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 Origins and motivations of irreg. Hodge theory Deligne, 1984. 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? 푗 1  ∙ Typical example: “ 푒 푥 ” on 픸 ⟶ ℙ 1 , i.e., ( 푗 ∗ 풪 픸 1 , d + d 푥 ) . ∙ 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 ) ∇ ℙ 1 ⊗퐹 푝 −1 ( 푗 ∗ 푉 ) → 0} → Ω 1 ← ← ← ← ← ← ← ← ← ← ← ← ← ∙ In these examples, degeneration at 퐸 1 , i.e., 푯 1 ( ℙ 1 , 퐹 푝 DR( 푗 ∗ 푉 , ∇))  ⟶ 푯 1 ( ℙ 1 , 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 Mirror symmetry for Fano’s. ∙ Need to consider a pair ( 푋, 푓 ) , 푓 ∶ 푋 → 픸 1 , 푋 smooth lead to Hodge theory? A hope it that it imposes bounds to 푝 -adic valuations of eigenvalues of Frobenius. quasi-proj., as possible mirror of a Fano mfld. ∙ ⟿ Various cohomologies 퐻 ∙( 푋, 푓 ) attached to ( 푋, 푓 ) , e.g. Adolphson-Sperber, 1987–89. ∙ Lower bound of the 푝 -adic Newton polygon of the 퐿 -function ∙ dual of Betti homology (Lefschetz thimbles), ∙ de Rham cohomology: hypercohom of (Ω∙ attached to a nondeg. Laurent pol. 푓 ∈ ℤ [ 푥 ±1 푋 , d + d 푓 ) , 1 , … , 푥 ±1 푛 ] given by a Newton polygon attached to 푓 . ∙ Periodic cyclic homology, ∙ Exponential motives. ∙ ⟿ Answers Deligne’s hope, but no Hodge filtration. ∙ (Would like to interpret this as “Newton above Hodge”.) Questions on the Hodge theory of Landau-Ginzburg models. Simpson, 1990. ∙ If ( 푋, 푓 ) is mirror of a Fano mfld 푌 , what is the Hodge filtra- ∙ Non abelian Hodge theory on curves. Correspondence be- tion on 퐻 ∙( 푋, 푓 ) corresponding to that of 퐻 ∙( 푌 ) ? tween ( 푉 , ∇) with reg. sing. (tame) at ∞ and stable tame par- ∙ If 푌 is a Fano orbifold (e.g. toric, like ℙ ( 푤 0 , … , 푤 푛 ) ), 퐻 ∙ orb ( 푌 ) abolic Higgs bdles. (Chen-Ruan) has rational exponents (corresponding to “twisted ∙ Simpson suggests it would be possible to extend this corre- sectors”). Natural to expect that 퐹 ∙ for ( 푋, 푓 ) is indexed by spondence to ( 푉 , ∇) wild (i.e., with irreg. sing.). 퐴 + ℕ , 퐴 ⊂ [0 , 1) ∩ ℚ . ∙ ⟿ Positive answer on curves by CS and Biquard-Boalch ∙ If 푌 is a Fano mfld, how to translate to 퐹 ∙ 퐻 푛 ( 푋, 푓 ) Hard (2000 ± 휀 ). Lefschetz for 푐 1 ( 푇 푌 ) ? ∙ Positive answer (any dimension) by T. Mochizuki (2011). ∙ Drawback: no Hodge filtration.

퐸 1 -degeneration Theorem (Esnault-S.-Yu, Katzarkov-Kontsevich-Pantev, M. Saito, T. Mochizuki) . Hodge realization for a pair ( 푋, 푓 ) . ∙ The spectral seq. for 퐹 ∙(Ω∙ 푋 (∗ 퐷 ) , d + d 푓 ) , equivalently for ∙ 푋 smooth quasi-proj. 퐹 ∙(Ω∙ 푋 (log 퐷, 푓 ) , d + d 푓 )) , degenerates at 퐸 1 . ∙ Choose a compact. 푓 ∶ 푋 → ℙ 1 of 푓 s.t. 퐷 = 푋 ∖ 푋 ncd. ∙ ⟿ Irreg. Hodge filtr. 퐹 ∙ 퐻 푘 dR ( 푋, 푓 ) . ∙ 푃 ∶= 푓 ∗ (∞) , | 푃 | ⊂ 퐷 . ∙ Four different proofs: { 푯 푘 ( 푋, (Ω∙ 푋 (∗ 퐷 ) , d + d 푓 )) , ∙ M. Saito uses a comparison with nearby cycles of 푓 along 퐻 푘 dR ( 푋, 푓 ) ≃ 푯 푘 ( 푋, (Ω∙ 푋 (log 퐷, 푓 ) , d + d 푓 )) 푓 ∗ (∞) and Steenbrink/Schmid limit theorems. { } ∙ K-K-P use reduction to char. 푝 à la Deligne-Illusie. But Ω 푘 휔 ∈ Ω 푘 푋 (log 퐷 ) ∣ d 푓 ∧ 휔 ∈ Ω 푘 +1 푋 (log 퐷, 푓 ) ∶ = 푋 (log 퐷 ) { } need assumption that 푓 ∗ (∞) is reduced. 휔 ∈ Ω 푘 푋 (log 퐷 ) ∣ (d + d 푓 ∧) 휔 ∈ Ω 푘 +1 푋 (log 퐷 ) = 1 by pushing forward by 푓 ∙ E-S-Y use reduction to 푋 = 픸 and previous results on CS extending the original construc- ∙ Quasi-isomorphic filtered complexes: 퐹 ∙(Ω∙ ∙ Yu: 푋 (∗ 퐷 ) , d + d 푓 ) , tion of Deligne on curves by means of twistor D-modules . ∙ K-K-P: 퐹 ∙(Ω∙ 푋 (log 퐷, 푓 ) , d + d 푓 )) . ∙ T. Mochizuki uses the full strength of twistor D-modules in 퐹 푝 (Ω∙ 푋 (log 퐷, 푓 ) , d) ∶= {0 → Ω 푝 (log 퐷, 푓 ) → ⋯ → Ω 푛 (log 퐷, 푓 ) → 0} arbitrary dimensions. ∙ Recall: for 푋 quasi-projective (and 푓 ≡ 0 ) 푝 ∙ Can take into account multiplicities of 푓 ∗ (∞) to refine 퐹 ∙ and index it by 퐴 + ℕ , { } Theorem (Degeneration at 퐸 1 , Deligne (Hodge II, 1972)) . 퓁 ∕ 푚 푖 ∣ 0 ⩽ 퓁 < 푚 푖 , 푚 푖 = mult. of a component of 푓 ∗ (∞) 퐴 = . 푯 ∙( 푋, 퐹 푝 (Ω∙ ⟶ 푯 ∙( 푋, (Ω∙ 푋 (log 퐷 ) , d)) ≃ 퐻 ∙ ( 푋, ℂ ) . 푋 (log 퐷 ) , d)) 

Computation of Hodge numbers by means of Synopsis. ∙ Motivations. Series of papers by Broadhurst-Roberts: some irregular Hodge theory Feynman integrals expressed as period integrals ∙ Standard course of calculus: often easier to compute convolu- ∞ ∫ 퐼 0 ( 푡 ) 푎 퐾 0 ( 푡 ) 푏 푡 푐 d 푡 ( 퐼 0 , 퐾 0 ∶ “modified Bessel functions”). tion 푓 ⋆ 푔 by applying Fourier transformation . 0 ∙ Same idea for Hodge nbrs. ⟿ various conjectures on 퐿 fns of Kloosterman moments. ∙ Arithmetic motivation: Functional equation for the 퐿 -function ∙ On Sym 푘 Kl 2 , ∇ has a regular sing. at 푧 = 0 , but an irregular attached to symmetric power moments of Kloosterman sums. one at ∞ , hence does not underlie a PVHS (Griffiths th.). ∙ Complex analogue of the Kloosterman sums: modified Bessel dR ( 픾 m , Sym 푘 Kl 2 ) has a motivic interpretation: this explains ∙ 퐻 1 differential equation on 픾 m . the MHS. ( 0 푧 ) ⋅ d 푧 ∙ Sym 푘 Kl 2 underlies a variation of irregular Hodge structure ∙ Kl 2 ∶ ( 풪 2 픾 m , ∇) , 푧 . ∇( 푣 0 , 푣 1 ) = ( 푣 0 , 푣 1 ) ⋅ 1 0 (i.e., an irregular mixed Hodge module on ℙ 1 ⊃ 픾 m ). ∙ For 푘 ⩾ 1 , want to consider Sym 푘 Kl 2 : dR ( 픾 m , Sym 푘 Kl 2 ) endowed with an irregular Hodge ∙ ⟹ 퐻 1 ∙ free ℂ [ 푧, 푧 −1 ] -mod. rk 푘 + 1 with connection, filtration . and its de Rham cohomology ∙ We prove that this irreg. Hodge filtr. coincides with the Hodge [ ] ∇ ∶ Sym 푘 Kl 2 ⟶ Sym 푘 Kl 2 ⊗ d 푧 dR ( 픾 m , Sym 푘 Kl 2 ) = coker 퐻 1 filtr. of the MHS. 푧 ∙ We compute this irreg. Hodge filtration by toric methods of Theorem (Fresán-S-Yu) . Assume 푘 odd for simplicity. Adolphson-Sperber & Yu. (Irreg. analogue of Danilov-Khovanski dR ( 픾 m , Sym 푘 Kl 2 ) canonically endowed with a MHS of weights ∙ 퐻 1 computation for toric hypersurfaces). 푘 + 1 & 2 푘 + 2 . dR ( 픾 m , Sym 푘 Kl 2 ) 푝,푞 = 1 if 푝 + 푞 = 푘 + 1 and 푝 = ∙ dim 퐻 1 2 , … , 푘 − 1 or 푝 = 푞 = 푘 + 1 , and 0 otherwise.