픸

1 푢 an

∙ ∀훼 ∈ [0, 1), (ℳ훼, ∇): vect. bdle on ℙ1, extending (ℳ, ∇) s.t. ∇ has a log. sing. at 푣 = 0, with residues having real part in [훼, 훼 + 1) (Deligne’s extension). ∙ HN푝(ℳ훼): Harder-Narasimhan filtr. ∙ 퐹 푝−훼

irr 퐻 ∶= HN푝(ℳ훼)|1 ⊂ 퐻.

Irregular Hodge structure.

• Definition. Category 햨헋헋햬햧햲: subcategory of integr. mixed twistor

structures with good limit properties w.r.t. the rescaling 푢 ↦ 휆 ⋅ 푢 (휆 → ∞). Example. ∙ 푋 smooth quasi-projective and 푓 ∶ 푋 → 픸

1 proper or tame.

∙ 퐻 = 퐻푘

dR(푋, 푓).

• Theorem. 퐻푘

dR(푋, 푓) underlies a pure object of 햨헋헋햬햧햲, with

(ℳ, ∇푢) = (퐻푘

dR,rel.(푋 × 픸 1 푢, 푓∕푢), ∇푢)

and 퐹∙

irr퐻 = 퐹∙퐻푘 dR(푋, 푓)

∙ for 휔 ∈ Ω푘

푋×픸

1 푢∕픸 1 푢:

∇푋휔 = 푒−푓∕푢 ⋅ d푋 ⋅ 푒푓∕푢(휔), ∇푢휔 = 푒−푓∕푢 ⋅ 휕

휕푢 ⋅ 푒푓∕푢(휔) = −푓

푢2 휔 + 휕푢휔. ∙ ℳ: hypercohomology on 푋 of ⋯ ⟶ Ω푘−1

푋 [푢]

∇푋 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → Ω푘

푋[푢]

∇푋 ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → Ω푘+1

푋 [푢] ⟶ ⋯

Irregular Hodge-Tate structures

∙ (풯, ∇) pure irreg. MHS of some weight, ∙ 퐹∙

irr퐻: irreg. Hodge filtr.

∙ Jumps of 퐹∙

irr퐻 are integers ⟺ unipotent monodromy on

ker ∇|ℂ∗. ∙ unipotent monodromy ⟿ Jakobson-Morosov filtr. 푀∙퐻 as- sociated to its nilpotent part.

• Definition. (풯, ∇) is irreg. Hodge-Tate if

∀푝, dim gr푀

2푝퐻 = dim gr푝 퐹irr퐻

and gr푀

2푝+1퐻 = 0

Conjecture (K-K-P, 2017). If (푋, 푓) is the Landau-Ginzburg model mirror to a projective Fano mfld 푌 , then the irreg. MHS 퐻푛(푋, 푓) (푛 = dim 푋) is pure and irregular Hodge-Tate. Many works on the conjecture. ∙ Lunts, Przyjalkowski, Harder ∙ Shamoto

The toric case. ∙ Lattices 푀 ⊂ ℝ푛, 푁 = 푀 ∨. ∙ Δ ⊂ ℝ푛: reflexive simplicial polyhedron with vertices in 푀, s.t. 0 is the only integral point in Δ. ∙ Δ∗: dual polyhedron (vertices in 푁 and of the same kind as Δ). ∙ Σ: fan dual to Δ, = cone (0, Δ∗). ∙ 푌 = ℙΣ assumed smooth, hence toric Fano (Batyrev). ∙ Chow ring 퐴∗(푌 ) ≃ 퐻2∗(푌 , ℤ) generated by div. classes 퐷푣, 푣 ∈ Vertices(Δ∗) =∶ 푉 (Δ∗). ∙ 푐1(퐾∨

푌 ) = ∑ 푣∈푉 (Δ∗) 퐷푣 satisfies Hard Lefschetz on 퐻2∗(푌 , ℚ).

∙ Coordinates 푥1, … , 푥푛 s.t. ℂ[푁] = ℂ[푥, 푥−1]. 푋 ∶= Spec ℂ[푥, 푥−1], 푓 ∶ 푋 ⟶ 픸

1,

푓(푥) = ∑

푣∈푉 (Δ∗)

푥푣 퐻푛

dR(푋, 푓) = Ω푛 푋∕(d + d푓∧)Ω푛−1 푋 ≃[ℂ[푥, 푥−1]∕(휕푓)] ⋅ d푥1

푥1 ∧ ⋯ ∧ d푥푛 푥푛 ∙ Newton filtration N∙ on the Jacobian ring ℚ[푥, 푥−1]∕(휕푓) ∙ Borisov-Chen-Smith: 퐻2∗(푌 , ℚ) ≃ grN

∙

(ℚ[푥, 푥−1]∕(휕푓)) ∙ Hard Lefschetz ⟹ ∀푘 s.t. 0 ⩽ 푘 ⩽ 푛∕2, 푓 푛−2푘 ∶ grN

푘

(ℚ[푥, 푥−1]∕(휕푓))

∼

⟶ grN

푛−푘

(ℚ[푥, 푥−1]∕(휕푓)) ∙ Idea of Varchenko from Singularity theory (Doklady, 1981): interpret multipl. by 푓 as the nilpotent part of a monodromy

• perator.

∙ Adapt and apply this idea to 퐻푛

dR(푋, 푓)

∙ ⟹ irreg. Hodge-Tate property.