Log singularities

Consider a variety X together with a Q-divisor D = ∑

idiDi, where di ∈ Q and D_i are prime divisors of X. If 0 ≤ di ≤ 1 (resp. di ≤ 1), then we call D a boundary (resp. subboundary). If KX + D is Q-Cartier, then we call (X, D) a log pair. We usually assume that X is normal, and D is a boundary unless stated otherwise. By Hironaka’s desingularization theorem, there is a proper birational morphism f : Y → X such that Y is smooth, and the proper transform of ∑

D_i and the exceptional divisors are simple normal crossing (SNC, for short). Denote

by ˜D_i the proper transform of D_i, and set ˜D =∑

d_iD˜_i. We may write

K_Y + ˜D = f^∗(K_X + D) +∑

j

a(X, D; E_j)E_j for some a(X, D; E_j)∈ Q. Here Ej are exceptional divisors of f .

For a non-exceptional divisor E, define

a(X, D; E) =







−di, if E = ˜D_i, 0, otherwise.

It is well known that a(X, D; E) is independent of log resolutions, and is called the discrepancy of E with respect to (X, D). We write X only instead of (X, 0) if D = 0. Discrepancies make a nice measure of singularities of pairs as follows.

Definition 2.19. Given a log pair (X, D), we introduce the following:

i) (X, D) has only log canonical singularities or is log canonical, denoted lc, if a(X, D; E)≥ −1 for all E, and for all log resolutions.

ii) (X, D) has only Kawamata log terminal singularities or is klt if a(X, D; E) >

−1 for all E, and for all log resolutions. In particular, 0 ≤ di < 1 .

iii) (X, D) has only pure log terminal singularities or is plt if a(X, D; E) > −1 for all exceptional E, and for all log resolutions.

iv) The log canonical threshold (lct) of (X, D) is defined by

lct(X, D) = sup{λ|(X, λD) has only log canonical singularities.}

Remark 2.20. i) The definition for log canonical and klt singularities are actually independent of resolutions, which can be seen by using a common resolution.

ii) Being log canonical, klt, or plt is a local property. For an open cover {Uα} of X, (X, D) is log canonical (resp. klt, plt) if and only if (U_α, D|Uα) is log canonical (resp. klt, plt) for all α. Similarly, lct(X, D) = inf_αlct(U_α, D|Uα). It also makes sense to talk about lct at a point p, namely,

lct_p = sup{λ|(U, λD|U) is lc, for some neighborhood U of p}.

We will need some facts of singularities of pairs. (cf.[13]).

Lemma 2.21. Let (X, D) and (Y, E) be log pairs. Suppose f : Y → X is a proper birational morphism, and K_Y + E = f^∗(K_X+ D) then (X, D) is log canonical (resp.

klt) if and only if (Y, E) is log canonical (resp. klt)

Proof. Take a log resolution of (Y, E), which is also a log resolution of (X, D). Then this follows directly by definition.

Lemma 2.22. Let (X, D) and (Y, E) be log pairs. Suppose f : Y → X is finite étale, and E = f^∗D then (X, D) is log canonical (resp. klt) if and only if (Y, E) is log canonical (resp. klt). Moreover, lct(X, D) = lct(Y, E).

Proof. First we take a resolution π : (Z, ˜D)→ (X, D), and write K_Z+ ˜D = π^∗(K_X + D) +∑

j

a_jE_j

Do the base change of π : (Z, ˜D)→ (X, D) by f, and we obtain π^′ : (W, ˜E)→ (Y, E) a resolution. Pulling back the above equation by f^′ : W → Z gives

K_W + ˜E = π^′∗(K_Y + E) +∑

j

a_jf^′∗E_j

Now that f^′∗E_j cannot be multiple for f^′ is étale. So every discrepancy remains invariant under pullback.

In general, being log canonical is preserved by finite morphisms.

Lemma 2.23. Let (X, D) and (Y, E) be log pairs. Suppose f : Y → X is a finite morphism such that K_Y + E ≡ f^∗(K_X + D). Then (X, D) is log canonical (resp.

klt) if and only if (Y, E) is log canonical (resp. klt).

Proof. First we prove the “if” part. Choose a log resolution g : Z → X, and define W to be the normalization of a component of Y ×X Z dominating X .

W ^{g′ //}

f^′



Y

f

Z ^{g //}X

Write K_Z+ D^′ = g^∗(K_X + D), where D^′ = g_∗⁻¹D−∑

a(X, D; D_j)D_j. Pulling back via f^′ gives f^′∗(K_Z+ D^′) = f^′∗g^∗(K_X + D) = g^′∗(K_Y + E)

Now write K_W + E^′ = f^′∗(K_Z + D^′). By Hurwitz formula, for a divisor E_i in E^′, with f^′(E_i) = D_j for some j, the coefficients satisfy (1 + a(X, D, D_j)) =

1

e(1 + a(Y, E, E_i))≥ 0 (resp. > 0), if (Y, E) is log canonical (resp. klt).

From this, for the “only if” part, we reduced to the case f is a Galois cover, i.e., X is a quotient variety of Y for some finite group action of G. We take now,

besides W , Z, a G-equivariant log resolution g^′ : W₁ → W , and let the quotient be Z1 = W^G. Such resolution exists by the functorial construction under smooth morphisms. Replace W, Z by W₁, Z₁. Note that Z₁ is Q-factorial, and we can calculate as before. For a divisor Ei in E^′ with f^′(Ei) = Dj, Hurwitz formula gives (1 + a(Y, E; E_i)) = e(1 + a(X, D; D_j)) ≥ 0 (resp. > 0), if (X, D) is log canonical (resp. klt).

Kawamata-Viehweg vanishing theorem is a useful theorem in birational geome-try.

Theorem 2.24 (Kawamata-Viehweg Vanishing Theorem). Suppose X is smooth, H is an ample Q-divisor. Then Hⁱ(X, KX +⌈H⌉) = 0 for i > 0.

We also have the following relative version. [11, Remark 1-2-6]

Theorem 2.25 (Relative Kawamata-Viehweg Vanishing Theorem). Suppose (X, D) is klt, H is a Q-divisor, and KX + D + H is an integral divisor.. If f : X → Y is a projective morphism such that H is f -nef and f -big then Rf_∗ⁱOX(K_X+ D + H) = 0 for i > 0.

Proof. Case 1. X is smooth, H is f -ample, and D =⌈H⌉ − H is SNC.

We may assume that X, Y projective. Let L be an ample Cartier divisor on Y . Replace H by H + f^∗L, we may assume that H ample by projection formula.

Consider the spectral sequence:

E₂^i,j = Hⁱ(Y, Rf_∗^j(OX(KX+⌈H⌉)⊗OY(Y, mL)))⇒ H^i+j(X,OX(KX+⌈H⌉+mf^∗L)) . By Serre vanishing, for m large, the spectral sequence degenerates as

H⁰(Y, Rf_∗^j(OX(K_X+⌈H⌉)⊗OY(Y, mL))) = H^j(X,OX(K_X+⌈H⌉+mf^∗L)) = 0 for j > 0 by Theorem2.24. So, Rf_∗^j(OX(K_X +⌈H⌉) = 0.

Case2. General case.

By Lemma 2.26 below, which is a corollary of Kodaira lemma, we take a res-olution g : Z → X of (X, D) such that g^∗H −∑

jδ_jF_j is (f ◦ g)-ample for some 0 < δ_j ≪ 1, and {Fj}, proper transform of D and exceptional divisors, are SNC.

We may apply Case 1 on g and h = f ◦ g, to the divisor H1 = g^∗H−∑

jδ_jF_j. Then for i > 0,

Rⁱg_∗OZ(K_Z+⌈H1⌉) = Rⁱh_∗OZ(K_Z+⌈H1⌉) = 0.

Hence by spectral sequence,

0 = Rⁱh_∗OZ(K_Z+⌈H1⌉) = Rⁱf_∗g_∗OZ(K_Z+⌈H1⌉).

On the other hand,

g_∗OZ(K_Z +⌈H1⌉) = g∗OZ(⌈KZ+ H₁⌉) = OX(K_X + D + H),

since⌈KZ− g^∗(K_X + D)⌉ is effective exceptional by the condition of being klt.

Lemma 2.26. Suppose f : X → Y is a proper surjective morphism of normal varieties, and H is an f -nef and f -big divisor. Then there is a resolution g : Z→ X such that g^∗H−∑

jδ_jF_j is f◦g-ample for small 0 < δj < 1and{Fj}, proper transform of D and exceptional divisors, are simple normal crossing.

Remark 2.27. Being f -nef and f -big is a numerical property. We easily derive the form that for an integral divisor D^′ ≡ KX + D + H, we have Rⁱf_∗(O(D^′)) = 0, for i > 0.

Here we introduce a special type of singularities called rational singularities.

Definition 2.28. X is said to have only rational singularities if for a resolution f : Y → X , Rⁱf_∗OY = 0 for all i > 0.

Remark 2.29. It is known that the definition of rational singularities is independent of resolutions.

Theorem 2.30. A surface with only quotient singularities is klt and thus has only rational singularities.

By Lemma 2.23, we see quotient singularities are klt. To prove klt singularities are rational, however, requires some work. We reproduce the proof in [14, Chap. 5]

here.

First we recall a coherent sheaf F is CM (Cohen-Macaulay) if all its stalks Fp

are CM modules. A scheme X is CM if the structure sheaf OX is CM. Projective CM varieties can be characterized as follows:

Lemma 2.31. [14, 5.72] For a projective variety X and an ample Cartier divisor D, X is CM if and only if Hⁱ(X,OX(−rD)) = 0 for i < n and large r.

We have the following alternative characterization of rational singularities.

Proposition 2.32. [13, 11.9] X has only rational singularities if and only if X is CM and for a resolution f : Y → X, we have f∗ω_Y = ω_X.

Proof of Theorem 2.30. Suppose (X, ∆) is klt, and we prove X has only rational singularities. Let f : Y → X be a resolution. Then it suffices to prove that f∗ωY = ω_X. Write K_Y = f^∗(K_X+ ∆) + E⁺−E⁻, where E⁺, E⁻≥ 0 are exceptional divisors without common components. Now⌈E⁺⌉ = KY − f^∗(KX+ ∆) + E⁻+{−E⁺}, and (Y, E⁻+{−E⁺}) is klt for Y is smooth and E⁻+{−E⁺} is SNC with coefficients in [0, 1). By Kawamata-Viehweg vanishing theorem, Rⁱf_∗OY(⌈E⁺⌉) = 0 for i > 0.

For any ample Cartier divisor D, by Larey spectral sequence E₂^p,q = H^p(X,OX(−D) ⊗ R^qf_∗OY(⌈

E⁺⌉

))⇒ H^p+q(Y,OY(⌈ E⁺⌉

− f^∗D), we get Hⁱ(X,OX(−D)) ∼= Hⁱ(Y,OY(⌈E⁺⌉ − f^∗D). Since this morphism factors through Hⁱ(Y,OY(−f^∗D), we get the injection:

Hⁱ(X,OX(−D)) ↩→ Hⁱ(Y,OY(−f^∗D)).

By Serre duality [14, 5.71] and Kawamata-Viehweg vanishing theorem, Hⁱ(Y,OY(−f^∗D)) = Hⁿ⁻ⁱ(Y, ω_Y(f^∗D)) = 0

for i < n, and thus Hⁱ(X,OX(−D)) = 0. This implies X is CM. On the other hand, for i = n Serre duality gives

H⁰(Y, ω_Y(f^∗D)) = H⁰(X, f_∗ω_Y(D)))  Hⁱ(X, ω_X(D)), which implies f_∗ω_Y → ωX is surjective, and hence an isomorphism.