Logarithmic derivations along coordinate hypersurfaces #
For a smooth hypersurface S = (x₁ = 0), Der_X(−log S) is the largest subsheaf of Der_X
mapping 𝒪_X(−S) into itself; in coordinates it is generated by x₁ ∂₁, ∂₂, …, ∂ₙ
[Kol07, 87]. For an ideal I, D(−log S)(I) := im[Der_X(−log S) × I → 𝒪_X] and
D^{r+1}(−log S)(I) := D(−log S)(D^r(−log S)(I)). Three properties: (87.1) restriction,
(D^r(−log S)(I))|_S = D^r(I|_S); the filtration D^s(−log S)(I) ⊂ D^{s−1}(−log S)(D(I)) ⊂ ⋯ ⊂ D^s(I); and (87.2) D^s(I) = D^s(−log S)(I) + D^{s−1}(−log S)(⟨∂I/∂x₁⟩) + ⋯ + ⟨∂^s I/∂x₁^s⟩, the
first j + 1 summands spanning D^{s−j}(−log S)(D^j(I)). Bierstone–Milman use the
several-divisor version D_E with E = V(x₁) + ⋯ + V(x_q) [BM08, §3].
Here the hypersurfaces are V(xᵢ), i ∈ E, for a finite set E of coordinates (logDerE,
DlogE), and Kollár's S = V(x_h) is E = {h} (logDer, Dlog, Dlogpow). The restriction
(87.1) is stated in R̄ = R/⟨x_h⟩ with the coordinate structure quotientCoords of
Hironaka/Algebra/Local/QuotientCoords.lean. The logarithmic derivative ideals are the local form
of the ideals whose birational transform is controlled in
Hironaka/Algebra/Local/TransformLogDeriv.lean.
Der(−log E) and its generators #
The ℚ-derivations of R preserving an ideal J (an R-submodule).
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∂ⱼ preserves ⟨xᵢ⟩ for j ≠ i (Leibniz and ∂ⱼ xᵢ = 0; pderiv_mem_span_x_of_ne of
Hironaka/Algebra/Local/QuotientCoords.lean for a general n).
Der(−log E), the derivations in the span of the ∂ᵢ that preserve every ⟨xᵢ⟩, i ∈ E
(the derivations preserving the ideal of the divisor E, [BM08, §3]; for E = {h}, Kollár's
Der_X(−log S), [Kol07, 87]).
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- c.logDerE E = Submodule.span R (Set.range c.pderiv) ⊓ ⨅ i ∈ E, IsLocalRing.RegularCoords.preservingDer (Ideal.span {c.x i})
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Der(−log S) for S = V(x_h) [Kol07, 87].
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δ = ∑ᵢ aᵢ ∂ᵢ preserves every ⟨xᵢ⟩, i ∈ E, iff aᵢ ∈ ⟨xᵢ⟩ for every i ∈ E.
∑ᵢ aᵢ ∂ᵢ ∈ Der(−log S) iff a_h ∈ ⟨x_h⟩.
The generators of Der(−log E): xᵢ ∂ᵢ for i ∈ E and ∂ⱼ for j ∉ E.
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Der(−log E) is generated by xᵢ ∂ᵢ (i ∈ E) and ∂ⱼ (j ∉ E) (Kollár's
Der_X(−log S) = ⟨x₁ ∂₁, ∂₂, …, ∂ₙ⟩ [Kol07, 87]; [BM08, §3]); for δ = ∑ᵢ cᵢ ∂ᵢ the condition
δ(⟨xᵢ⟩) ≤ ⟨xᵢ⟩ says exactly cᵢ ∈ ⟨xᵢ⟩.
Der(−log S), S = V(x_h), is generated by x_h ∂_h and the ∂ⱼ, j ≠ h [Kol07, 87].
The logarithmic derivative ideals #
The logarithmic derivative ideal
D_E(I) := I + ⟨xᵢ ∂ᵢ f : i ∈ E, f ∈ I⟩ + ⟨∂ⱼ f : j ∉ E, f ∈ I⟩ ([BM08, §3]; for E = {h},
Kollár's D(−log S)(I), [Kol07, 87]).
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The iterated logarithmic derivative ideal D_E^r(I).
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D(−log S)(I) for S = V(x_h) [Kol07, 87].
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Kollár's image definition D(−log E)(I) = im[Der(−log E) × I → 𝒪] [Kol07, 87] agrees with
the generator description.
Kollár's image definition of D(−log S)(I) [Kol07, 87] agrees with the generator
description.
D(−log S)(I) ≤ J iff J contains I, the x_h ∂_h f and the ∂ⱼ f (j ≠ h), f ∈ I.
D^{r+1}(−log S)(I) = D(−log S)(D^r(−log S)(I)) [Kol07, 87].
The decomposition of D(J) and the filtration #
D(J) = D(−log S)(J) + ⟨∂_h J⟩, the first step of (87.2) of [Kol07, 87].
One step of the filtration of [Kol07, 87]: D^{s+1}(−log S)(J) ≤ D^s(−log S)(D(J)).
The filtration
D^s(−log S)(I) ≤ D^{s−j}(−log S)(D^j(I)) ≤ D^{s−j−1}(−log S)(D^{j+1}(I)) ≤ ⋯ ≤ D^s(I) of
[Kol07, 87].
The case E = ∅ #
The boundary case showing that the equality of [BM08, Lemma 3.2(3)] needs E = ∅: for
h ∈ E, D_E(⟨x_h⟩) = ⟨x_h⟩ — the logarithmic derivative ideal of the divisor itself does not
grow.
The decomposition (87.2): ∂_h against the logarithmic derivative ideals #
For a derivation δ and an ideal K, the set {f ∈ K | δ f ∈ K} is an ideal (Leibniz);
the device for showing δ(J) ≤ K on generators of J.
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∂_h (⟨S⟩) ≤ ⟨S⟩ + ⟨∂_h S⟩ (Leibniz).
The key step for (87.2): on the generators of D_E(J), ∂_h f ∈ ⟨∂_h J⟩,
∂_h(xᵢ ∂ᵢ f) = xᵢ ∂ᵢ(∂_h f) + [i = h] ∂_h f, ∂_h ∂ⱼ f = ∂ⱼ ∂_h f; hence
∂_h (D_E(J)) ≤ D_E(⟨∂_h J⟩) + D_E(J).
Iterating the key step: ∂_h (D_E^t(J)) ≤ D_E^t(⟨∂_h J⟩) + D_E^t(J).
⟨∂_h^j I⟩, the ideal generated by the j-fold h-partials of the elements of I
(Kollár's ⟨∂^j I/∂x_1^j⟩ in (87.2) of [Kol07, 87]).
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- c.iterPderivIdeal h j I = Ideal.span ((⇑(c.pderiv h))^[j] '' ↑I)
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∂_h ⟨∂_h^j I⟩ ≤ ⟨∂_h^j I⟩ + ⟨∂_h^{j+1} I⟩ (Leibniz).
The filtration of [Kol07, 87] in one statement:
D^{s-i}(−log S)(D^i(I)) ≤ D^{s-j}(−log S)(D^j(I)) for i ≤ j ≤ s.
The inclusion ⊇ of (87.2): the summands D^{s-i}(−log S)(⟨∂_h^i I⟩), i ≤ j, lie in
D^{s-j}(−log S)(D^j(I)).
The inclusion ⊆ of (87.2), by induction on j: the step uses D_eq_Dlog_sup, the case
s = j of the induction hypothesis (D^j(I) = ∑_{i ≤ j} D^{j-i}(−log S)(⟨∂_h^i I⟩)), and the
key step ∂_h D^t(−log S)(J) ≤ D^t(−log S)(⟨∂_h J⟩) + D^t(−log S)(J).
The second clause of (87.2) of [Kol07, 87]: the first j + 1 summands span
D^{s-j}(−log S)(D^j(I)).
The decomposition (87.2) of [Kol07, 87]: D^s(I) = ∑_{j=0}^{s} D^{s-j}(−log S)(⟨∂_h^j I⟩)
(the sum of ideals is their supremum).
Restriction to the hypersurface, (87.1) #
One step of (87.1): the image of D(−log S)(J) in R̄ = R/⟨x_h⟩ is D_{R̄}(J̄) —
x_h ∂_h f ↦ 0, ∂ⱼ f ↦ ∂̄ⱼ f̄ (the coordinates quotientCoords of the hypersurface).
The restriction property (87.1) of [Kol07, 87] (compare the proof of [Wlo05, Lemma 2.10.4]):
the image of D^r(−log S)(I) in R̄ = R/⟨x_h⟩ is D^r_{R̄}(Ī).