Regular logarithmic connections Motivic Geometry CAS Oslo Sep 8, - - PowerPoint PPT Presentation

Regular logarithmic connections

Motivic Geometry CAS Oslo

Sep 8, 2020

Piotr Achinger

IMPAN Warsaw

I Regular connections

(after Deligne)

Regularity in dimension one

K = C((t)) ⊇ C[[t]] = O

• τ = t d

dt

MIC(K/C) = M fin. dim. over K with a C-linear ∇τ: M → M 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, “e1/t” = (K,1 → 1 t ) is not. 2 (Kn,∇τ) cyclic corresponding to a DE

• τn + an−1(t)τn−1 + ··· + a0(t)
• u = 0,

ai(t) ∈ K is regular iff all ai(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 ∇τ: M → M

• residue map

ρ: M0 → M0, M0 := M/tM Its eigenvalues∗ are the exponents of M.

Theorem (Canonical extension)

For M ∈ MIC(K/C) regular, there is a unique M = Mcan with exponents in {0 ≤ Re(z) < 1}. If M is obtained by base change from a meromorphic M ∈ MICmero(∆∗) on the punctured disc ∆∗, then the monodromy of M∇ is conjugate to exp(−2πiρcan).

Regularity in higher dimension

X/C smooth scheme MIC(X/C) =

• E ∈ Coh(X), ∇: E → E ⊗ Ω1

X integrable conn.

• Definition

E ∈ MIC(X/C) is regular (at infinity) if for every formal punctured disc s: SpecC((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 E ∈ MIC(X/C) is regular ⇐⇒ it extends to a log connection E → E ⊗ Ω1

X(log D),

and ∃! E = Ecan (“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 → Ean : MICreg(X/C) −→ MIC(Xan/C) ≃ LocSysC(Xan) is an equivalence.

Comparison Theorem

For E ∈ MICreg(X/C) we have H∗

dR(X, E)

≃ H∗

dR(Xan, Ean)

≃ H∗(Xan, E∇

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 = SpecC[P]/ Σ, MY induced by P → C[P] monoid monomial ideal Note: Ω1

Y ≃ Pgp ⊗ C[P]/Σ is free and spanned by d log p’s.

Log strata are locally described as torus orbits (T = Hom(P ,Gm)

• Y )

Examples.

1 X smooth, D ⊆ X sncd

• log scheme X with Ω1

X ≃ Ω1 X(log D). 2 Z ⊆ D stratum

• induced log str. on Z with Ω1

Z ≃ Ω1 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 smooth log scheme

• a manifold with corners Xlog

+ a proper map τX : Xlog → Xan Examples.

1 X = (A1,0)

Xlog = C = R≥0 × S1

• C = Xan

τX

(r,θ) → r · θ

2 X = SpecC[P]

• Xlog = Hom(P

,C)

τX

−→ Hom(P ,C) = Xan

3 (X, D) snc pair

• Xlog → Xan “oriented real blow-up”

Xlog is the Betti realization of X in the sense that H∗(Xlog,C) ≃ H∗

dR(X/C).

Logarithmic connections, Ogus’ theorem

X/C idealized log smooth log scheme or complex analytic space MIC(X/C) =

• E ∈ Coh(X) endowed with an

integrable connection ∇: E → E ⊗ Ω1

X

• Warning: E might not be locally free! E.g. OZ 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) ≃ Lcoh(Clog

X )

between MIC(X/C) and certain M

gp X ⊗ C-graded C[M gp X ]-modules on Xlog.

I I I Regular logarithmic connections

Splittings of log structures

X log scheme with MX locally constant

Definition

A splitting of the log structure on X is a homomorphism ǫ: MX → MX inducing a splitting of (⋆) 1 −→ O×

X −→ Mgp X −→ M gp X −→ 1.

Splittings of (⋆) form a torsor π: VX → X under torus TX = Hom(MX,Gm). ǫuniv := universal splitting of MX on VX

Splittings of log structures

Intuition: for MX locally constant, the log structure is a torus bundle

ε Xlog Xan

and a splitting ǫ is a section “ǫ”: X → X. For example, ǫ induces “ǫ∗”: Ω1

X → Ω1 X

and “ǫ∗”: MIC(X/C) → MIC(X/C).

Regular logarithmic connections. Definition

X/C idealized log smooth

Definition

1 Suppose that X is smooth and MX locally constant. Have π: VX → X

and ǫuniv on VX. 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: Xstrat → X be the (reduced) log stratification. Then

E ∈ MIC(X/C) is regular if σ∗E ∈ MIC(Xstrat/C) is regular in sense (1). Write MICreg(X/C) ⊆ MIC(X/C) for the full subcategory.

Regular logarithmic connections. Properties

1 X proper

⇒ MICreg(X/C) = MIC(X/C) (not obvious since VXstrat is usually not proper)

2 If X ⊇ X “good” compactification, then MICreg(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 E → Ean : MICreg(X/C) −→ MIC(Xan/C) τ∗

X

≃

Ogus

Lcoh(Clog

X )

is an equivalence.

Theorem 2 (Comparison theorem)

For E ∈ MICreg(X/C), we have H∗

dR(X, E)

≃ H∗

dR(Xan, Ean)

≃

Ogus

H∗(Xlog,τ∗

X(Ean)0).

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, MX = j∗MX. 3 Locally, (X, X) looks like

(SpecC[P][x1,..., xr], SpecC[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 = X0/C its log special fiber (is idealized log smooth) ◮ Ylog → (SpfO)0,log ≃ S1 its Kato–Nakayama space

generic fiber formal model special fiber Kato–Nakayama space

X

• X
• Y
• Ylog
• SpC((t))

SpfC[[t]]

SpecC

• S1
• Slogan: the topology Ylog reflects the topology of X with its monodromy

Homotopy types of rigid-analytic spaces

Theorem (A.–Talpo)

The homotopy type of Ylog/S1 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∗( Ylog,Z),

• Ylog = Ylog ×S1 R(1)

carry a natural MHS.

Riemann–Hilbert on rigid-analytic spaces

MIC(X/C) = {C-linear int. conn. on X} (so τ = t d

dt acts)

MICreg(X/C) ⊆ MIC(X/C) regular connections LocSysC(Ψ(X)) = C-local systems on Ylog (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: MICreg(X/C) ≃ LocSysC(Ψ(X)).

VMHS on rigid-analytic spaces

Definition (tentative)

A variation of mixed Hodge structure (VMHS) on X consists of ◮ V ∈ MICreg(X/C) with a Griffiths-transverse Hodge filtration F•V, ◮ V ∈ LocSysQ(Ψ(X)) with a weight filtration W•V, ◮ an isomorphism ι: RH(V) ≃ VC, such that for every classical point s: SpC((t1/N)) → X, the pull-back s∗(V, F•,V,W•,ι) is an “admissible limit VMHS.”