Regular logarithmic connections Motivic Geometry CAS Oslo Sep 8, 2020 Piotr Achinger IMPAN Warsaw
I Regular connections (after Deligne)
Regularity in dimension one τ = t d K = C (( t )) ⊇ C [[ t ]] = O � dt � M fin. dim. over K with a C -linear ∇ τ : M → M � MIC ( K / C ) = satisfying ∇ τ ( f m ) = τ ( f ) m + f ∇ τ ( m ) Definition M ∈ MIC ( K / C ) is regular if it admits a ∇ τ -stable O -lattice M ⊆ M . Examples. 1 “ t − λ ” = ( K ,1 �→ λ ) , λ ∈ C is regular, “ e 1 / t ” = ( K ,1 �→ 1 t ) is not. 2 ( K n , ∇ τ ) cyclic corresponding to a DE � � τ n + a n − 1 ( t ) τ n − 1 + ··· + a 0 ( t ) u = 0, a i ( t ) ∈ K is regular iff all a i ( t ) ∈ O . (N.B. Every M ∈ MIC ( K / C ) is cyclic.)
Residue and monodromy M ∈ MIC ( K / C ) regular, M ⊆ M a ∇ τ -stable O -lattice residue map � ∇ τ : M → M ρ : M 0 → M 0 , M 0 : = M / tM Its eigenvalues ∗ are the exponents of M . Theorem (Canonical extension) For M ∈ MIC ( K / C ) regular, there is a unique M = M can with exponents in { 0 ≤ Re ( z ) < 1 } . If M is obtained by base change from a meromorphic M ∈ MIC mero ( ∆ ∗ ) on the punctured disc ∆ ∗ , then the monodromy of M ∇ is conjugate to exp ( − 2 π i ρ can ) .
Regularity in higher dimension X / C smooth scheme � � E ∈ Coh ( X ) , ∇ : E → E ⊗ Ω 1 MIC ( X / C ) = X integrable conn. Definition E ∈ MIC ( X / C ) is regular (at infinity) if for every formal punctured disc s : Spec C (( t )) → X , the induced connection s ∗ E ∈ MIC ( C (( t )) / C ) is regular. If X is a smooth compactification of X with D = X \ X sncd, then it extends to a log connection E ∈ MIC ( X / C ) is regular ⇐⇒ E → E ⊗ Ω 1 X ( log D ) , and ∃ ! E = E can (“canonical extension”) with exponents in { 0 ≤ Re ( z ) < 1 } .
Regularity in higher dimension Existence Theorem For a smooth scheme X / C , the analytification functor E �→ E an : MIC reg ( X / C ) −→ MIC ( X an / C ) ≃ LocSys C ( X an ) is an equivalence. Comparison Theorem For E ∈ MIC reg ( X / C ) we have H ∗ H ∗ H ∗ ( X an , E ∇ dR ( X , E ) ≃ dR ( X an , E an ) ≃ an ) .
I I Logarithmic connections (Kato, Nakayama, Illusie, Ogus)
Log schemes X / C idealized log smooth log scheme ... that is, X étale locally looks like Y = Spec C [ P ] / Σ , M Y induced by P → C [ P ] monoid monomial ideal Y ≃ P gp ⊗ C [ P ] / Σ is free and spanned by d log p ’s. Note: Ω 1 Log strata are locally described as torus orbits ( T = Hom ( P , G m ) � Y ) Examples . � log scheme X with Ω 1 X ≃ Ω 1 1 X smooth, D ⊆ X sncd X ( log D ) . � induced log str. on Z with Ω 1 Z ≃ Ω 1 2 Z ⊆ D stratum X ( log D ) | Z . In general, log strata of a log smooth scheme will be idealized log smooth, which allows for inductive arguments.
Betti realization (Kato–Nakayama) X / C idealized log a manifold with corners X log � smooth log scheme + a proper map τ X : X log → X an Examples. 1 X = ( A 1 ,0 ) τ X � X log = C = R ≥ 0 × S 1 C = X an ( r , θ ) �→ r · θ τ X � 2 X = Spec C [ P ] X log = Hom ( P , C ) −→ Hom ( P , C ) = X an � 3 ( X , D ) snc pair X log → X an “oriented real blow-up” X log is the Betti realization of X in the sense that H ∗ ( X log , C ) ≃ H ∗ dR ( X / C ) .
Logarithmic connections, Ogus’ theorem X / C idealized log smooth log scheme or complex analytic space � � E ∈ Coh ( X ) endowed with an MIC ( X / C ) = integrable connection ∇ : E → E ⊗ Ω 1 X Warning: E might not be locally free! E.g. O Z for a log stratum Z ⊆ X . Theorem (Ogus’ logarithmic Riemann–Hilbert correspondence) For an idealized log smooth log analytic space X there is an equivalence MIC ( X / C ) ≃ L coh ( C log X ) gp gp between MIC ( X / C ) and certain M X ⊗ C -graded C [ M X ] -modules on X log .
I I I Regular logarithmic connections
Splittings of log structures X log scheme with M X locally constant Definition A splitting of the log structure on X is a homomorphism ǫ : M X → M X inducing a splitting of gp X −→ M gp 1 −→ O × ( ⋆ ) X −→ M X −→ 1. Splittings of ( ⋆ ) form a torsor π : V X → X under torus T X = Hom ( M X , G m ) . ǫ univ : = universal splitting of M X on V X
Splittings of log structures Intuition: for M X locally constant, the log structure is a torus bundle X log ε X an and a splitting ǫ is a section “ ǫ ”: X → X . For example, ǫ induces “ ǫ ∗ ”: Ω 1 X → Ω 1 “ ǫ ∗ ”: MIC ( X / C ) → MIC ( X / C ) . and X
Regular logarithmic connections. Definition X / C idealized log smooth Definition 1 Suppose that X is smooth and M X locally constant. Have π : V X → X and ǫ univ on V X . Then E ∈ MIC ( X / C ) is regular (at infinity) if “ ǫ ∗ univ ” ( π ∗ E ) ∈ MIC ( V X / C ) is regular at infinity in the classical sense. 2 In general, let σ : X strat → X be the (reduced) log stratification. Then E ∈ MIC ( X / C ) is regular if σ ∗ E ∈ MIC ( X strat / C ) is regular in sense (1). Write MIC reg ( X / C ) ⊆ MIC ( X / C ) for the full subcategory.
Regular logarithmic connections. Properties ⇒ MIC reg ( X / C ) = MIC ( X / C ) 1 X proper (not obvious since V X strat is usually not proper) 2 If X ⊇ X “good” compactification, then MIC reg ( X / C ) is the essential image of MIC ( X / C ) , and there is a “canonical extension.” 3 Regularity is étale local, and “birational”: if U ⊆ X contains associated primes of E , then E | U regular ⇒ E regular. 4 “Cut-by-curves” criterion: E is regular iff its restriction to every formal log punctured disc is regular.
Main theorems Theorem 1 (Existence theorem) The analytification functor τ ∗ X L coh ( C log E �→ E an : MIC reg ( X / C ) −→ MIC ( X an / C ) ≃ X ) Ogus is an equivalence. Theorem 2 (Comparison theorem) For E ∈ MIC reg ( X / C ) , we have H ∗ H ∗ H ∗ ( X log , τ ∗ dR ( X , E ) ≃ dR ( X an , E an ) ≃ X ( E an ) 0 ) . Ogus
About the proofs. Good compactifications Theorem (Toroidal compactification, version of Włodarczyk 2020) X / C log smooth, i.e. a toroidal embedding ( X , D ) . Then étale locally X admits a good compactification j : X � → X , i.e. 1 X is log smooth, i.e. a toroidal embedding ( X , D ) . 2 D = ( closure of D ) + ( X \ X ) . In particular, M X = j ∗ M X . 3 Locally, ( X , X ) looks like ( Spec C [ P ][ x 1 ,..., x r ] , Spec C [ P ][ x ± 1 1 ,..., x ± 1 r ]) . A good compactification (especially the form (3)) allows us to perform canonical extensions from X to X and invoke GAGA on X .
I V Hodge theory on rigid spaces (work in progress)
� � � � � � � � Rigid-analytic spaces Setup. ◮ K = C (( t )) ⊇ C [[ t ]] = O ◮ X / K smooth qcqs rigid-analytic space ◮ X / O a (generalized) semistable formal model of X ◮ Y = X 0 / C its log special fiber (is idealized log smooth) ◮ Y log → ( Spf O ) 0,log ≃ S 1 its Kato–Nakayama space generic fiber formal model special fiber Kato–Nakayama space � X X Y Y log � Spf C [[ t ]] S 1 Sp C (( t )) Spec C Slogan: the topology Y log reflects the topology of X with its monodromy
Homotopy types of rigid-analytic spaces Theorem (A.–Talpo) The homotopy type of Y log / S 1 does not depend on the choice of X . This gives rise to a functor Ψ : { smooth rigid-analytic spaces over K } −→ ( ∞ -category of spaces). Theorem (Stewart–Vologodsky, Berkovich) The cohomology groups H ∗ ( � Ψ ( X ) , Z ) : = H ∗ ( � � Y log , Z ) , Y log = Y log × S 1 R ( 1 ) carry a natural MHS.
Riemann–Hilbert on rigid-analytic spaces (so τ = t d MIC ( X / C ) = { C -linear int. conn. on X } dt acts) MIC reg ( X / C ) ⊆ MIC ( X / C ) regular connections LocSys C ( Ψ ( X )) = C -local systems on Y log (indep. of model X ) “Theorem” 3 (Riemann–Hilbert for rigid-analytic spaces) Let X be a smooth qcqs rigid-analytic space over K = C (( t )) . There is an equivalence of categories RH: MIC reg ( X / C ) ≃ LocSys C ( Ψ ( X )) .
VMHS on rigid-analytic spaces Definition (tentative) A variation of mixed Hodge structure (VMHS) on X consists of ◮ V ∈ MIC reg ( X / C ) with a Griffiths-transverse Hodge filtration F • V , ◮ V ∈ LocSys Q ( Ψ ( X )) with a weight filtration W • V , ◮ an isomorphism ι : RH ( V ) ≃ V C , such that for every classical point s : Sp C (( t 1 / N )) → X , the pull-back s ∗ ( V , F • , V , W • , ι ) is an “admissible limit VMHS.”
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