Riemann-Hilbert correspondence for irregular holonomic D -modules - PowerPoint PPT Presentation

Riemann-Hilbert correspondence for irregular holonomic D -modules (joint work with Masaki KASHIWARA) Andrea D’AGNOLO Universit` a di Padova – Italy Winter School on Higher Structures in Algebraic Analysis Padova, 18 February 2014 A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 1 / 12

The classical RH problem Hilbert’s 21st problem (1900) “A problem that Riemann himself may have in mind” “To show that there always exists a linear differential equation of the Fuchsian class, with given singular points and monodromic group” A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 2 / 12

Fuchsian ODEs P ( z , ∂ z ) = a m ( z ) ∂ m z + · · · + a 1 ( z ) ∂ z + a 0 ( z ) , a j ∈ O C Definition z 0 ∈ C Fuchsian singularity: a m ( z 0 ) = 0, ord z = z 0 a m − m ≤ ord z = z 0 a j − j ∀ j Basis of m local solutions at z 0 of the form: u ( z ) = ( z − z 0 ) λ v ( z ) + (log terms) , λ ∈ C , v ∈ O C , z 0 λ � monodromy Corollary { u ∈ O C ; Pu = 0 } is a local system outside of the singular points A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 3 / 12

D -modules X : complex manifold D X : sheaf of linear differential operators Definition M a D X -module � S ol ( M ) = R H om D X ( M , O X ) Example P ∈ D X � M = D X / D X P H 0 S ol ( M ) = { u ∈ O X ; Pu = 0 } holonomic D X -module � ODE regular holonomic D X -module � Fuchsian ODE C -constructible sheaf � local system A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 4 / 12

� � � � � � � � � � � Regular RH correspondence D b ( D X ) : bounded derived category of D X -modules D b ( C X ) : bounded derived category of sheaves of C -vector spaces Theorem (Kashiwara 1984) S ol � D b ( C X ) D b ( D X ) op D b hol ( D X ) op D b rh ( D X ) op D b C - c ( C X ) ∼ analysis topology There is also an explicit reconstruction functor: D b rh ( D X ) ∋ M � F = S ol ( M ) ∈ D b C - c ( C X ) � R H om ( F , O t X ) ≃ M A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 5 / 12

Subanalytic sheaves [Kashiwara-Schapira 2001] X sa the subanalytic site: ◮ open subanalytic subsets of X ◮ locally finite covers Mod ( C X sa ) subanalytic sheaves ( � ind-sheaves) Tempered distributions: � � D b t ( U ) = image D b X ( X ) − → D b X ( U ) O t X = Dolbeault complex with coefficients in D b t X S ol t ( M ) = R H om D X ( M , O t X ) Example E 1 / z = D C e 1 / z = D C / D C P , P ( z , ∂ z ) = z 2 ∂ z − 1 not Fuchsian C H 0 S ol t ( E 1 / z ) = “ lim C { Re ( 1 / z ) < c } → ” C − c − → + ∞ Caveat: S ol t ( E 1 / z ) ≃ S ol t ( E 2 / z ) C C A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 6 / 12

Irregular ODEs P ( z , ∂ z ) = a m ( z ) ∂ m z + · · · + a 0 ( z ) , z 0 ∈ C not Fuchsian For w = ( z − z 0 ) 1 / r , basis of m formal solutions: ϕ ∈ C [ w − 1 ] , λ ∈ C , v ∈ � u ( w ) = e ϕ ( w ) w λ � � v ( w )+ (log terms) , O C , z 0 ∀ direction θ , ∃ analytic solution u with u ∼ � u on a sector S Caveat: u + u 1 ∼ � u if Re ϕ 1 < Re ϕ at θ � Stokes phenomenon Theorem ([Deligne] and [Malgrange] in the 80s) Irregular RH in dimension one, for fixed singular locus Idea: order as above the exponents, so that u is well defined in the graded part Caveat: difficult to extend in higher dimensions, cf [Sabbah 2013] A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 7 / 12

Enhanced sheaves X × R ∞ = bordered subanalytic site ◮ open subanalytic subsets of X × P 1 ( R ) included in X × R ◮ locally finite covers Definition (influenced by [Tamarkin 2008]) E b ( C X ) = D b ( C X × R ∞ ) / { K : K ≃ π − 1 R π ∗ K } π : X × R ∞ − → X sa It is a commutative tensor category + ⊗ K 2 = R µ !! ( q − 1 1 K 1 ⊗ q − 1 K 1 2 K 2 ) convolution unit: C { t = 0 } = C X ×{ 0 } + ⊗ , I hom + , R f ∗ , R f !! , f − 1 , f ! Six operations: Lemma F �→ C E X ⊗ π − 1 F, C E D b ( C X ) ֒ → E b ( C X ) X = “ lim → ” C { t ≥ c } − c − → + ∞ A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 8 / 12

Reconstruction of exponential modules � � ∂ t − 1 D b E D b t → D b t = H om D R ∞ ( E t R ∞ , D b t X = − − − X × R ∞ )[ 1 ] X × R ∞ X × R ∞ 0 O E X = (Dolbeault complex with coefficients in D b E X ) [ 1 ] S ol E ( M ) = R H om D X ( M , O E X ) ∼ S ol t ( M ⊠ E t R ∞ )[ 1 ] Theorem S ol E ( E ϕ E ϕ X = D X e ϕ ( ∗ D ) , ϕ ∈ O X ( ∗ D ) X ) ≃ “ lim → ” C { t + Re ϕ ( z ) ≥ c } − c − → + ∞ Generalizes the example of E 1 / z C Theorem E ϕ C { t + Re ϕ ( z ) ≥ c } , O E R H om ∼ R π ∗ I hom + X ≃ R H om ( “ lim X ) , → ” − c − → + ∞ Related to [D’A 2013] A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 9 / 12

Structure of holonomic modules Key result from [Mochizuki 2011] and [Kedlaya 2011] Lemma A statement Q X ( M ) is true for any M ∈ D b hol ( D X ) and any X if: ⇒ Q U i ( M| U i ) ∀ i ∈ I, for X = � Q X ( M ) ⇐ i ∈ I U i an open cover. Q X ( M ) = ⇒ Q X ( M [ n ]) ∀ n ∈ Z . ⇒ Q X ( M ) , for M ′ − → M ′′ + 1 Q X ( M ′ )& Q X ( M ′′ ) = → M − − − → a d.t. Q X ( M ⊕ M ′ ) = ⇒ Q X ( M ) . Q X ( M ) = ⇒ Q Y ( D f ∗ M ) , for f : X − → Y projective. Q X ( M ) holds for M with a normal form along a n.c. divisor. M has normal form if it is a direct sum of exponential D -modules on polysectors along the divisor. A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 10 / 12

� � � � � � � � � � RH correspondence Theorem S ol E : D b hol ( D X ) op − → E b ( C X ) is fully faithful S ol E ( M ) is R -constructible Reconstruction holds: hol ( D X ) ∋ M � K = S ol E ( M ) ∈ E b D b R - c ( C X ) � R H om ( K , O E X ) ≃ M Compatibility with the regular case: R H om ( ∗ , O E X ) hol ( D X ) op � � S ol E � E b � D b ( D X ) op D b R - c ( C X ) R H om ( ∗ , O t X ) S ol t � D b D b rh ( D X ) op D b rh ( D X ) op C - c ( C X ) ∼ ∼ A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 11 / 12

Stokes phenomenon ϕ j ϕ 1 , ϕ 2 ∈ O C ( ∗ 0 ) � K j = S ol E ( E C ) ≃ “ lim C { t + Re ϕ j ( z ) ≥ c } → ” − c − → + ∞ M flat meromorphic connection ∀ θ : M ∼ θ E ϕ 1 C ⊕ E ϕ 2 C � K = S ol E ( M ) ∼ θ K 1 ⊕ K 2 { Re ( ϕ 1 − ϕ 2 ) = 0 } = ⊔ ℓ n � L n Stokes lines Lemma S an open sector � b ± , S ⊂ {± Re ( ϕ 1 − ϕ 2 ) > 0 } End E b ( C C ) ( π − 1 C S ⊗ ( K 1 ⊕ K 2 )) ≃ t , S ⊃ L n 0 , S ∩ L n = ∅ , n � = n 0 ( b ± upper/lower triangular in M 2 ( C ) , t = b + ∩ b − ) A. D’Agnolo (Padova) Riemann-Hilbert correspondence Padova, 18 February 2014 12 / 12