LOG-CANONICAL FORMS AND LOG CANONICAL SINGULARITIES

HUBERT FLENNER AND MIKHAIL ZAIDENBERG

Dedicated to H. Grauert on occasion of his 70’s birthday

Abstract. For a normal subvariety V of ℂ^{n} with a good ℂ^{∗}-action we give a simple characterization for when it has only log canonical, log terminal or rational singularities. Moreover we are able to give formulas for the plurigenera of isolated singular points of such varieties and of the logarithmic Kodaira dimension of V \{0}. For this purpose we introduce sheaves of m-canonical and L^{2,m}-canonical forms on normal complex spaces. For the case of affine varieties with good ℂ^{∗}-action we give an explicit formula for these sheaves in terms of the grading of the dualizing sheaf and its tensor powers.

Contents

Introduction 1 1. Logarithmic m-canonical forms on singular spaces 3 1.1. Logarithmic m-canonical forms on smooth varieties 3 1.2. Logarithmic m-canonical forms on singular varieties 4 1.3. Logarithmic m-canonical forms versus log canonical, log terminal and

rational singularities 9

1.4. Plurigenera of isolated singularities 11 2. Logarithmic m-canonical forms on quasihomogeneous varieties 12 2.1. Graded rings and associated schemes 12 2.2. Characterizing log canonical forms in terms of gradings 14 2.3. Log terminal and log canonical singularities of quasihomogeneous

varieties 17

2.4. Plurigenera of quasihomogeneous singularities 19 2.5. Log-Kodaira dimension of quasihomogeneous varieties 20 References 21

Introduction

Let X be a normal complex space and D a reduced Weil divisor on X. In this paper we will associate to the pair (X, D) two sheaves L_{X,D}^{m} and L_{X,D}^{2,m}, which we

Date: December 18, 2000. This work was supported by stays of the first author at the University of Grenoble and by

both authors at the Max Planck Institute of Mathematics at Bonn; we would like to thank these institutions for their hospitality.

1991 Mathematics Subject Classification: 14B05, 14B15, 14E15, 14J17, 32S20, 32S25. Key words: rational singularity, log canonical singularity, log terminal singularity, quotient sin-

gularity, plurigenus, quasihomogeneous singularity, ℂ^{∗}-action, graded algebra, Kodaira dimension.

call the sheaves of logarithmic m-, resp. L^{2}-m-canonical forms. The construction is in brief as follows: let π : X′ → X be a resolution of singularities such that D′ := π^{−1}(D ∪ Sing X)_{red} is an SNC (=simple normal crossing) divisor. The sheaves

L_{X,D}^{m} := π_{∗}(O_{X′}(m(K_{X′} + D′))) and L_{X,D}^{2,m} := π_{∗}(O_{X′}(mK_{X′} + (m − 1)D′))

are then independent of the choice of resolution. For instance, the L^{2}-m-canonical forms are just those m-canonical forms on X\(D ∪ Sing X) that are locally L^{2} at the points of D ∪ Sing X, see [17]. The sheaf L_{X}^{2,1} := L_{X,0}^{2,1} was previously studied in the paper of Grauert-Riemenschneider [7] (it is called there the canonical sheaf of X).

A motivation to study these sheaves is that they allow simple characterizations for when a singularity is rational, log terminal or log canonical. As an example, by a result of Kempf [11] a normal complex algebraic variety X has rational singularities if and only if it is Cohen-Macaulay and satisfies L_{X}^{2,1} ≅ O_{X}(K_{X}). In analogy with this result we will show: if (X, x) is a normal complex singularity and D is a reduced Weil divisor with K_{X} +D being ℚ-Cartier, then (X, D) has a log canonical singularity at x if and only if the stalks (L_{X,D}^{m})_{x} and O_{X}(m(K_{X} + D))_{x} are equal for all m ≥ 1. In the case when D = 0 a similar characterization holds for log terminal singularities, see 1.17.

The main application of these log canonical sheaves is to the case of affine varieties V = Spec A for which the coordinate ring A = ⨁_{i≥0} A_{i} is a non-negatively graded ℂ-algebra. Let D ⊆ V be a reduced Weil divisor and assume for simplicity that D is given by an equation P = 0, where P is homogeneous of degree d. The modules of sections

L_{A,D}^{m} := H^{0}(V, L_{V,D}^{m}) and L_{A,D}^{2,m} := H^{0}(V, L_{V,D}^{2,m})

as well as the reflexive hull ω_{A}^{[m]} of the m-th tensor power of the dualizing module ω_{A} then carry natural gradings. We will show: if V^{∗} := V \V (A_{+}) is smooth and D ∩ V^{∗} is an SNC divisor then

L_{A,D}^{m} = (ω_{A}^{[m]})_{≥md} and L_{A,D}^{2,m} = (ω_{A}^{[m]})_{>(m−1)d};

see 2.15 for a more general statement. This implies for instance that (V, D) has log canonical singularities if and only if ω_{A}^{[m]} has no elements of degree < md. Similar characterizations hold for the properties log terminal and rational, see 2.16. For the latter case this is a result of [4] and [19].

As another application we obtain formulas for the plurigenera of the singularities of V, where V is as above. For instance, the plurigenus δ_{m} introduced in [20] is given by

δ_{m}(X, p) = dim_{ℂ} (ω_{A}^{[m]})_{≤0},

see 2.22. Applying this to complete intersections we recover a result of Morales [15]. A final application concerns the logarithmic plurigenera and the Kodaira dimension of V \D. With the assumptions on (V, D) as above, assume moreover that A_{0} ≅ ℂ so that the ℂ^{∗}-action corresponding to the grading is good. Then the logarithmic plurigenus p̄_{m}(V \D) is given by the dimension of ω_{md}^{[m]}, see 2.26.

The paper is organized as follows. In Section 1 we introduce the sheaves of log canonical forms and study their basic properties. In particular we show the characterization of log canonical and log terminal singularities in terms of log canonical forms mentioned above.

Sections 2.2–2.5 contain the applications to affine varieties with ℂ^{∗}-action as described above In Section 2.1 we provide some material concerning equivariant completions and weighted blowups of quasihomogeneous affine varieties.

In this paper we work in the category of complex spaces and varieties over ℂ. However, the principal results remain valid for algebraic varieties over any field of characteristic zero.

1. Logarithmic m-canonical forms on singular spaces

1.1. Logarithmic m-canonical forms on smooth varieties.

Notation 1.1. Let X be a complex manifold and D ⊆ X a divisor with simple normal crossings (SNC in brief). Consider the sheaf of logarithmic m-canonical forms

L_{X,D}^{m} := O_{X}(mK_{X} + mD)

and the sheaf of logarithmic L^{2}-m-canonical forms

L_{X,D}^{2,m} := O_{X}(mK_{X} + (m − 1)D) = L_{X,D}^{m}(−D) .

If x_{1}, …, x_{n} are local coordinates around a point, say, p ∈ X with D = {x_{1}·...·x_{k} = 0}, then near p the O_{X}-module L_{X,D}^{m} is generated by

ω_{m} = (\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{k}}{x_{k}} ∧ dx_{k+1} ∧ ⋯ ∧ dx_{n})^{⊗m} ,

whereas the sheaf L_{X,D}^{2,m} is generated by x_{1} · ... · x_{k} · ω_{m}. As explained in [17, Thm. 2.1], the forms in H^{0}(X, L_{X,D}^{2,m}) are just the meromorphic m−canonical forms on X\D which locally at the points of D belong to L^{2/m}.

If D = 0 is the zero divisor, we write L_{X}^{m} and L_{X}^{2,m} instead of L_{X,D}^{m}, resp. L_{X,D}^{2,m}.

1.2. Let π : Y → X be a morphism of complex manifolds of the same dimension such that π^{−1}(D)_{red} is contained in an SNC divisor E on Y. Pulling back differential forms induces natural homomorphisms

π_{m}^{∗} : L_{X,D}^{m} → L_{Y,E}^{m}  and  π_{2,m}^{∗} : L_{X,D}^{2,m} → L_{Y,E}^{2,m}

(see [9, §11.1.c] for the case of π_{m}^{∗}; the case of π_{2,m}^{∗} is similarly). They are injective, if π is dominant [9, Prop. 11.2].

For later purposes we need the following simple observation.

Lemma 1.3. Let π : Y → X be a proper surjective morphism of complex manifolds of the same dimension, and let D ⊆ X be an SNC divisor such that E = π^{−1}(D)_{red} is also an SNC divisor. For a section η ∈ H^{0}(X\D, ω_{X}^{⊗m}) the following hold:

η ∈ H^{0}(X, L_{X,D}^{m}) ⇔ π^{∗}η ∈ H^{0}(Y, L_{Y,E}^{m})

and

η ∈ H^{0}(X, L_{X,D}^{2,m}) ⇔ π^{∗}η ∈ H^{0}(Y, L_{Y,E}^{2,m}) .

Proof. The implication ‘⇒’ was already observed before. To show ‘⇐’, assume that π^{∗}η ∈ H^{0}(Y, L_{Y,E}^{m}). Clearly, η is a holomorphic section of L_{X,D}^{m} if it is locally holomorphic outside an analytic subset of X of codimension 2. Hence we may assume that π is finite and that there are local coordinates x_{1}, …, x_{n} on X, resp. y_{1}, …, y_{n} on Y such that D is given locally by x_{1} = 0 and π is given locally by

π : (y_{1}, y_{2}, …, y_{n}) ⟼ (y_{1}^{k}, y_{2}, …, y_{n})

for some k ∈ ℕ. The differential form ω = \frac{dx_{1}}{x_{1}} ∧ dx_{2} ∧ ⋯ ∧ dx_{n} is then a local generator of the invertible sheaf O_{X}(K_{X} + D) and so ω^{⊗m} generates L_{X,D}^{m}. Moreover,

π^{∗}(ω^{⊗m}) = (k \frac{dy_{1}}{y_{1}} ∧ dy_{2} ∧ ⋯ ∧ dy_{n})^{⊗m}

locally generates L_{Y,E}^{m}, and π^{∗}(x_{1}^{a}ω^{⊗m}) = y_{1}^{ak}π^{∗}(ω^{⊗m}) is a local section of L_{Y,E}^{2,m} if and only if a ≥ 1. This easily implies both statements of the lemma.

Applying 1.3 to local sections yields the following corollary (cf. [9, Thm. 11.1], [17, Thm. 1.1]).

Corollary 1.4. For a bimeromorphic proper morphism π : Y → X of complex manifolds and D, E as above we have

π_{∗}(L_{Y,E}^{m}) = L_{X,D}^{m}  and  π_{∗}(L_{Y,E}^{2,m}) = L_{X,D}^{2,m} .

1.2. Logarithmic m-canonical forms on singular varieties. In virtue of 1.4 the definition of the sheaves L_{X,D}^{m} and L_{X,D}^{2,m} can be extended as follows.

Definition 1.5. Consider a normal complex space X and a closed analytic subset D ⊆ X. Let σ : X′ → X be a resolution of singularities such that D′ := σ^{−1}(D ∪ Sing X)_{red} is an SNC divisor. We call

L_{X,D}^{m} := σ_{∗}(L_{X′,D′}^{m}) and L_{X,D}^{2,m} := σ_{∗}(L_{X′,D′}^{2,m})

the sheaf of logarithmic L^{m}-canonical forms, resp. logarithmic L^{2,m}-canonical forms on X.

Because of 1.4 and by standard arguments, this is independent of the choice of resolution of singularities. As before, if D = ∅ then we write in brief L_{X}^{m} and L_{X}^{2,m} instead of L_{X,D}^{m} and L_{X,D}^{2,m}, respectively.

Remark 1.6. Clearly, D ⊆ D_{1} implies that

L_{X,D}^{m} ⊆ L_{X,D_{1}}^{m}  and  L_{X,D}^{2,m} ⊆ L_{X,D_{1}}^{2,m} .

In most of our considerations D will be a Weil divisor. This is justified by the following lemma.

Lemma 1.7. With X, D as in (1.5) let divD denote the (reduced) union of all divisorial components of D. Then the following hold.

(a) L_{X,D}^{m} = L_{X,divD}^{m} and L_{X,D}^{2,m} = L_{X,divD}^{2,m} .

(b) There are natural inclusions

L_{X,D}^{m} ⊆ O_{X} (m(K_{X} + divD)) and L_{X,D}^{2,m} ⊆ O_{X} (mK_{X} + (m − 1)divD)

with equality outside the set Sing X ∪ Sing divD.

Proof. (a) We confine ourselves to the proof of the first equality, the proof of the second one being similarly. By 1.6 we have L_{X,divD}^{m} ⊆ L_{X,D}^{m}. To show the converse inclusion, let π : X′ → X be a resolution of singularities such that Δ′ := π^{−1}(divD∪ Sing X)_{red} and D′ := π^{−1}(D ∪ Sing X)_{red} are SNC divisors on X′. Let η be a local section of L_{X,D}^{m} in a neighbourhood, say, U of a point p ∈ X. Its restriction to U\(D ∪Sing X) is a holomorphic section of ω_{X}^{⊗m} and so extends holomorphically to an m-form on U\(divD ∪ Sing X). Thus π^{∗}(η) has no poles along the components of D′\Δ′, so is a section of O_{X′}(m(K_{X′} + Δ′)), whence η is a section of L_{X,divD}^{m}, as required.

In order to show (b), in view of (a), we may assume that D is a reduced divisor. The sheaves L_{X,D}^{m} and O_{X} (m(K_{X} + D)) are then equal outside the set B := Sing X ∪ Sing D. As the latter sheaf is reflexive and codimB ≥ 2 this implies that

L_{X,D}^{m} ⊆ O_{X} (m(K_{X} + divD)).

The same argument also gives the second inclusion in (b).

Remark 1.8. More generally one can introduce m- and L^{2,m}-canonical forms for any effective ℚ-divisor D = ∑ a_{i}D_{i} with 0 < a_{i} ≤ 1 ^{1}. Again, the definition is given in two steps. If X is a manifold and D_{red} := ∑ D_{i} is an SNC divisor, we set

L_{X,D}^{m} := O_{X}(mK_{X} + ⌊mD⌋) and L_{X,D}^{2,m} := O_{X}(mK_{X} + ⌊(m − 1)D⌋).

As a_{i} ≤ 1, with the same arguments as above the construction is functorial under generically finite maps of manifolds of the same dimension.

In the case of a normal variety X we choose a resolution of singularities σ : X′ → X such that σ^{−1}(D ∪ Sing X)_{red} is a simple normal crossing divisor. Let D′ denote the divisor D^{pr} + ∑_{i} E_{i}, where E_{i} are the exceptional divisors and D^{pr} denotes the proper transform of D. Now one can introduce as in 1.5 the sheaves

L_{X,D}^{m} := σ_{∗}(L_{X′,D′}^{m}) and L_{X,D}^{2,m} := σ_{∗}(L_{X′,D′}^{2,m}).

As before one can show that this definition does not depend on the choice of the resolution.

The following proposition indicates certain functorial properties of L^{m}- and L^{2,m}- canonical forms (cf. [9, Prop. 11.3]).

Proposition 1.9. Let π : Y → X be a generically finite morphism of normal connected complex spaces of the same dimension. Let D ⊆ X be an analytic subset, and assume that E ⊆ Y is an analytic subset with divE = divπ^{−1}(D ∪ Sing X). Then the following hold.

(a) There are natural injections

L_{X,D}^{m} → π_{∗}(L_{Y,E}^{m})  and  L_{X,D}^{2,m} → π_{∗}(L_{Y,E}^{2,m}) .

(b) If moreover π is proper, then for a form η ∈ H^{0}(X\D, O_{X}(mK_{X})) we have

\begin{aligned}η ∈ H^{0}(X, L_{X,D}^{m}) & ⇔ & π^{∗}η ∈ H^{0}(Y, L_{Y,E}^{m}) and \\ η ∈ H^{0}(X, L_{X,D}^{2,m}) & ⇔ & π^{∗}η ∈ H^{0}(Y, L_{Y,E}^{2,m}) . \end{aligned}

(c) If π is proper and birational, then

L_{X,D}^{m} = π_{∗}(L_{Y,E}^{m})  and  L_{X,D}^{2,m} = π_{∗}(L_{Y,E}^{2,m}) .

1In [5] such a divisor is called a subboundary.

Proof. Because of 1.7 we may assume that E = π^{−1}(D ∪ Sing X). As before we consider only the case of L^{m}-forms, the case of L^{2,m}-forms being similarly. Consider resolutions of singularities Y′ → Y and X′ → X that fit into a diagram

[figure: Y′; π′; X′; Y; q; π; X; p]

and such that

E′ := q^{−1}(E ∪ Sing Y)_{red} and D′ := p^{−1}(D ∪ Sing X)_{red}

are SNC divisors in Y′ resp. X′. As π^{′−1}(D′) is contained in E′ the morphism π′ induces an injection L_{X′,D′}^{m} → π′_{∗}(L_{Y′,E′}^{m}) (see 1.2). Applying p_{∗} gives the desired injection in (a).

(c) is an immediate consequence of (b). To deduce (b), note first that π(Sing Y) ⊆ X is a closed analytic subset of codimension at least 2. By 1.7 the sheaves L_{X,D}^{m} and L_{X,D_{1}}^{m} are equal, where D_{1} := D ∪ π(Sing Y). Moreover Sing Y is contained in E_{1} := π^{−1}(D_{1}) and L_{Y,E}^{m} ⊆ L_{Y,E_{1}}^{m}. Hence it is sufficient to prove (b) for D_{1} instead of D. In other words, we may assume that E contains Sing Y.

Let now q : Y′ → Y and p : X′ → X be resolutions of singularities as above. By 1.3

p^{∗}η ∈ H^{0}(X′, L_{X′,D′}^{m}) ⇔ (π′p)^{∗}η ∈ H^{0}(Y′, L_{Y′,E′}^{m}).

As by definition p_{∗}(L_{X′,D′}^{m}) = L_{X,D}^{m} and q_{∗}(L_{Y′,E′}^{m}) = L_{Y,E}^{m} , (b) follows.

For our purposes it is useful to introduce certain sheaves that are invariants of the singularities.

Notation 1.10. Let X, D be as in 1.5. Because of 1.7 (b) we may form the quotient sheaves

Λ_{X,D}^{m} := O_{X} (m(K_{X} + divD))/L_{X,D}^{m}

and

Δ_{X,D}^{m} := O_{X} (mK_{X} + (m − 1)divD)/L_{X,D}^{2,m} .

Note that by 1.7 (a) Λ_{X,D}^{m} = Λ_{X,divD}^{m} and Δ_{X,D}^{m} = Δ_{X,divD}^{m}. Moreover, by 1.7 (b) these sheaves are concentrated on Sing X ∪ Sing divD. As before, in the case D = ∅ we write in brief Λ_{X}^{m} and Δ_{X}^{m} instead of Λ_{X,D}^{m}, resp. Δ_{X,D}^{m}.

Later on we will need the following fact.

Lemma 1.11. For analytic subsets D_{1} ⊆ D_{2} of X the natural maps

Λ_{X,D_{1}}^{m} ⟶ Λ_{X,D_{2}}^{m} and Δ_{X,D_{1}}^{m} ⟶ Δ_{X,D_{2}}^{m}

are injective.

Proof. We restrict to the proof of the first inclusion the other one being similarly. By 1.7 (a) we may assume that D_{1} and D_{2} are reduced divisors. We need to show that

L_{X,D_{1}}^{m} = L_{X,D_{2}}^{m} ∩ O_{X}(m(K_{X} + D_{1})).

The inclusion ‘⊆’ follows from 1.6 and 1.7 (b). To show the converse inclusion, let π : X′ → X be a resolution of singularities such that D′_{i} := π^{−1}(D_{i} ∪ Sing X)_{red}, i = 1, 2, are SNC divisors. If η is a section of L_{X,D_{2}}^{m} ∩ O_{X}(m(K_{X} + D_{1})) defined over some open subset U of X, then π^{∗}(η) is a form in ω_{X′}^{⊗m} that has at most logarithmic poles along the irreducible components of D′_{2} and is holomorphic along the components of the proper transform of D_{2} − D_{1}. Hence π^{∗}(η) is a section of O_{X′}(m(K_{X′} + D′_{1})) so that η ∈ H^{0}(U, L_{X,D_{1}}^{m}), as required.

In the next proposition we describe the behavior of L^{m}- and L^{2,m}-canonical forms and the sheaves Λ^{m} and Δ^{m} under taking Cartesian products.

Proposition 1.12. Let X_{1}, X_{2} be normal complex spaces, let D_{i} ⊆ X_{i}, i = 1, 2, be closed analytic subsets and let D denote the closed analytic subset X_{1}×D_{2}∪D_{1}×X_{2} of the product X := X_{1} × X_{2}. With p_{i} : X → X_{i} being the canonical projection (i = 1, 2) the following hold.

(a) The sheaves L^{m}, L^{2,m}, Λ^{m} and Δ^{m} (m ≥ 1) are compatible with taking products, i.e. with H any one of these sheaves we have

H_{X,D} ≅ p_{1}^{∗}(H_{X_{1},D_{1}}) ⊗ p_{2}^{∗}(H_{X_{2},D_{2}}).

(b) If D_{1} and D_{2} are divisors then for all k, m ∈ ℤ

O_{X}(mK_{X} + kD) ≅ p_{1}^{∗}(O_{X_{1}}(mK_{X_{1}} + kD_{1})) ⊗ p_{2}^{∗}(O_{X_{2}}(mK_{X_{2}} + kD_{2})) .

(c) If D_{1}, D_{2}, are divisors then D is ℚ-Cartier if and only if D_{1} and D_{2} are ℚ-Cartier.

Proof. (b) is obvious on the regular part of X. The sheaves on both sides of (b) are reflexive and so are determined by their restrictions to the regular parts, whence (b) follows.

In order to deduce (a), let π_{i} : X′_{i} → X_{i} (i = 1, 2) be resolutions of singularities such that D′_{i} := π_{i}^{−1}(D_{i} ∪ Sing X_{i}) are SNC divisors in X′_{i}. The product X′ := X′_{1} × X′_{2} then provides a resolution of singularities π : X′ → X such that D′ := π^{−1}(D ∪ Sing X) = X′_{1} × D′_{2} ∪ D′_{1} × X′_{2} is an SNC divisor. Let q_{i} : X′ → X′_{i} denote the canonical projection (i = 1, 2). By (b)

L_{X′,D′}^{m} ≅ q_{1}^{∗}(L_{X′_{1},D′_{1}}^{m}) ⊗ q_{2}^{∗}(L_{X′_{2},D′_{2}}^{m}) and L_{X′,D′}^{2,m} ≅ q_{1}^{∗}(L_{X′_{1},D′_{1}}^{2,m}) ⊗ q_{2}^{∗}(L_{X′_{2},D′_{2}}^{2,m}).

Applying π_{∗} and using the Künneth formula gives (a) for the cases H = L^{m} and H = L^{2,m}. Using (b) also the remaining two cases follow by taking quotients.

(c) is an immediate consequence of the fact that for n ∈ ℕ the divisor nD is Cartier if and only if nD_{1} and nD_{2} are Cartier.

For our applications to quasihomogeneous singularities it is important to study the behaviour of L^{m}- and L^{2,m}-canonical forms under finite group actions.

Proposition 1.13. Let G be a finite group acting on a normal complex space Y, and let π : Y → X := Y/G be the canonical morphism onto the orbit space. Let D ⊆ X be an analytic subset and assume that π is unramified in codimension one outside E := π^{−1}(D)_{red}. Then the following hold.

(a) L_{X,D}^{m} = π_{∗}(L_{Y,E}^{m})^{G} and L_{X,D}^{2,m} = π_{∗}(L_{Y,E}^{2,m})^{G} .

(b) Λ_{X,D}^{m} = (π_{∗}Λ_{Y,E}^{m})^{G} and Δ_{X,D}^{m} = (π_{∗}Δ_{Y,E}^{m})^{G}.

(c) If D is a divisor, then π^{∗}(D) is ℚ-Cartier if and only if D itself is ℚ-Cartier.

Proof. In order to show (a) we may assume that D, and then also E, are Weil divisors, see 1.7. With E′ := π^{−1}(D ∪ Sing X), 1.9 (b) implies that L_{X,D}^{m} = π_{∗}(L_{Y,E′}^{m})^{G} and L_{X,D}^{2,m} = π_{∗}(L_{Y,E′}^{2,m})^{G}. As E and E′ are equal in codimension 1, (a) follows from 1.7.

(a) implies in particular that

(1)  O_{X} (m(K_{X} + D)) = π_{∗}(O_{Y} (m(K_{Y} + E)))^{G}
(2)  O_{X} (mK_{X} + (m − 1)D) = π_{∗}(O_{Y} (mK_{Y} + (m − 1)E))^{G} .

Indeed, the involved sheaves are reflexive and so it is sufficient to verify the equalities on the part where X and Y are both smooth. Now (b) follows from (a) in view of (1) and (2). The statement of (c) is well known (see e.g., [18, 6]).

Recall the following notion.

Definition 1.14. A morphism of complex spaces π : Y → X is called nondegenerate if for every point y ∈ Y we have dim_{y} Y = dim_{y} π^{−1}(π(y)) + dim_{π(y)} X.

For instance, every finite surjective morphism is non-degenerate. The following proposition will be used in the sequel.

Proposition 1.15. Let π : (Y, y) → (X, x) be a non-degenerate morphism of normal complex space germs, and let D ⊆ X be a reduced Weil divisor with preimage E := π^{−1}(D)_{red}. Then there are (in general non-canonical) injections of O_{X,x}- modules

(Λ_{X,D}^{m})_{x} ↪ (Λ_{Y,E}^{m})_{y} (3)

and

(Δ_{X,D}^{m})_{x} ↪ (Δ_{Y,E}^{m})_{y} . (4)

Proof. We restrict ourselves to the proof (4), the other one being similarly. First we treat the special case that π is finite. Pulling back differential forms induces an injective map

π^{∗} : O_{X}(mK_{X} + (m − 1)D) ↪ O_{Y} (mK_{Y} + (m − 1)E). (5)

The analytic sets E and π^{−1}(D ∪ Sing X)_{red} are equal in codimension 1 and so by 1.9 (b) for a local section, say, η of H^{0}(U, O_{X}(mK_{X} + (m − 1)D)) over some open subset U ⊆ X

η ∈ H^{0}(U, L_{X,D}^{2,m}) if and only if π^{∗}(η) ∈ H^{0}(π^{−1}(U), L_{Y,E}^{2,m}).

Thus (5) induces an injective map as in (4).

In the general case we can find functions f_{1}, …, f_{d} ∈ O_{Y,y} vanishing at y, where d := dim_{y} π^{−1}(x), such that f := (f_{1}, …, f_{d}) restricts to a finite map of germs (π^{−1}(x), y) → (ℂ^{d}, 0). Thus, letting Z := X × ℂ^{d} and z := (π(y), 0), we can factorize π into two maps

π : (Y, y) ⟶^{π×f} (Z, z) ⟶^{pr_{1}} (X, x),

where π × f is finite. With D_{Z} := D × ℂ^{d} we have (see 1.12 (a))

(Δ_{Z,D_{Z}}^{m})_{z} ≅ (Δ_{X,D}^{m})_{x} ⊗_{O_{X,x}} O_{Z,z}, (6)

Applying the first part of the proof, (4) follows.

1.3. Logarithmic m-canonical forms versus log canonical, log terminal and rational singularities. Recall the following notions.

Definition 1.16. Let X be a normal complex space and let D be a reduced effective Weil divisor on X such that the divisor K_{X} + D is ℚ-Cartier. Let σ : X′ → X be a resolution of singularities such that D′ = σ^{−1}(D∪Sing X)_{red} is an SNC divisor. Write

σ^{∗}(K_{X} + D) = (K_{X′} + D′) − ∑_{i} a_{i}E_{i} with a_{i} ∈ ℚ ∀i,

where the summation is taken over the set of all divisorial irreducible components E_{i} of the exceptional set of the blowup σ. (The number a_{i} is the so called log discrepancy of E_{i} [5, 2.5.3].) One says^{2} that the pair (X, D) has

1. log canonical singularities if a_{i} ≥ 0 ∀i.

2. log terminal singularities^{3} if a_{i} > 0 ∀i for every resolution σ : X′ → X. Following [13] we will simply say that (X, D) is log canonical (lc, for short) resp.,

log terminal (lt, for short). In the case when D is the zero divisor one says that X (instead of (X, 0)) has log canonical resp. log terminal singularities.

3. (Artin [1]) X is said to have rational singularities if for a resolution of singularities σ : X′ → X the higher direct image sheaves R^{i}σ_{∗}(O_{X}), i ≥ 1, vanish.

In the next proposition we recall Kempf’s characterization of rational singularities in terms of canonical forms. Moreover, we characterize log canonical and log terminal singularities in terms of logarithmic m-canonical forms.

Proposition 1.17. (a) (Kempf [11, Prop. on p. 50]) X has rational singularities

if and only if X is Cohen-Macaulay and L_{X}^{2,1} ≅ O_{X}(K_{X}) i.e., Δ_{X}^{1} = 0. Assume further that K_{X} + D is a ℚ-Cartier divisor. Then

(b) (X, D) is lc if and only if L_{X,D}^{m} = O_{X}(m(K_{X} + D)) for all m ≥ 1 or, equivalently, Λ_{X,D}^{m} = 0.

(c) X is lt if and only if L_{X,D}^{2,m} = O_{X}(mK_{X} + (m − 1)D) for all m ≥ 1 or, equivalently, Δ_{X,D}^{m} = 0.

Proof. (a) The original result of Kempf (formulated only for algebraic singularities) generalizes to the complex analytic setting in view of the fact that the Grauert-Riemenschneider vanishing theorem [7] also holds for non-algebraic singularities. The latter follows from a more general result of Moriwaki [16, Thm. 3.2].

(b) By assumption k(K_{X} + D) is a Cartier divisor for some k ∈ ℕ. Fix a resolution of singularities σ : X′ → X such that D′ := σ^{−1}(D ∪ Sing X)_{red} is an SNC divisor. By 1.16, (X, D) is lc if and only if

O_{X′} (σ^{∗}(k(K_{X} + D))) ⊆ O_{X′} (k(K_{X′} + D′)) . (7)

Thus for a local section ω of O_{X}(m(K_{X} +D)) the tensor power ω^{⊗k} pulls back to a section σ^{∗}ω^{⊗k} of O_{X′}(km(K_{X′} + D′)), whence σ^{∗}(ω) is a section of O_{X′}(m(K_{X′} + D′)) and so by definition (see 1.5)

O_{X}(m(K_{X} + D)) ⊆ L_{X,D}^{m} for all m ≥ 1 .

2See e.g. [14, Def. 2.34], [13, Def. 3.5].

3In [14] these singularities are called purely log terminal.

Since L_{X,D}^{m} ⊆ O_{X}(m(K_{X} + D)) (see 1.7 (b)), the latter means that

Λ_{X,D}^{m} = 0 for all m ≥ 1 . (8)

Conversely, if Λ_{X,D}^{k} = 0 then (7) holds, proving (b). The proof of (c) is similarly and left to the reader.

For further purposes, it is convenient to introduce the following definition.

Definition 1.18. We will say that the pair (X, D) is L^{2}-log terminal (L^{2}-lt, for short) if Δ_{X,D}^{m} = 0 ∀m ≥ 1.

Proposition 1.19. Let X be a normal complex space and D be a divisor on X such that Sing X ⊆ D. Assume that both K_{X} and D are ℚ-Cartier. If (X, D) is lc, then it is L^{2}-lt.

Proof. Let as above σ : X′ → X be a resolution of singularities such that D′ := σ^{−1}(D)_{red} = σ^{−1}(D∪Sing X)_{red} is an SNC divisor. We need to show that O_{X} (mK_{X}+ (m − 1)D) ⊆ L_{X,D}^{2,m} for all m ∈ ℕ or, equivalently, that for every local section ω of O_{X} (mK_{X} + (m − 1)D) ⊆ O_{X} (m(K_{X} + D)) the form σ^{∗}(ω) extends to a section in L_{X′,D′}^{m}(−D′). By assumption, (8) holds and therefore σ^{∗}(ω) gives a section in L_{X′,D′}^{m}. Choose k such that kD is a Cartier divisor, so that kD is locally given on X by one equation, say f = 0. The form ω^{⊗k} is a section in O_{X} (kmK_{X} + k(m − 1)D) = L_{X,D}^{km}(−kD), and so it can be written locally as f · η for some section η in L_{X,D}^{km}. Hence σ^{∗}(ω^{⊗k}) = σ^{∗}(fη) = σ^{∗}(f) · σ^{∗}(η) becomes a section of L_{X′,D′}^{km} that vanishes along D′. It follows that π^{∗}(ω) as a section of the sheaf L_{X′,D′}^{m} vanishes along D′ as well, which implies the assertion.

Remark 1.20. (1) Thus for pairs (X, D) satisfying the assumptions of 1.19 the following inclusions hold:

(lt) ⊆ (lc) ⊆ (L^{2} − lt) ,

whereas for D = 0 by 1.17 (c) we have:

(lt) = (L^{2} − lt) ⊆ (lc) .

The latter equality is no longer true for pairs (X, D) if D ≠ 0. The simplest example is given by the union D of two smooth curves on a smooth surface X meeting transversally. The pair (X, D) is not lt although it is lc and hence (by 1.19) it is L^{2}-lt.

(2) Using the definition of log canonical forms given in 1.8 one can also characterize log canonical singularities if D = ∑ a_{i}D_{i} is a ℚ-divisor with 0 < a_{i} ≤ 1. Moreover, using this characterization one can extend the notion of log canonical singularities without requiring that K_{X} + D is ℚ-Cartier.

Combining 1.9 (c) and 1.17 gives the following corollary.

Corollary 1.21. Let π : Y → X be a proper surjective bimeromorphic morphism of connected normal complex spaces and let D ⊆ X be an analytic subset. Denote by E the union of divisorial components of the analytic subset π^{−1}(D ∪ Sing X)_{red}.

(a) If (Y, E) is lc then

L_{X,D}^{m} = π_{∗}(O_{Y} (m(K_{Y} + E))) .

(b) If (Y, E) is L^{2}-lt then

L_{X,D}^{2,m} = π_{∗}(O_{Y} (mK_{Y} + (m − 1)E)) .

From 1.13 and 1.17 we obtain the following corollary.

Corollary 1.22. Let π : Y → X = Y/G, D ⊆ X and E ⊆ Y be as in 1.13.

a) If (Y, E) is L^{2}-lt then so is (X, D).

b) If D is a reduced divisor and π^{∗}(K_{X} + D) is ℚ-Cartier then the same is true for the property ‘lc’.

The next corollary follows from 1.12 (a),(c) using 1.17 (b).

Corollary 1.23. Under the assumptions as in 1.12 if D_{1}, D_{2} are divisors then (X, D) := (X_{1} × X_{2}, X_{1} × D_{2} ∪ D_{1} × X_{2}) is lc (resp., L^{2}-lt) if and only if (X_{1}, D_{1}) and (X_{2}, D_{2}) are lc (resp., L^{2}-lt).

The following corollary is an immediate consequence of 1.15 and 1.17. Part (a) was shown by Bingener and Storch, whereas part (b) generalizes a result of Ishii and Kollár.

Corollary 1.24. For a non-degenerate surjective morphism of normal complex spaces π : Y → X the following hold.

(a) [2, 5.7] If Y has rational singularities and X is Cohen-Macaulay, then X has also rational singularities.

(b) (cf. [10, 1.7.II], [5, 20.3.3]) Let D ⊆ X be a reduced Weil divisor with preimage E := π^{−1}(D)_{red}. If (Y, E) is L^{2}-lt then so is (X, D). If K_{X} +D is ℚ-Cartier, then the same holds for the property ‘lc’.

1.4. Plurigenera of isolated singularities. Let X be a normal complex space. Recall the following notions and facts [20, 12, 15].

Definition 1.25. By [20], the sections of the sheaf L_{X}^{2,m} over an open subset U ⊆ X are just the sections in H^{0}(U_{reg}, ω_{X}^{⊗m}) which are locally L^{2/m} on U. If (X, x) is an isolated singularity then

δ_{m} (X, x) = dim_{ℂ} [ω_{X, x}^{[m]}/(L_{X}^{2,m})_{x}] = dim_{ℂ} (Δ_{X}^{m})_{x}

is the m-th L^{2}-plurigenus as defined in [20]; here ω_{X}^{[m]} := O_{X}(mK_{X}). Similarly, the λ-plurigenera [15] are given by

λ_{m} (X, x) = dim_{ℂ} [ω_{X, x}^{[m]}/(L_{X}^{m})_{x}] = dim_{ℂ} (Λ_{X}^{m})_{x} .

1.26. Note that by Kempf’s criterion 1.17, an isolated singularity (X, x) is rational if and only if it is Cohen-Macaulay and δ_{1}(X, x) = 0. Moreover, if (X, x) is ℚ-Gorenstein^{4} then by 1.17 (X, x) is lt (resp., lc) if and only if all plurigenera δ_{m} (resp. λ_{m}), m ≥ 1, vanish.

In analogy with 1.24 we have the following result.

4Recall [10, 1.3-1.4] that (X, x) is Gorenstein iff it is Cohen-Macaulay and the canonical sheaf ω_{X} is invertible at x. It is called ℚ-Gorenstein if the sheaf ω_{X}^{[m]} is invertible at x for some m ∈ ℕ.

Corollary 1.27. If π : (Y, y) → (X, x) is a non-degenerate surjective morphism of germs of normal isolated singularities then

δ_{m}(X, x) ≤ δ_{m}(Y, y) and λ_{m}(X, x) ≤ λ_{m}(Y, y) ∀m ≥ 1 .

Moreover, if dim_{y} Y > dim_{x} X then

δ_{m}(X, x) = λ_{m}(X, x) = 0 ∀m ≥ 1 .

Proof. The first part follows from the inclusions (3) and (4) in 1.15. To show the second assertion, consider a factorization (Y, y) → (Z, z) → (X, x) as in the proof of 1.15. Assuming that δ_{m}(X, x) ≠ 0, by (6) in the proof of loc.cit. the module (ω_{Z}^{[m]}/L_{Z}^{2,m})_{z} = (Δ_{Z}^{m})_{z} has infinite dimension and is contained in the finite dimensional vector space (ω_{Y}^{[m]}/L_{Y}^{2,m})_{y} = (Δ_{Y}^{m})_{y}. Thus we get a contradiction, and so δ_{m}(X, x) vanishes. As λ_{m} ≤ δ_{m}, the λ-plurigenera vanish as well.

2. Logarithmic m-canonical forms on quasihomogeneous varieties

In this section we show how to compute L^{m}- and L^{2,m}-canonical forms on affine varieties with ℂ^{∗}-action, and we apply this to characterize different types of singularities.

2.1. Graded rings and associated schemes. For the convenience of the reader we recall some facts about projective schemes associated to graded rings which will be useful in the sequel (see [4, sect. 2]).

Notation 2.1. Let K denote a field of characteristic 0 containing all roots of unity. Recall that for a finitely generated graded K-algebra A = ⨁_{ν≥0} A_{ν} the associated projective scheme Proj A is defined by the set of all homogeneous prime ideals 𝔭 of A with A_{+} ⊈ 𝔭, where A_{+} = ⨁_{ν>0} A_{ν} is the augmentation ideal. The scheme Proj A is separated and proper over Spec A_{0} [8]. Furthermore, Proj A is covered by the affine open subsets

D_{+}(f) = D_{+}(fA) := {𝔭 ∈ Proj A : f ∉ 𝔭} ≅ Spec A_{(f)} ,

where f ∈ A_{+} is a homogeneous element and A_{(f)} := (A_{f})_{0} denotes the degree zero part of the localization A_{f}. Denote also V = Spec A and V^{∗} = V \V (A_{+}) (where, as usual, V (I) is the zero set of an ideal I, whereas for a homogeneous ideal I ⊆ A, V_{+}(I) stands for its zero set in Proj A). There is a natural surjective morphism V^{∗} → Proj A.

The multiplicative group 𝔾_{m} of the field K acts on A via t.a = t^{ν}a for t ∈ K^{∗} and a ∈ A_{ν}. If A = A_{0}[A_{1}] is generated as an A_{0}-algebra by the elements of degree 1 then V^{∗} → Proj A is a locally trivial 𝔾_{m}-bundle. In general, we have the following well known fact.

Lemma 2.2. Proj A ≅ V^{∗}/𝔾_{m}.

Proof. In lack of a reference we provide the simple argument: V^{∗} is covered by the 𝔾_{m}-invariant affine open subsets D(f) := Spec A_{f}, where f ∈ A_{d} with d > 0. As the ring of invariants (A_{f})^{𝔾_{m}} is just A_{(f)} = (A_{f})_{0} we obtain D_{+}(f) = D(f)/𝔾_{m} and so the lemma follows.

2.3. To describe the situation more closely, for f ∈ A_{d} with d > 0 denote F = F (f) = A/(f − 1) resp., Y = Y (f) = Spec F and consider the homomorphism of graded rings

μ : A_{f} → F [T, T^{−1}] given by a/f^{k} ⟼ \overline{a} · T^{deg a−kd} ,

where F [T, T^{−1}] is graded via F [T, T^{−1}]_{0} = F and deg T = 1. Clearly μ is degree preserving; in particular μ((A_{f})_{0}) ⊆ F. There is a commutative diagram

[figure: A_{(f)} =; (A_{f})_{0}; F; A_{f}; i; μ; F [T, T^{−1}]; i]

where i stands for the natural inclusions.

The cyclic group ℤ_{d} ≅ ⟨ζ⟩ generated by a primitive d-th root of unity ζ acts (homogeneously) on F [T, T^{−1}]; namely for b = \overline{a}T^{k} ∈ F [T, T^{−1}] with a ∈ A homogeneous we let ζ.b = ζ^{deg a−k}b . This action restricts to F = F [T, T^{−1}]_{0}; the next lemma describes the quotients.

Lemma 2.4. [4, 2.1-2.2] (a) μ provides isomorphisms A_{f} ≅ F [T, T^{−1}]^{ℤ_{d}} resp., A_{(f)} = (A_{f})_{0} ≅ F^{ℤ_{d}} onto the rings of invariants, and the horizontal arrows (i.e., the orbit maps) in the induced commutative diagram

Spec F [T, T^{−1}]=Y × 𝔾_{m} \overline{^{\underline{μ} ; /ℤ_{d}}} D(f) = Spec A_{f}

[figure: Spec F; =; Y; pr_{1}; /ℤ_{d}; D_{+}(f); /𝔾_{m}; =; Spec A_{(f)}]

are cyclic coverings; moreover, \underline{μ} is an étale covering.

(b) Furthermore, μ induces isomorphisms

(A_{f})_{≥0} ≅ F [T]^{ℤ_{d}} and (A_{f})_{≤0} ≅ F [T^{−1}]^{ℤ_{d}} ,

and so it provides ramified cyclic coverings

Y × 𝔸_{K}^{1} ⟶^{/ℤ_{d}} Spec (A_{f})_{≥0} resp., Y × 𝔸_{K}^{1} ⟶^{/ℤ_{d}} Spec (A_{f})_{≤0} .

The following two constructions will be important in our applications below.

Example 2.5. (Weighted blowup) Let S be an indeterminate of degree −1. Consider the graded subring

Â:= A[S]_{≥0} ≅ ⨁_{ν≥μ} A_{ν}S^{μ}

of the ring A[S]. By definition, the weighted blowup of V = Spec A is the scheme V′ := Proj Â. Note that for every element fS^{μ} ∈ Â with f ∈ A_{ν} and ν > μ we can write (fS^{μ})^{n} = f · (f^{n−1}S^{μn}), where deg f^{n−1}S^{μn} ≥ 0 for n ≫ 0. Therefore D_{+}(fS^{μ}Â) ⊆ D_{+}(fÂ), and so V′ is covered by the affine open subsets U_{f} := D_{+}(fÂ) = Spec Â_{(f)}, where

Â_{(f)} = ⨁_{ν≥0} (A_{f})_{ν} S^{ν} ≅ (A_{f})_{≥0} ≅ F [T]^{ℤ_{d}}

with T being an indeterminate of degree 1. Thus, U_{f} ≅ (Y × 𝔸_{K}^{1})/ℤ_{d}, see 2.4 (b). In particular, if K_{Y} is a ℚ-Cartier divisor then so is K_{U_{f}}, see 1.13 (c). Notice also that the blowup morphism σ : V′ → V = Spec A restricted to U_{f} is induced by the inclusion A ↪ (A_{f})_{≥0}, and that the exceptional divisor E = σ^{−1}(V (A_{+})) is isomorphic to Proj A under the natural morphism V′ = Proj Â → Proj A.

Example 2.6. (Weighted completion) Let T be an indeterminate with deg T = 1, and consider the projective scheme V̄ = Proj A[T]. The scheme V = Spec A is naturally isomorphic to the affine open subset D_{+}(T) ⊆ V̄ as

A[T]_{(T)} = ⨁_{ν≥0} A_{ν}T^{−ν} ≅ A .

Thus V̄ = V ∪ D_{∞} (where D_{∞} := {T = 0} ≅ Proj A is the ‘divisor at infinity’) is indeed proper over Spec A_{0}; we call V̄ the weighted completion of V. The divisor D_{∞} is covered by the affine open subsets U^{f} := D_{+}(fA[T]) ≃ Spec(A_{f})_{≤0} with f ∈ A_{d}, d > 0 (indeed,

A[T]_{(f)} = (A_{f}[T])_{0} = ⨁_{ν≥0}(A_{f})_{−ν}T^{ν} ≅ (A_{f})_{≤0} ).

Furthermore, by 2.4 (b) we have (A_{f})_{≤0} ≅ F [S]^{ℤ_{d}}, where this time S = T^{−1} is an indeterminate of degree −1, and again U^{f} ≃ (Y × 𝔸_{K}^{1})/ℤ_{d}.

Remark 2.7. If, more generally, A is a graded (but not necessarily positively graded) ring, then the above construction provides a partial completion V̄ of V.

From 2.1-2.4 we obtain the following proposition.

Proposition 2.8. With the assumptions of 2.1, if V^{∗} = V \V (A_{+}) is smooth then Proj A as well as the weighted blowup Proj A[S]_{≥0}, deg S = −1, have at most cyclic quotient singularities. Similarly, the subset V̄ \V_{+}(A_{+}A[T]) of the weighted completion V̄ = Proj A[T], deg T = 1, has at most cyclic quotient singularities.

Proof. For a homogeneous element f ∈ A_{d}, d > 0, the morphism

\underline{μ} : Y × 𝔾_{m} → Spec A_{f} = V \D_{f} ⊆ V^{∗}

is étale (see 2.4 (a)), and so Y = Spec F is smooth. Hence Spec A_{(f)} ≅ Spec (F^{ℤ_{d}}) = Y/ℤ_{d} has at most cyclic quotient singularities. As Proj A is covered by the affine open subsets D_{+}(f) = Spec A_{(f)}, this scheme has at most cyclic quotient singularities as well. The proof of the remaining cases is similarly using the descriptions of affine open coverings given in 2.5 and 2.6.

2.2. Characterizing log canonical forms in terms of gradings. We fix the following notation.

Notation 2.9. Let now K = ℂ, and let A be a normal ℂ-algebra of finite type with a grading A = ⨁_{ν≥0} A_{ν} (which corresponds to a ℂ^{∗}-action on A via t.a = t^{deg a}a, where t ∈ ℂ^{∗} and a ∈ A is homogeneous). In the sequel such an affine variety V = Spec A with ℂ^{∗}-action will be referred to as a quasihomogeneous variety (indeed,

there is a closed affine embedding V ↪^{i} ℂ^{n} given by a set of homogeneous generators of A and equivariant with respect to a diagonal ℂ^{∗}-action on ℂ^{n}). Note that the vertex set V (A_{+}) is just the fixed point set of the ℂ^{∗}-action on V.

Fix a ℂ^{∗}-invariant divisor D on V. In the next theorem we compute the Amodules

L_{A,D}^{m} := H^{0}(V, L_{V,D}^{m}) and L_{A,D}^{2,m} := H^{0}(V, L_{V,D}^{2,m})

as submodules of

H^{0}(V^{∗}, L_{V,D}^{m}) respectively H^{0}(V^{∗}, L_{V,D}^{2,m}),

where as before V^{∗} = V \V (A_{+}). Note that all these modules have an induced ℂ^{∗}-action and so they carry natural gradings.

Theorem 2.10. If V = Spec A with A = ⨁_{ν≥0} A_{ν} is a normal quasihomogeneous variety and D ⊆ V is a ℂ^{∗}-invariant divisor containing the divisorial part of V (A_{+}) then

L_{A,D}^{m} = H^{0}(V^{∗}, L_{V,D}^{m})_{≥0} and L_{A,D}^{2,m} = H^{0}(V^{∗}, L_{V,D}^{2,m})_{>0} . (9)

2.11. Observe first that if B := (A_{f})_{≥0} with f ∈ A_{+} homogeneous, then (B_{f})_{≥0} = B. Bearing this in mind we start with the following special case.

Lemma 2.12. With the notation and assumptions as in 2.10, suppose furthermore that there is an element f ∈ A_{d}, d > 0, such that A = (A_{f})_{≥0}. Then V^{∗} = Spec A_{f}, and (9) holds, i.e.

L_{A,D}^{m} = (L_{A_{f} ,D^{∗}}^{m})_{≥0}  and  L_{A,D}^{2,m} = (L_{A_{f} ,D^{∗}}^{2,m})_{>0}

with D^{∗} := D ∩ V^{∗}.

Proof. As in 2.4, consider the homogeneous homomorphism

μ : A = (A_{f})_{≥0} ⟶ F [T] with a/f^{k} ⟼ ā · T^{deg a−dk},

where T is an indeterminate with deg T = 1 and ā is the residue class of a in F = A/(f −1). By 2.4 this homomorphism induces an étale morphism A_{f} → F [T, T^{−1}], and

A_{f} ≅ F [T, T^{−1}]^{ℤ_{d}} , A = (A_{f})_{≥0} ≅ F [T]^{ℤ_{d}} , (A_{f})_{+} ≅ T F [T]^{ℤ_{d}}

with ℤ_{d} := ℤ/dℤ. Geometrically this means that, with Y = Spec F, the morphism

\underline{μ} : Y × ℂ → V = Spec A

gives a cyclic covering (whence V ≅ (Y × ℂ)/ℤ_{d}) non-ramified off \underline{μ}^{−1}(V (A_{+})) = Y × {0}, so that the restriction

\underline{μ}|_{Y ×ℂ^{∗}} : Y × ℂ^{∗} → V^{∗} = V \V (A_{+}) = Spec A_{f}

is an étale covering.

To compute L_{A,D}^{m} as a graded submodule of L_{A_{f} ,D^{∗}}^{m}, notice first that by 1.12 (a)

L_{F [T],_{D}}^{m} ≅ L_{F,D∩Y}^{m} ⊗ L_{ℂ[T],{0}}^{m},

where

D̃ := \underline{μ}^{−1}(D) = (D ∩ Y) × ℂ ∪ Y × \{0\} ⊆ Y × ℂ

(note that by our assumptions V (A_{+}) is a divisor contained in D). The module L_{ℂ[T],{0}}^{m} is equal to ℂ[T] · (dT/T)^{⊗m} and so it embeds into L_{ℂ[T,T^{−1}]}^{m} as the submodule of elements of degree ≥ 0. It follows that

L_{F [T],_{D}}^{m} = (L_{F [T,T^{−1}],(D∩Y)×ℂ^{∗}}^{m})_{≥0} .

Taking invariants with respect to ℤ_{d} and using 1.13 (a) we obtain that L_{A,D}^{m} = (L_{A_{f} ,D^{∗}}^{m})_{≥0}. The proof of the equality L_{A,D}^{2,m} = (L_{A_{f} ,D^{∗}}^{2,m})_{>0} is similarly using the fact that L_{ℂ[T],{0}}^{2,m} is generated by T (dT/T)^{⊗m}.

Proof of Theorem 2.10. Let S be an indeterminate with deg S = −1. Consider the weighted blowup (see 2.5)

σ : V′ := Proj(A[S]_{≥0}) → V

with exceptional divisor E ⊆ V′, and denote D′ := σ^{−1}(D)_{red} ∪E ⊆ V′. According to 1.9 (c) H^{0}(V′, L_{V′,D′}^{m}) ≅ L_{A,D}^{m}, and so we need to show that

H^{0}(V′, L_{V′,D′}^{m}) ≅ H^{0}(V^{∗}, L_{V,D}^{m})_{≥0} .

By 2.5 the affine open subsets

U_{f} = D_{+}(fÂ) = Spec(A_{f})_{≥0} ⊆ V′

with f ∈ A_{+} homogeneous form a covering of V′. If ω ∈ H^{0}(V^{∗}, L_{V,D}^{m}) is a homogeneous form, then σ^{∗}(ω) belongs to H^{0}(V′, L_{V′,D′}^{m}) if and only if, for all f as above, the form ω|_{D(f)} ∈ L_{A_{f} ,D∩D(f)}^{m} (where D(f) = Spec A_{f} ⊂ V^{∗}, see 2.2) extends to a form in L_{(A_{f})_{≥0},D∩U_{f}}^{m}. As by 2.11 and 2.12

L_{(A_{f})_{≥0},D∩U_{f}}^{m} = (L_{A_{f} ,D∩D(f)}^{m})_{≥0} ,

the result for L^{m}-forms follows. The proof in the case of L^{2,m}-forms is analogously and left to the reader.

2.13. In the next proposition we give a dual version of 2.10 which allows to control the L^{m} and L^{2,m}-forms at infinity (this will be useful later on, see 2.25). With the assumptions as in 2.10, consider the weighted completion V̄ = Proj A[T] with deg T = 1 (see 2.6). The subset V_{∞} := V̄ \V_{+}(A_{+}A[T]) contains the divisor at infinity D_{∞} ≅ Proj A of V̄. Let D̄ denote the union D ∪ D_{∞}.

Proposition 2.14. With the above notation we have

\begin{matrix}H^{0}(V_{∞}, L_{V_{∞},D̄}^{m}) = H^{0}(V^{∗}, L_{V,D}^{m})_{≤0} \\ H^{0}(V_{∞}, L_{V_{∞},D̄}^{2,m}) = H^{0}(V^{∗}, L_{V,D}^{2,m})_{<0} . \end{matrix}

Proof. The variety V_{∞} is covered by the affine open subsets U^{f} = Spec (A_{f})_{≤0}, where f ∈ A_{+} is homogeneous (see 2.6). Applying 2.12 to (A_{f})_{≤0} with the grading reversed we obtain

L_{(A_{f})_{≤0},D̄}^{m} ≅ (L_{A_{f} ,D̄}^{m})_{≤0} and L_{(A_{f})_{≤0},D̄}^{2,m} ≅ (L_{A_{f} ,D̄}^{2,m})_{<0} .

Now we can proceed as in the proof of 2.10; we leave the details to the reader.

2.3. Log terminal and log canonical singularities of quasihomogeneous varieties. With the notations as in 2.9, let ω_{A} := H^{0}(V, O_{V} (K_{V})) be the dualizing module of A and let

ω_{A}^{[m]}(kD) := H^{0}(V, O_{V} (mK_{V} + kD)).

We have seen in 1.7 (b) that L_{A,D}^{m} and L_{A,D}^{2,m} are in a natural way submodules of

ω_{A}^{[m]}(mD) resp. ω_{A}^{[m]}((m − 1)D). In the next theorem we identify the modules

Λ_{A,D}^{m} := H^{0}(V, Λ_{V,D}^{m}) and Δ_{A,D}^{m} := H^{0}(V, Δ_{V,D}^{m})

with certain graded pieces of ω_{A}^{[m]}(mD) resp. ω_{A}^{[m]}((m − 1)D).

Theorem 2.15. With the notation and assumptions as in 2.10 the following hold.

(a) If (V^{∗}, D^{∗}) is lc then

L_{A,D}^{m} ≅ ω_{A}^{[m]}(mD)_{≥0} and Λ_{A,D}^{m} ≅ ω_{A}^{[m]}(mD)_{<0} .

(b) Similarly, if (V^{∗}, D^{∗}) is L^{2}-lt then

L_{A,D}^{2,m} ≅ ω_{A}^{[m]}((m − 1)D)_{>0} and Δ_{A,D}^{m} ≅ ω_{A}^{[m]}((m − 1)D)_{≤0} .

(c) If (V^{∗}, D^{∗}) is lc, respectively L^{2}-lt then so is the weighted blowup (V′, D′) (see 2.5).

Proof. (a) Observe that in virtue of 1.17 (b) Λ_{V^{∗},D^{∗}}^{m} = 0 or, equivalently, L_{V^{∗},D^{∗}}^{m} = O_{V^{∗}}(m(K_{V^{∗}} + D^{∗})). Hence 2.10 implies that

L_{A,D}^{m} ≅ H^{0}(V^{∗}, O_{V} (m(K_{V} + D)))_{≥0}.

The module on the right contains H^{0}(V, O_{V} (m(K_{V} +D)))_{≥0} = ω_{A}^{[m]}(mD)_{≥0}, and by 1.7 (b) ω_{A}^{[m]}(mD) contains L_{A,D}^{m} whence (ω_{A}^{[m]}(mD))_{≥0} contains L_{A,D}^{m} = (L_{A,D}^{m})_{≥0}. Thus L_{A,D}^{m} and ω_{A}^{[m]}(mD)_{≥0} are equal, as required. The proof of (b) is similarly and left to the reader.

In order to show (c) consider first the special case treated in 2.12 so that A = (A_{f})_{≥0}. Then U_{f} = D_{+}(fÂ) = Spec (A_{f})_{≥0} = V and σ|_{U_{f}} = id, whence V′ = V = (Y × ℂ)/ℤ_{d} (see 2.4 (b)). If (V^{∗}, D^{∗}) is lc then (with the notation as in the proof of 2.12) the étale covering

(Y × ℂ^{∗}, \underline{μ}^{−1}(D ∩ V^{∗})) = (Y × ℂ^{∗}, (D ∩ Y) × ℂ^{∗})

of (V^{∗}, D^{∗}) is also lc. By 1.12 (Y, D∩Y) is lc and so, applying 1.12 again, (Y ×ℂ, D̃) (with D̃ = (D ∩ Y) × ℂ ∪ Y × \{0\} = \underline{μ}^{−1}(D) ⊆ Y × ℂ) is lc as well. As \underline{μ}|_{(Y ×ℂ^{∗})} is unramified, the divisor \underline{μ}^{∗}(K_{V} + D) on Y × ℂ is equal to K_{Y ×ℂ} + D̃ + λ(Y × \{0\}) for some λ ∈ ℤ; in particular, it is ℚ-Cartier. Taking quotients and using 1.13 and 1.17 (b) we deduce that (V, D) = (V′, D′) is also lc.

The general case follows easily from this with the same reasoning as in the proof of 2.10. The proof in the L^{2}-lt case is similarly and left to the reader.

Corollary 2.16. With the assumptions as in 2.10, the following hold.

(a) If (V^{∗}, D^{∗}) is lc and K_{V} + D is ℚ-Cartier then (V, D) is lc if and only if

Λ_{A,D}^{m} = ω_{A}^{[m]}(mD)_{<0} = 0 ∀m ≥ 1 .

(b) If (V^{∗}, D^{∗}) is L^{2}-lt then (V, D) is L^{2}-lt if and only if

Δ_{A,D}^{m} = ω_{A}^{[m]}((m − 1)D)_{≤0} = 0 ∀m ≥ 1 .

(c) ([4]) If V^{∗} has rational singularities and V is Cohen-Macaulay then V has rational singularities if and only if (ω_{A})_{≤0} = 0.

Proof. As the coherent sheaves Λ_{V,D}^{m} resp., Δ_{V,D}^{m} on the affine variety V are globally generated, under the assumptions of (a) resp., (b) they vanish. Now (a) and (b) follow immediately from 2.15 (a), (b) and 1.17 (b)-1.18. To prove (c) observe that ω_{A} = H^{0}(V, O_{V} (K_{V})). Thus (c) is a consequence of 2.15 (b) and 1.17 (a).

2.17. Assume now that ω_{A} is a free A-module so that

ω_{A} ≅ A[N_{A}]

for some N_{A} ∈ ℤ (where as usual A[N] is the module A equipped with the new grading A[N]_{i} := A_{i+N}). Note that by the homogeneous version of Nakayama’s lemma this assumption is satisfied if, for instance, A is a Gorenstein ring and A_{0} is a local ring. If moreover D is given by a homogeneous equation of degree d then

\begin{matrix}H^{0}(V, O_{V} (m(K_{V} + D))) ≅ A[m(N_{A} + d)] and \\ H^{0}(V, O_{V} (mK_{V} + (m − 1)D)) ≅ A[mN_{A} + (m − 1)d] . \end{matrix}

From 2.16 we obtain the following characterizations.

Corollary 2.18. Let (A, D) be as in 2.10. If ω_{A} ≅ A[N_{A}] and D is given by a homogeneous equation of degree d then the following hold.

(a) If (V^{∗}, D^{∗}) is lc then (V, D) is lc if and only if N_{A} + d ≤ 0.

(b) If (V^{∗}, D^{∗}) is L^{2}-lt then (V, D) is L^{2}-lt if and only if d > 0 and N_{A} + d ≤ 0 or d = 0 and N_{A} < 0.

(c) [4] If V^{∗} has rational singularities and A is Cohen-Macaulay then V has rational singularities if and only if N_{A} < 0.

Example 2.19. In particular, let V = Spec A be a normal complete intersection of dimension n given in ℂ^{n+s} by polynomials p_{1}, …, p_{s} ∈ ℂ^{n+s} which are quasihomogeneous of degrees d_{1}, …, d_{s} with respect to weights w_{j} ≥ 0, j = 1, …, n + s, so that

p_{i}(λ^{w_{1}}x_{1}, …, λ^{w_{n+s}}x_{n+s}) = λ^{d_{i}}p(x_{1}, …, x_{n+s}), i = 1, …, s .

It is well known (see e.g. [4, p.42]) that A is Gorenstein (whence Cohen-Macaulay) and

ω_{A} = A[N_{A}] with N_{A}(= N_{V}) := ∑_{i=1}^{s} d_{i} − ∑_{j=1}^{n+s} w_{j} . (10)

Example 2.20. As a concrete example, consider the polynomial x^{a}y^{d}+u^{b}+v^{c} = 0 with a, b, c, d ≥ 2. It is weighted homogeneous of degree abc with weights, say, deg x = bc, deg y = 0, deg v = ab and deg u = ac. The associated quasihomogeneous variety V = {p = 0} ⊆ ℂ^{4} has singularities only along the x- and y-axes. By 2.18 the singularities along the x-axis off the origin are log canonical iff 1/b + 1/c + 1/d ≥ 1; they are rational and, moreover, log terminal iff 1/b+1/c+1/d > 1 (indeed, locally near a point (x_{0}, 0, 0, 0) ∈ V^{∗} with x_{0} ≠ 0 the mapping

V ∋ (x, y, u, v) ⟼ (x, x^{a/d}y, u, v) ∈ Ṽ := {y^{d} + u^{b} + v^{c} = 0} ⊆ ℂ^{4}

is well defined and biholomorphic). Letting D be the divisor {x = 0} it follows that (V, D) is lc if and only if 1/b + 1/c ≥ 1. Moreover the singularities of V are rational (and log terminal) if and only if 1/a + 1/b + 1/c > 1 and 1/b + 1/c + 1/d > 1.

As another example we study varieties that are given by the maximal Pfaffians of a skew symmetric matrix (cf. [3]).

Example 2.21. Let R = ⨁_{i≥0} R_{i} be a finitely generated graded ℂ-algebra which is Gorenstein with ω_{R} ≅ R[N_{R}]. Consider a skew symmetric (2n + 1) × (2n + 1)- matrix (a_{ij}) of homogeneous elements of R and assume that deg a_{ij} = d_{i} + d_{j} − N, where N and d_{i}, 1 ≤ i ≤ 2n + 1, are positive integers satisfying ∑_{i} d_{i} = nN. The maximal Pfaffians generate a homogeneous ideal, say, I of R. Assume that the quotient A := R/I has dimension dim R − 3. Then by [3] the minimal resolution of A has the form

0 → R[−N] → ⨁_{j=1}^{2n+1} R[−N + d_{j}] ⟶^{(a_{ij})} ⨁_{i=1}^{2n+1} R[−d_{i}] → R → A → 0 .

Taking Ext_{R}(−, ω_{R}) of this sequence it follows that ω_{A} ≅ A[N_{A}] with N_{A} := N + N_{R}. Hence 2.18 applies in this situation.

2.4. Plurigenera of quasihomogeneous singularities. The results above provide the following explicit formulas for the plurigenera of isolated singularities of quasihomogeneous varieties, where as usual ω_{A}^{[m]} denotes the reflexive hull of the module ω_{A}^{⊗m} of Kählerian m-differentials on A.

Proposition 2.22. Let A = ⨁_{ν≥0} A_{ν} be a normal ℂ-algebra of finite type and assume that the corresponding affine variety V = Spec A has dim V ≥ 2 and at most isolated singularities. Then the following hold.

(a) If A_{0} ≠ ℂ (in particular, if V has at least two singular points) then δ_{m}(V, p) = λ_{m}(V, p) = 0 for all m ≥ 1 and p ∈ Sing V.

(b) If A_{0} = ℂ and V has a unique singular point p then for every m ≥ 1 we have

δ_{m}(V, p) = dim_{ℂ} (ω_{A}^{[m]})_{≤0}  and  λ_{m}(V, p) = dim_{ℂ} (ω_{A}^{[m]})_{<0} .

Proof. (a) Let D denote the divisorial part of V (A_{+}). Using 1.11 resp. 2.15 (b) we have an inclusion

Δ_{A}^{m} ⊆ Δ_{A,D}^{m} ≅ (ω_{A}^{[m]})((m − 1)D)_{≤0}.

By our assumption, dim_{ℂ} A_{0} = ∞. Since (ω_{A}^{[m]})((m − 1)D) is a torsion-free A_{0}- module, its A_{0}-submodule Δ_{A}^{m} is also torsion-free over A_{0}, and so either Δ_{A}^{m} = 0 or dim_{ℂ} Δ_{A}^{m} = ∞. On the other hand,

dim_{ℂ} Δ_{A}^{m} = ∑_{p∈Sing V} δ_{m}(V, p) < ∞ .

It follows that Δ_{A}^{m} = 0, and so δ_{m}(V, p) vanishes for all m ≥ 1 and p ∈ Sing V. Consequently λ_{m}(V, p) = 0, and (a) follows.

(b) Since by our assumption V_{+} = Spec A_{0} = {p} and so div V_{+} = ∅, by 2.15 (a), (b) (with D = 0) we have

Λ_{A}^{m} ≅ (ω_{A}^{[m]})_{<0}  and  Δ_{A}^{m} ≅ (ω_{A}^{[m]})_{≤0} .

Therefore (since p is the unique singular point of V) we obtain

δ_{m}(V, p) = ∑_{x∈Sing V} δ_{m}(V, x) = dim_{ℂ} Δ_{A}^{m} = dim_{ℂ} (ω_{A}^{[m]})_{≤0} ,

and similarly λ_{m}(V, p) = dim_{ℂ} (ω_{A}^{[m]})_{<0}, proving (b).

Example 2.23. Let V = Spec A be a complete intersection as in 2.19 and assume moreover that V has an isolated singularity at the origin 0 ∈ ℂ^{n+s}. As ω_{A} = A[N_{A}] we have ω_{A}^{[m]} = A[mN_{A}]. Now 2.22 (a) implies the result of [15] which says that

δ_{m} = dim ∑_{i≤0} A_{i+mN_{A}}  and  λ_{m} = dim ∑_{i<0} A_{i+mN_{A}} .

2.5. Log-Kodaira dimension of quasihomogeneous varieties. Recall the following notions.

Definition 2.24. [9] Let V be a smooth quasi-projective variety, and let V̄ be a smooth compactification of V by an SNC divisor D̄ = V̄ \V. The logarithmic plurigenera p̄_{m}(V) are defined by

p̄_{m}(V) = dim H^{0}(V̄ , OV̄ (m(KV̄ + D̄))) , m ≥ 1.

The logarithmic Kodaira dimension k̄ (V) of V is

k̄ (V) = \{ \begin{matrix}−∞ & if p̄_{m}(V) = 0 ∀m ∈ ℕ \\ min \{k ∈ ℕ : lim sup_{m→∞} p̄_{m}(V)/m^{k} < ∞\} & otherwise. \end{matrix}

Taking the intersection of the modules in 2.10 (or 2.15) and 2.14 we obtain the following proposition.

Proposition 2.25. Let A = ⨁_{i≥0} A_{i} be a normal graded ℂ-algebra of finite type with A_{0} = ℂ (so that V (A_{+}) = {p}) and let V = Spec A be the corresponding quasihomogeneous variety. If D is a homogeneous reduced divisor on V then

p̄_{m}(V \(D ∪ Sing V)) = dim (L_{A,D}^{m})_{0} .

In particular, k̄(V \(D ∪ Sing V)) = −∞ if and only if (L_{A,D}^{m})_{0} = 0 for all m ≥ 1.

Summarizing the preceding results gives the following theorem.

Theorem 2.26. With the assumptions as in 2.25 above, suppose in addition that the divisor D is given by a homogeneous equation of degree d and that the pair (V^{∗}, D^{∗}) is lc (where as before, V^{∗} := V \{p} and D^{∗} = D\{p} with {p} = V (A_{+}) being the vertex of V). Then the following hold.

(a) p̄_{m}(V \(D ∪ Sing V)) = dim (ω_{A}^{[m]})_{md} .

(b) If moreover V has an isolated singularity at p, then k̄(V^{∗}) = −∞ if and only if (ω_{A}^{[m]})_{0} = 0 for all m ≥ 1 or, equivalently, if and only if δ_{m}(V, p) = 0 for all m ≥ 1.

(c) Suppose that K_{V} is a ℚ-Cartier divisor, so that for a certain m_{0} ≥ 1 we have ω_{A}^{[m_{0}]} ≅ A[N] with some N ∈ ℤ. Then

k̄(V \(D ∪ Sing V)) = \{ \begin{matrix}−∞ & if N + m_{0}d < 0 \\ 0 & if N + m_{0}d = 0 \\ dim V − 1 & if N + m_{0}d > 0 . \end{matrix}

(d) In particular, with the assumptions as in (c) k̄(V \(D ∪ Sing V)) ≤ 0 if and only if (V, D) has a log canonical singularity at p. If moreover (V, p) is an isolated singularity then (V, p) is lt if and only if k̄(V^{∗}) = −∞.

Proof. By 2.15 (a) we have L_{A,D}^{m} ≅ ω_{A}^{[m]}[md]_{≥0} and so (a) follows from 2.25. Under the assumptions as in (b), by 2.25 we have p_{m}(V^{∗}) = (ω_{A}^{[m]})_{0}. Thus the first equivalence in (b) is an immediate consequence of (a). To show the second equivalence, choose a non-zero homogeneous element g ∈ A of some degree, say k > 0. If η is a non-zero form of degree s < 0 in ω_{A}^{[m]}, then g^{s}η^{k} is a non-vanishing form of degree 0 in ω_{A}^{[mk]}. In other words, (ω_{A}^{[m]})_{0} = 0 for all m ≥ 1 if and only if (ω_{A}^{[m]})_{≤0} = 0 for all m ≥ 1. In view of 2.22 (a) this proves (b).

In order to deduce (c) notice that (in virtue of (a))

p̄_{m}(V \(D ∪ Sing V)) = (ω_{A}^{[km_{0}]})_{km_{0}d} = A_{k(N+m_{0}d)} = 0 for all k ≥ 1

if and only if N + m_{0}d < 0. In the case N + m_{0}d = 0 we have (ω_{A}^{[km_{0}]})_{km_{0}d} = A_{0} = ℂ for all k ≥ 1. If N + m_{0}d > 0 then we may choose a number r such that dim (ω_{A}^{[rkm_{0}]})_{rkm_{0}d} = dim A_{rk(N+m_{0}d)} grows asymptotically like k^{dim A−1} = k^{dim Proj A}. (d) follows from 1.26 and 2.18 (a) above.

Remark 2.27. Let V = Spec A be a normal quasihomogeneous variety with a vertex set V_{0} = Spec A_{0} of positive dimension (that is, A_{0} ≠ ℂ). If V has isolated singularities then k̄(V \ Sing V) = −∞ (cf. 2.22 (b)). Indeed, the general fibre of the canonical projection q : V → V_{0} is a smooth quasihomogeneous variety with a vertex set of dimension 0, whence it is isomorphic to ℂ^{k} for some k > 0 (see e.g., [21, 8.5]).

Corollary 2.28. Let A := ℂ[X_{1}, …, X_{n+s}]/(p_{1}, …, p_{s}) be a normal quasihomogeneous complete intersection with weights w_{1}, …, w_{n+s} > 0 (see 2.19). With V = Spec A being the associated affine variety and D being a homogeneous degree d divisor on V, we have

k̄(V \(D ∪ Sing V)) = \{ \begin{matrix}−∞ & if N_{A} + d < 0 \\ 0 & if N_{A} + d = 0 \\ dim V − 1 & if N_{A} + d > 0 . \end{matrix}

Proof. As the weights are all positive we have A_{0} = ℂ, moreover ω_{A} = A[N_{A}] (see (10) in 2.19), and so 2.26 (c) (with m_{0} = 1) applies.

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Fakultät für Mathematik, Ruhr Universität Bochum, Geb. NA 2/72, Universitätsstr. 150, 44780 Bochum, Germany

   E-mail address: Hubert.Flenner@ruhr-uni-bochum.de

Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d’Hères cédex, France

   E-mail address: zaidenbe@ujf-grenoble.fr