SLIDE 1
Quadratic relations for periods of connections Claude Sabbah Joint - - PowerPoint PPT Presentation
Quadratic relations for periods of connections Claude Sabbah Joint - - PowerPoint PPT Presentation
Quadratic relations for periods of connections Claude Sabbah Joint work with Javier Fresn (CMLS, Palaiseau) and Jeng-Daw Yu (NTU, Taipei) Centre de mathmatiques Laurent Schwartz CNRS, cole polytechnique, Institut polytechnique de Paris
SLIDE 2
SLIDE 3
Sketch of proof. 퐻푛
dR(푋an) ⊗ 퐻푛 dR(푋an)
햰푛
- P푛 ≀
- ℂ
퐻푛(푋an) ⊗ 퐻푛
dR(푋an)
햯푛
- P푛
≀
- ℂ
퐻푛(푋an) ⊗ 퐻푛(푋an) 햡푛
ℂ
∙ P푛 ∶= Poincaré isomorphism. ∙ Compatibility proved by de Rham by realizing 퐻푛(푋an) as currents. ∙ In term of matrices (e.g. 햰푛(휔, 휔′) = 푡휔 ⋅ 햰푛 ⋅ 휔′):
푡P푛 ⋅ 햯푛 = 햰푛,
햡푛 ⋅ P푛 = 햯푛. ⟹ 햡푛 = 햯푛 ⋅ (P푛)−1 = 햯푛 ⋅ (푡햰푛)−1 ⋅ 푡햯푛. ∙ Use
푡햰푛 = (−1)푛햰푛.
- Quadratic relations for periods of vector bundles
with log connection
Vector bundles with log connection. ∙ 푋 connected smooth quasi-projective, (푉 , ∇): alg. vect. bdle
- n 푋 with flat connection having reg. sing. at ∞ on 푋.
∙ 퐻푘
dR(푋, (푉 , ∇)), 퐻푘 dR,c(푋, (푉 , ∇)):
∙ Choose (푋, 퐷) smooth proj. 퐷 = ncd, 푋 = 푋 ∖ 퐷. ∙ Deligne’s canonical extension (푉0, ∇): ∗ 푉0: vect. bdle on 푋 extending 푉 : ∗ ∇ ∶ 푉0 → Ω1
푋(log 퐷) ⊗ 푉0 extending ∇
∗ eigenvalues of res퐷푖∇ have real part in [0, 1). ∙ 퐻푘
dR(푋, (푉 , ∇)) ≃ 푯푘(푋, (Ω∙ 푋(log 퐷) ⊗ 푉0, ∇)),
∙ 퐻푘
dR,c(푋, (푉 , ∇)) ≃ 푯푘(푋, (Ω∙ 푋(log 퐷) ⊗ 푉0(−퐷), ∇)).
∙ Assume given pairing ⟨∙ , ∙⟩ ∶ 푉 ⊗ 푉 → 풪푋 s.t. ∙ nondegener. i.e., induces 푉
∼
⟶ 푉 ∨, ∙ ±-symmetric, i.e., ⟨푤, 푣⟩ = ±⟨푣, 푤⟩, ∙ compatible with ∇, i.e., d⟨푣, 푤⟩ = ⟨∇푣, 푤⟩ + ⟨푣, ∇푤⟩. ∙ ⟿ 햲푚 ∶ 퐻푚
dR,c(푋, (푉 , ∇))⊗퐻2푛−푚 dR
(푋, (푉 , ∇)) ⟶ 퐻2푛
dR,c(푋, (풪푋, d)) Tr
≃ ℂ
SLIDE 4
Intersection pairings between flat sections. ∙ 풱 = 푉 an,∇ loc. cst. sheaf of horiz. sections. ∙ ⟿ ±-sym. nondeg. pairing ⟨∙ , ∙⟩ ∶ 풱 ⊗ 풱 → ℂ푋 . ∙ Assume defined over ℚ: ∙ 풱 = ℂ ⊗ℚ 풱
ℚ,
∙ ⟨∙ , ∙⟩ ∶ 풱
ℚ ⊗ 풱 ℚ → ℚ푋.
∙ ⟿ 퐻푚(푋an, 풱
ℚ), 퐻 BM 푚 (푋an, 풱 ℚ),
∙ ⟿ 햡푚 ∶ 퐻푚(푋an, 풱
ℚ) ⊗ 퐻 BM 2푛−푚(푋an, 풱 ℚ) → ℚ.
Period pairings. ∙ Two period pairings (by using ⟨∙ , ∙⟩): 햯푚 ∶ 퐻푚(푋an, 풱
ℚ) ⊗ 퐻2푛−푚 dR
(푋, (푉 , ∇)) ⟶ ℂ 햯BM
푚 ∶ 퐻 BM 푚 (푋an, 풱 ℚ) ⊗ 퐻2푛−푚 dR,c (푋, (푉 , ∇)) ⟶ ℂ
Theorem (Matsumoto & al., 1994). ∙ 햯푚 and 햯BM
푚 are nondeg.
∙ “Quadratic relations” e.g. for 푚 = 푛: ±(−2휋헂)푛햡푛 = 햯푛 ⋅ (햲푛)−1 ⋅ 푡햯BM
푛 .
Middle quadratic relations. ∙ 퐻푚
dR,mid(푋, (푉 , ∇)) ∶= im
[ 퐻푚
dR,c(푋, (푉 , ∇)) → 퐻푚 dR(푋, (푉 , ∇))
] , ∙ 퐻mid
푚 (푋an, 풱 ℚ) ∶= im
[ 퐻푚(푋an, 풱
ℚ) → 퐻 BM 푚 (푋an, 풱 ℚ)
] ∙ ⟿ Nondeg. ±-sym. pairings, e.g. for 푚 = 푛: 햲mid ∶ 퐻푛
dR,mid(푋, (푉 , ∇)) ⊗ 퐻푛 dR,mid(푋, (푉 , ∇)) ⟶ ℂ,
햡mid ∶ 퐻mid
푛
(푋an, 풱
ℚ) ⊗ 퐻mid 푛
(푋an, 풱
ℚ) ⟶ ℚ,
햯mid ∶ 퐻mid
푛
(푋an, 풱
ℚ) ⊗ 퐻푛 dR,mid(푋, (푉 , ∇)) ⟶ ℂ.
Corollary (Quadratic relations). ±(−2휋헂)푛햡mid = 햯mid ⋅ (햲mid)−1 ⋅ 푡햯mid Example (Matsumoto, 1994). ∙ Quadratic relations for generalized hypergeometric functions (Appell, Lauricella...).
SLIDE 5
A conjecture of Broadhurst and Roberts
Bessel moments and Bernoulli matrices. ∙ Bessel moments: ∙ Special values of some Feynman integrals expressed as pe- riod of Laurent polynomials. E.g. 푓(푥, 푦, 푧) = (1 + 푥 + 푦 + 푧)(1 + 푥−1 + 푦−1 + 푧−1). ∙ These periods are also expressed as 푘-moments of the “mod- ified Bessel functions” 퐼0(푡), 퐾0(푡) (e.g. 푘 = odd integer): BM푘(푖, 푗) = ⋆ ∫
∞
퐼푖
0(푡)퐾푘−푖
(푡) ⋅ 푡2푗 d푡 푡 . ∙ Bernoulli matrix (퐵푛 ∶= 푛th Bernoulli nbr): B푘(푖, 푗) = (−1)푘−푖 (푘 − 푖)!(푘 − 푗)!) 푘! ⋅ 퐵푘−푖−푗−1 (푘 − 푖 − 푗 − 1)!. Conjecture (B-R, by computation, e.g. 푘 odd). Set 푘′ = (푘−1)∕2. Consider the 푘′ × 푘′ matrices BM푘 = (BM푘(푖, 푗))1⩽푖,푗⩽푘′ and B푘 = (B푘(푖, 푗))1⩽푖,푗⩽푘′. There exists D푘 ∈ GL푘′(ℚ) defined by an explicit algorithm s.t. (−2휋헂)푘+1B푘 = BM푘 ⋅D푘 ⋅ 푡BM푘 . ¿ Interpret the conj. in terms of quadratic relations for periods ? Generalization of the quadratic relations (F-S-Y). ∙ Since 퐼0, 퐾0 are sols of a diff. eq. with irreg. sing. need to extend quadratic relations to this case. ∙ ⟿ Consider (Kl2, ∇) rk 2 vect. bdle on 픾m ⟷ “modified Bessel diff. eq.” and (Sym푘 Kl2, ∇). ∙ ⟿ Nondegen. de Rham pairing 햲푘 ∶ 퐻1
dR,c(픾m, Sym푘 Kl2) ⊗ 퐻1 dR(픾m, Sym푘 Kl2) ⟶ ℂ.
∙ ⟿ Rapid decay and moderate twisted homology and 퐻mid
1
(픾m, Sym푘 Kl2) ∶= im [ 퐻rd
1 (픾m, Sym푘 Kl2) → 퐻mod 1
(… ) ] . ∙ ⟿ Nondegen. Betti intersection pairing: 햡푘 ∶ 퐻rd
1 (픾m, Sym푘 Kl2) ⊗ 퐻mod 1
(픾m, Sym푘 Kl2) ⟶ ℚ. ∙ ⟿ Nondegen. Period pairings 햯rd,mod
푘
∶ 퐻rd
1 (픾m, Sym푘 Kl2) ⊗ 퐻1 dR(픾m, Sym푘 Kl2) ⟶ ℂ
햯mod,rd
푘
∶ 퐻mod
1
(픾m, Sym푘 Kl2) ⊗ 퐻1
dR,c(픾m, Sym푘 Kl2) ⟶ ℂ.
∙ ⟿ Middle quadratic relations: (−2휋헂)푘+1햡mid
푘
= 햯mid
푘
⋅ (햲mid
푘 )−1 ⋅ 푡햯mid 푘
SLIDE 6
Theorem (Fresán-S-Yu, 2020). ∙ There exists an explicit basis of 퐻mid
1
(픾m, Sym푘 Kl2) such that 햡mid
푘
= B푘. ∙ There exists an explicit basis of 퐻1
dR,mid(픾m, Sym푘 Kl2) s.t.
햯mid
푘
= BM푘 . ∙ The de Rham matrix 햲mid
푘
∈ GL푘′(ℚ) has an algorithmic com- putation (the matrix (햲mid
푘 )−1 checked to agree with the matrix
D푘 suggested by Broadhurst-Roberts for 푘 ⩽ 22). ∙ (햲mid
푘 , 햡mid 푘 , 햯mid 푘 ) also enter in a quadratic relation for a motive
(hence the Bessel moments are periods). In other words, the period structure (퐻1
dR,mid(픾m, Sym푘 Kl2), 퐻mid 1
(픾m, Sym푘 Kl2), 햯mid
푘 )
coincides with the period structure of a Nori motive. 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, Yu, 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).
SLIDE 7
Quadratic relations for irregular periods
- Irreg. singularities.
∙ 푋 smooth quasi-proj., (푉 , ∇) on 푋 with possibly irreg. sing. at ∞ on 푋 ∙ ⟹ ∄(푉0, ∇) log. connection on (푋, 퐷) extending (푉 , ∇). ∙ But (Kedlaya-Mochizuki, 2011): ∃ (푋, 퐷), 퐷 = strict ncd and ∃ (푉0, ∇) good Deligne-Malgrange lattice: ∗ ∀푥 ∈ 퐷, ∃Φ ⊂ 풪푋,푥(∗퐷) finite, ∗ ∀휑 ∈ Φ, ∃ (푅휑, ∇) with reg. sing. on (nb(푥), 퐷), ∗ (풪̂
푥 ⊗ 푉0, ∇) ≃ ⨁ 휑∈Φ
[ (풪̂
푥, d + d휑) ⊗ (푅휑,0, ∇)
] . ∙ 푗 ∶ 푋 → 푋, ∙ ∀푖 ⩾ 1, 푉푖 ∶= 푉푖−1 + Θ푋(− log 퐷) ⋅ 푉푖−1 ⊂ 푗∗푉 . ∙ ⟿ ∇ ∶ 푉푖−1 → Ω1
푋(log 퐷) ⊗ 푉푖
De Rham cohomologies (T. Mochizuki, Esnault-S). 퐻푘
dR(푋, (푉 , ∇)) ≃ 푯푘(푋, (Ω∙ 푋(log 퐷) ⊗ 푉∙, ∇))
퐻푘
dR,c(푋, (푉 , ∇)) ≃ 푯푘(푋, (Ω∙ 푋(log 퐷) ⊗ 푉∙(−퐷), ∇))
Rapid decay and moderate homologies. ∙ 휛 ∶ ̃ 푋 ⟶ 푋 real oriented blow up of the components of 퐷. ∙ ̃ 횥 ∶ 푋an ⟶ ̃ 푋 ∙ 풱 = ker ∇ loc. cst. sheaf on 푋an, e.g. defined over ℚ. ∙ 풱rd ⊂ 풱mod ⊂ ̃ 횥∗풱: ℝ-constructible sheaves on ̃ 푋. ∙ 퐻rd
푚 (푋an, 풱) ∶= 퐻푚( ̃
푋, 풱rd), 퐻mod
푚
(푋an, 풱) ∶= 퐻푚( ̃ 푋, 풱mod)
- Pairings. ⟨∙ , ∙⟩: nondeg. ±-sym. pairing on (푉 , ∇).