Minimal model theory for log surfaces in Fujiki's class $\mathcal C$

We establish the minimal model theory for $\mathbb Q$-factorial log surfaces and log canonical surfaces in Fujiki's class $\mathcal C$.


Introduction
A log surface (X, ∆) in Fujiki's class C consists of a compact normal analytic surface X that is bimeromorphically equivalent to a compact Kähler manifold and a Q-divisor ∆ on X whose coefficients are in [0, 1] ∩ Q such that K X + ∆ is Q-Cartier, that is, there exists a positive integer m such that m∆ is integral and ω ⊗m X ⊗ O X (m∆) * * is locally free, where ω X is the canonical sheaf of X. In this paper, we establish the following theorem, which is a generalization of the minimal model theory for projective Q-factorial log surfaces obtained in [Fn4] (for some related topics, see [FT], [T], [Ha1], [Li], [Mi], and [Fn7,Section 4.10]).
(ii) (Mori fiber space). There exists a surjective morphism g : X * → W onto a normal projective variety W with connected fibers such that −(K X * + ∆ * ) is g-ample, dim W < 2, and the relative Picard number ρ(X * /W ) is one.
We note that (1) if X i 0 is projective for some i 0 then X i is automatically projective for every i, and (2) if X i 0 has only rational singularities for some i 0 then all the singularities of X i are rational for every i. We note that the above sequence of contraction morphisms is nothing but the minimal model program for projective Q-factorial log surfaces established in [Fn4] when X is projective and that X is automatically projective when κ(X, K X + ∆) = −∞ or 2.
Theorem 1.1 is not difficult to check once we know the minimal model theory for projective Q-factorial log surfaces in [Fn4], the Enriques-Kodaira classification of compact complex surfaces (see [BHPV, Chapter VI]), and some basic results on complex analytic spaces. We note that Theorem 1.1 includes the abundance theorem for Q-factorial log surfaces in Fujiki's class C.
Theorem 1.2 (Abundance theorem for Q-factorial log surfaces in Fujiki's class C, see Theorem 7.2). Let (X, ∆) be a Q-factorial log surface in Fujiki's class C. Assume that (K X + ∆) · C ≥ 0 for every curve C on X. Then K X + ∆ is semi-ample.
From the minimal model theoretic viewpoint, it is very natural to treat log canonical surfaces (X, ∆) in Fujiki's class C. Unfortunately, X is not necessarily Q-factorial in this case. So we can not directly apply Theorem 1.1 to log canonical surfaces in Fujiki's class C. In order to establish the minimal model theory for log canonical surfaces in Fujiki's class C, we prove the following theorem. Theorem 1.3 (Projectivity of log canonical surfaces in Fujiki's class C with negative Kodaira dimension, see Theorem 9.1). Let (X, ∆) be a log canonical surface in Fujiki's class C. Assume that κ(X, K X + ∆) = −∞ holds. Then X is projective.
The proof of Theorem 1.3 is much more difficult than we expected. We prove it with the aid of the classification of two-dimensional log canonical singularities. We note that there are non-projective normal complete rational surfaces (see [Ng,Section 4]). Fortunately, such surfaces do not appear under the assumption of Theorem 1.3. Since Nagata's example in [Ng,Section 4] is not log canonical, we explicitly construct some examples of complete non-projective log canonical algebraic surfaces in Section 12 for the reader's convenience. Our construction, which was suggested by Kento Fujita, is arguably simpler than Nagata's original and classical one (see [Ng,Section 4]). Here, we explain the most interesting example. Thus, by using Theorem 1.3, we have the following minimal model theory for log canonical surfaces in Fujiki's class C.
Theorem 1.5 (Minimal model theory for log canonical surfaces in Fujiki's class C). Let (X, ∆) be a log canonical surface in Fujiki's class C. Then we can construct a finite sequence of projective bimeromorphic morphisms starting from (X, ∆): such that (X i , ∆ i ), where ∆ i := ϕ i−1 * ∆ i−1 , is a log canonical surface in Fujiki's class C and that Exc(ϕ i ) =: C i ≃ P 1 and −(K X i + ∆ i ) · C i > 0 for every i. The final model (X * , ∆ * ) satisfies one of the following conditions.
(ii) (Mori fiber space). There exists a surjective morphism g : X * → W onto a normal projective variety W with connected fibers such that −(K X * + ∆ * ) is g-ample, dim W < 2, and the relative Picard number ρ(X * /W ) is one. We note that (1) if X i 0 is projective for some i 0 then X i is automatically projective for every i, (2) if X i 0 has only rational singularities for some i 0 then all the singularities of X i are rational for every i, and (3) if X i 0 is Q-factorial for some i 0 then so is X i for every i. We note that the above sequence of contraction morphisms is nothing but the usual minimal model program for projective log canonical surfaces (see [Fn4]) when X is projective and that X is automatically projective when κ(X, K X + ∆) = −∞ by Theorem 1.3.
In a series of papers (see [HP1], [HP2], and [CHP]), Campana, Höring, and Peternell established the minimal model program and the abundance theorem for Kähler threefolds (see also [HP3]). Their approach is essentially analytic. On the other hand, our approach is much more elementary than theirs and is not analytic. Although we mainly treat compact analytic surfaces in Fujiki's class C, we do not discuss Kähler forms (or currents) on singular surfaces (see [Fk]).
In Section 11, which is an appendix, we treat some vanishing theorems for proper bimeromorphic morphisms between analytic surfaces. They play an important role in this paper. Although they are more or less known to the experts, we explain the details for the reader's convenience because we can find no suitable references. We think that the results are useful for other applications. The most useful formulation is Theorem 11.3 (2). Theorem 1.6 (see Theorem 11.3). Let X be a normal analytic surface and let ∆ be an effective Q-divisor on X whose coefficients are less than one such that K X + ∆ is Q-Cartier. Let f : X → Y be a proper bimeromorphic morphism onto a normal analytic surface Y . Let L be a line bundle on X and let D be a Q-Cartier Weil divisor on X.
We explain the organization of this paper. In Section 2, we collect some basic definitions and results. In Section 3, we explain a very easy version of the basepoint-free theorem for projective bimeromorphic morphisms between surfaces (see Theorem 3.11). In Section 4, we collect some useful projectivity criteria for Q-factorial compact analytic surfaces. In Section 5, we discuss the minimal model program for Q-factorial log surfaces based on Sakai's contraction theorem, which is a slight generalization of Grauert's famous contraction theorem. Then we prove Theorem 1.1 except for the semi-ampleness of K X * + ∆ * . In Section 6, we briefly discuss the finite generation of log canonical rings of Q-factorial log surfaces, which is essentially contained in [Fn4], and some related topics. In Section 7, we prove the non-vanishing theorem and the abundance theorem. Precisely speaking, we explain how to modify the arguments in [Fn4] for Q-factorial log surfaces in Fujiki's class C. In Section 8, we discuss a contraction theorem for log canonical surfaces. A key point is that the exceptional curve automatically becomes Q-Cartier. This simple fact plays a crucial role in our minimal model theory for log canonical surfaces. Section 9 is devoted to the proof of Theorem 1.3, that is, the projectivity of log canonical surfaces in Fujiki's class C with negative Kodaira dimension. Our proof needs the classification of two-dimensional log canonical singularities. In Section 10, we prove Theorem 1.5, that is, the minimal model theory for log canonical surfaces in Fujiki's class C. In Section 11, which is an appendix, we discuss some vanishing theorems for proper bimeromorphic morphisms between normal analytic surfaces. Fortunately, we need no deep analytic methods except for the theorem on formal functions for proper morphisms between analytic spaces. In Section 12, which is also an appendix, we construct some complete non-projective log canonical algebraic surfaces. Numbers JP16H03925, JP16H06337. He would like to thank Kenta Hashizume, Haidong Liu, and Hiromu Tanaka for very useful comments and pointing out some mistakes. He also would like to thank Kento Fujita very much for useful discussions and advice, and for allowing him to use his ideas on complete non-projective algebraic surfaces in Section 12. Finally, he thanks Seiko Hashimoto for her help and the referees for many comments.
We will use the minimal model theory for projective log surfaces defined over C, the complex number field, established in [Fn4]. We will freely use the basic notation of the minimal model theory as in [Fn3] and [Fn7].

Preliminaries
In this section, we collect some basic definitions and results.
Definition 2.2 (Operations for Q-divisors). Let D be a Q-divisor on a normal analytic space. Then ⌈D⌉ (resp. ⌊D⌋) denotes the round-up (resp. round-down) of D. We put {D} := D − ⌊D⌋ and call it the fractional part of D.
Definition 2.3 (Algebraic dimensions). Let X be an irreducible compact analytic space. Let M(X) be the field of meromorphic functions on X. Then the transcendence degree of M(X) over C is called the algebraic dimension of X and is denoted by a(X). It is well known that 0 ≤ a(X) ≤ dim X holds. If a(X) = dim X holds, then we say that X is Moishezon. We note that if X is Moishezon then X is an algebraic space which is proper over C (see [U,Remark 3.7]).
For the basic properties of a(X), we recommend the reader to see [U,Section 3].
Definition 2.4. Let X be an irreducible compact normal analytic space such that X is Moishezon. Then we can obtain the perfect pairing induced from the intersection pairing of curves and line bundles as in the case where X is projective. We note that ρ(X) := dim R N 1 (X) < ∞ always holds. We call ρ(X) the Picard number of X. When X is an algebraic variety, NE(X) (⊂ N 1 (X)) denotes the Kleiman-Mori cone of X.
In this paper, we do not consider N 1 (X) when 0 ≤ a(X) < dim X. When X is a complete non-projective singular algebraic variety, NE(X) does not always behave well (see [Fn1], [FP], and Section 12).
Remark 2.5. Let X be a compact smooth analytic surface whose algebraic dimension a(X) is zero. Then it is well known that there are only finitely many curves on X (see Chapter IV, [BHPV,(8.2) Theorem]).
In the subsequent sections, we will repeatedly use the following well-known negativity lemma and its consequences without mentioning them explicitly. For the proof of Lemma 2.6, see [Ma,.
Lemma 2.6 (Negativity lemma). Let P ∈ Y be a germ of normal analytic surface and let f : X → Y be a proper bimeromorphic morphism from a smooth analytic surface X. Then f −1 (P ) is connected and has a negative definite intersection form.
Let us quickly recall the definition of the Iitaka dimension κ. For the details of κ, see [Nk] and [U].
Definition 2.7 (Iitaka dimensions). Let X be an irreducible compact normal analytic space and let L be a line bundle on X. Then we set κ(X, L) := lim sup m→∞ log dim C H 0 (X, L ⊗m ) log m and call it the Iitaka dimension of L. It is well known that κ(X, L) ∈ {−∞, 0, 1, 2, . . . , dim X} holds. We can define κ(X, D) for Q-Cartier Q-divisors D on X similarly.
We close this section with an easy lemma on rational singularities.
Lemma 2.8 (see [Fn6,Lemma 3.1]). Let ϕ : X → Y be a proper bimeromorphic morphism between normal analytic spaces. If R i ϕ * O X = 0 for every i > 0, then X has only rational singularities if and only if so does Y .
Proof. The problem is local. So we can freely shrink Y around an arbitrary given point. Let us consider a common resolution: By assumption and the Leray spectral sequence, we have R i p * O W ≃ R i q * O W for every i. This implies the desired statement.

Log surfaces
In this section, we define Q-factorial log surfaces and log canonical surfaces in Fujiki's class C and discuss a very easy version of the basepoint-free theorem for proper bimeromorphic morphisms between normal analytic surfaces.
In this paper, we adopt the following definition of analytic spaces in Fujiki's class C.
Definition 3.1 (Fujiki's class C). Let X be an irreducible compact analytic space. If X is bimeromorphically equivalent to a compact Kähler manifold, then we say that X is in Fujiki's class C.
Remark 3.2. Let X be an irreducible compact analytic space. We note that if X is Moishezon then X is automatically in Fujiki's class C.
We have a useful characterization of surfaces in Fujiki's class C.
Lemma 3.3. Let X be an irreducible compact normal analytic surface. Then X is in Fujiki's class C if and only if there exists a resolution of singularities f : Y → X such that Y is Kähler, that is, Y is a two-dimensional compact Kähler manifold and f is a bimeromorphic morphism.
Proof. Note that a compact smooth analytic surface S is Kähler if and only if the first Betti number b 1 (S) is even (see [BHPV,Chapter IV,(3.1) Theorem]). We also note that the first Betti number is preserved under blow-ups. Thus we can easily check the statement.
As an easy consequence of Lemma 3.3 and its proof, we have: Corollary 3.4. Let X be a compact normal analytic surface in Fujiki's class C. Let f : Y → X be any resolution of singularities. Then Y is a compact Kähler manifold.
Let us define canonical sheaves.
Definition 3.5 (Canonical sheaves). Let X be a normal analytic surface and let SingX denote the singular locus of X. Then we have codim X SingX ≥ 2 since X is normal. Let ω U be the canonical bundle of U := X \ SingX. We put ω X := ι * ω U , where ι : U ֒→ X is the natural open immersion, and call ω X the canonical sheaf of X.
Remark 3.6. Some normal analytic surface X does not admit any non-zero meromorphic section of ω X . However, if there is no risk of confusion, we use the symbol K X as a formal divisor class with an isomorphism O X (K X ) ≃ ω X and call it the canonical divisor of X.
In this paper, we adopt the following definition of log surfaces.
Definition 3.7 (Log surfaces). Let X be an irreducible compact normal analytic surface and let ∆ be a boundary Q-divisor on X. Assume that K X + ∆ is Q-Cartier. Note that this means that there exists a positive integer m such that m∆ is integral and that ω ⊗m X ⊗ O X (m∆) * * is locally free. Then the pair (X, ∆) is called a log surface. We say that a log surface (X, ∆) is in Fujiki's class C when X is in Fujiki's class C. Let (X, ∆) be a log surface. Then we usually call κ(X, K X + ∆) the Kodaira dimension of (X, ∆).
We need to define log canonical surfaces.
Definition 3.8 (Log canonical surfaces). Let (X, ∆) be a log surface and let f : Y → X be a proper bimeromorphic morphism from a smooth analytic surface Y . Then we can If the coefficients of ∆ Y are less than or equal to one for every f : Y → X, then (X, ∆) is called a log canonical surface.
The notion of Q-factoriality plays a crucial role in this paper.
Definition 3.9 (Q-factoriality). Let X be an irreducible compact normal analytic surface and let D be a Q-divisor on X. Then we say that D is Q-Cartier if there exists a positive integer m such that mD is Cartier. If every Weil divisor on X is Q-Cartier, then we say that X is Q-factorial.
Lemma 3.10 is well known.
Lemma 3.10. Let X be an irreducible compact normal analytic surface. Assume that X has only rational singularities. Then X is Q-factorial.
We close this section with a very easy version of the basepoint-free theorem for projective bimeromorphic morphisms between normal analytic surfaces (see also Remark 11.4 below).
Theorem 3.11. Let (X, ∆) be a log surface and let ϕ : X → Y be a projective bimeromorphic morphism onto a normal analytic surface Y . Assume that C := Exc(ϕ) is Q-Cartier, C ≃ P 1 , and −C and −(K X + ∆) are ϕ-ample. Let L be a line bundle on X with L · C = 0. Then there exists a line bundle L Y on Y such that L ≃ ϕ * L Y holds. In particular, X is Q-factorial if and only if so is Y .
Proof. In Step 1, we will prove the existence of L Y . In Steps 2 and 3, we will see that X is Q-factorial if and only if so is Y .
Step 1. Since C ≃ P 1 and L·C = 0, L| C is trivial. Since −C and −(K X +∆) are ϕ-ample, we may assume that C ≤ ∆ by increasing the coefficient of C in ∆. Let us consider the following short exact sequence: holds. Therefore, by Theorem 11.3 below, we get R 1 ϕ * (L ⊗ O X (−C)) = 0. Thus, we have the following short exact sequence: Step 2. Assume that X is Q-factorial. We take a prime divisor D on Y . Let D ′ be the strict transform of D on X. Then we can take a ∈ Q and a divisible positive integer m such that m(D ′ + aC) is Cartier and m(D ′ + aC) · C = 0. We put L = O X (m(D ′ + aC)) and apply the result obtained in Step 1 to L. Then mD = ϕ * (m(D ′ + aC)) is Cartier. This means that Y is Q-factorial.
Step 3. Assume that Y is Q-factorial. We take a prime divisor D on X.
We complete the proof of Theorem 3.11.

Projectivity criteria
Let us start with an easy but very useful projectivity criterion.
Lemma 4.1 (Projectivity of Q-factorial compact analytic surfaces). Let X be a Q-factorial compact analytic surface. Assume that the algebraic dimension a(X) of X is two, that is, X is Moishezon. Then X is projective.
Proof. By the assumption a(X) = 2, we can construct a proper bimeromorphic morphism f : Y → X from a smooth projective surface Y . By the assumption a(X) = 2 again, X is an algebraic space which is proper over C by Artin's GAGA (see [U,Remark 3.7]). We take a very ample effective Cartier divisor H on Y . We put A = f * H. Since X is Q-factorial, A is a Q-Cartier divisor. Then we have A · C = H · f * C > 0 for every curve C on X. In particular, we have A 2 > 0. Therefore, A is ample by Nakai-Moishezon's ampleness criterion for algebraic spaces (see [P, (1.4) Theorem]). This implies that X is projective.
The following corollary is obvious by Lemma 4.1.
Corollary 4.2. Let X be a Q-factorial compact analytic surface. Assume that there exists a line bundle L such that κ(X, L) = 2, that is, L is a big line bundle. Then X is projective.
Proof. By the assumption that L is big, we see that the algebraic dimension a(X) of X is two. Therefore, X is projective by Lemma 4.1.
By combining Lemma 4.1 with the Enriques-Kodaira classification, we obtain the following projectivity criterion.
Proof. We take the minimal resolution f : Y → X. We put K Y +∆ Y := f * (K X +∆). Then we see that ∆ Y is effective by the negativity lemma and that κ( Therefore, by the Enriques-Kodaira classification (see [BHPV, Chapter VI]), Y is a smooth projective surface. Thus, by Lemma 4.1, X is projective.
We will repeatedly use the above projectivity criteria throughout this paper.
We note that the statement of Theorem 1.3 looks very similar to that of Lemma 4.3. However, a log canonical surface is not necessarily Q-factorial. Therefore, Theorem 1.3 is much harder to prove than Lemma 4.3 (see Section 9).

Minimal model program for Q-factorial log surfaces
By repeatedly using Grauert's contraction theorem, we can easily run a kind of the minimal model program for Q-factorial log surfaces (X, ∆). We note that X is not assumed to be in Fujiki's class C in Theorem 5.1. A key point of Theorem 5.1 is the assumption that X is Q-factorial.
Theorem 5.1. Let (X, ∆) be a compact Q-factorial log surface. We assume that κ(X, K X + ∆) ≥ 0. Then we can construct a finite sequence of projective bimeromorphic morphisms Proof. Since κ(X, K X + ∆) ≥ 0, we can take an effective Cartier divisor D ∈ |m(K X + ∆)| for some large and divisible positive integer m. If m(K X + ∆) · C = D · C ≥ 0 for every curve C on X, then we set (X * , ∆ * ) := (X 0 , ∆ 0 ) = (X, ∆). So we assume that there exists some irreducible curve C on X such that D · C < 0. Then C is an irreducible component of Supp D and C 2 < 0. By Sakai's contraction theorem (see [S,Theorem (1.2)]), which is a slight generalization of Grauert's famous contraction theorem, we get a bimeromorphic morphism ϕ 0 : X = X 0 → X 1 that contracts C to a normal point of X 1 . We take a divisible positive integer l such that lC is Cartier. Then O X (−lC) is a ϕ 0 -ample line bundle on X. In particular, ϕ 0 is a projective morphism. By construction, Claim 1. C is isomorphic to P 1 .
Proof of Claim 1. We consider the following exact sequence: where I C is the defining ideal sheaf of C on X. As we saw above, R 1 ϕ 0 * O X = 0 holds. Since C is a curve, R 2 ϕ 0 * I C = 0 holds by the theorem on formal functions for proper morphisms between analytic spaces (see [BS,Chapter VI,Corollary 4.7]). Thus we get H 1 (C, O C ) = R 1 ϕ 0 * O C = 0 by the above exact sequence. This implies that C is isomorphic to P 1 . Therefore, by Theorem 3.11, we obtain that (X 1 , ∆ 1 ) is a Q-factorial log surface. Since Supp D has only finitely many irreducible components, we get a desired sequence of contraction morphisms and finally obtain (X * , ∆ * ) with (K X * + ∆ * ) · C ≥ 0 for every curve C on X * .
We note the following well-known lemma on extremal rays of projective surfaces.
Lemma 5.2. Let X be a normal projective surface and let C be a Q-Cartier irreducible curve on X with C 2 < 0. Then the numerical equivalence class [C] of C spans an extremal ray of the Kleiman-Mori cone NE(X) of X.
Proof. This is obvious. For the proof, see [KM,Lemma 1.22].
By Lemma 5.2, if X is projective in Theorem 5.1, then the minimal model program in Theorem 5.1 is nothing but the minimal model program for projective Q-factorial log surfaces formulated and established in [Fn4]. We also note that X is projective in Theorem 5.1 if the algebraic dimension a(X) of X is two by Lemma 4.1.
Let us prove Theorem 1.1 except for the semi-ampleness of K X * + ∆ * .
Proof of Theorem 1.1. If (K X +∆)·C ≥ 0 for every curve C on X, then we put (X * , ∆ * ) := (X, ∆). We will see that K X * + ∆ * is semi-ample in Theorem 7.2. We note that X is in Fujiki's class C. If κ(X, K X + ∆) = −∞, then X is projective by Lemma 4.3. Therefore, we can run the minimal model program for Q-factorial projective log surfaces in [Fn4] and finally get a Mori fiber space. Therefore, we may further assume that κ(X, K X + ∆) ≥ 0. Then we can apply Theorem 5.1 and finally get a model (X * , ∆ * ) such that (K X * +∆ * )·C ≥ 0 for every curve C on X * . In this case, by the abundance theorem: Theorem 7.2, we will see that K X * + ∆ * is semi-ample.
Since we have R 1 ϕ i * O X i = 0 (see Theorem 11.3), X i has only rational singularities if and only if so does X i+1 by Lemma 2.8. Thus we have (2).
Since each contraction ϕ i is projective, X i is projective when so is X i+1 . On the other hand, if X i is projective then so is X i+1 because ϕ i is nothing but the usual contraction morphism associated to a (K X i + ∆ i )-negative extremal ray (see Lemma 5.2). Thus, we have (1).
We obtained Theorem 1.1 except for the semi-ampleness of K X * + ∆ * , which will be proved in Section 7.

Finite generation of log canonical rings
In this section, we briefly discuss the finite generation of log canonical rings of pairs for the reader's convenience.
The following theorem is the main result of this section, which is essentially contained in [Fn4].
Theorem 6.1 (Finite generation of log canonical rings). Let (X, ∆) be a compact Qfactorial log surface. Then the log canonical ring As an easy consequence of Theorem 6.1, we have: Corollary 6.2. Let (X, ∆) be a compact log canonical surface. Then the log canonical ring is a finitely generated C-algebra.
We note that X is not assumed to be in Fujiki's class C in Theorem 6.1 and Corollary 6.2.
Proof of Corollary 6.2. Let f : Y → X be the minimal resolution. We put K Y + ∆ Y := f * (K X + ∆). Since (X, ∆) is log canonical, we see that ∆ Y is a boundary Q-divisor by the negativity lemma. By Theorem 6.1, the log canonical ring of (Y, ∆ Y ) is a finitely generated C-algebra. This implies that the log canonical ring of (X, ∆) is a finitely generated Calgebra.
Before we prove Theorem 6.1, let us recall the following easy well-known lemma for the reader's convenience.
Lemma 6.3. Let X be an irreducible compact normal analytic space and let L be a line bundle on X such that κ(X, L) ≤ 1. Then the graded ring is a finitely generated C-algebra.
Sketch of Proof. If κ(X, L) = −∞ or 0, then it is very easy to see that R(X, L) is a finitely generated C-algebra. If κ(X, L) = 1, then we can reduce the problem to the case where X is a smooth projective curve and L is an ample line bundle on X by taking the Iitaka fibration (see [Mo,(1.12) Theorem]). Thus, R(X, L) is a finitely generated C-algebra when κ(X, L) ≤ 1.
Let us prove Theorem 6.1.
Proof of Theorem 6.1. By Lemma 6.3, we may assume that κ(X, K X + ∆) = 2. Then, by Corollary 4.2, X is projective. In this case, the log canonical ring m≥0 H 0 (X, O X (⌊m(K X + ∆)⌋)) of (X, ∆) is a finitely generated C-algebra by the minimal model theory for projective Q-factorial log surfaces established in [Fn4].
Let us quickly see some results and conjectures on log canonical rings of higher-dimensional pairs.
Theorem 6.4 ( [BCHM], [FM], and [Fn5,Theorem 1.8]). Let (X, ∆) be a kawamata log terminal pair such that ∆ is a Q-divisor on X and that X is in Fujiki's class C. Then the log canonical ring is a finitely generated C-algebra.
Conjecture 6.5. Let (X, ∆) be a log canonical pair such that ∆ is a Q-divisor on X and that X is in Fujiki's class C. Then the log canonical ring is a finitely generated C-algebra.
Conjecture 6.5 is still widely open even when X is projective (see [Fn2], [Fn5], [FG], [Ha2], and [FL]). When X is projective in Conjecture 6.5, it is essentially equivalent to the existence problem of good minimal models for lower-dimensional varieties (for the details, see [FG]). Note that Corollary 6.2 completely settled Conjecture 6.5 in dimension two.
We close this section with a naive question.
Question 6.6. Let X be an irreducible compact normal analytic surface such that K X is Q-Cartier. Then is the canonical ring a finitely generated C-algebra?
We do not know the answer even when X is projective.

Abundance theorem
In this section, we prove the abundance theorem for Q-factorial log surfaces in Fujiki's class C.
Let us start with the non-vanishing theorem.
Proof. Let f : Y → X be the minimal resolution. We put K Y + ∆ Y := f * (K X + ∆). Then ∆ Y is an effective Q-divisor by the negativity lemma. If κ(Y, K Y ) ≥ 0, then we have Therefore, from now on, we assume that κ(Y, K Y ) = −∞. By Lemma 4.3, Y is projective. Therefore, by Lemma 4.1, X is projective since X is Q-factorial by assumption. Thus, by [Fn4,Theorem 5.1], we get κ(X, K X + ∆) ≥ 0.
The following theorem is the main result of this section, which is the abundance theorem for Q-factorial log surfaces in Fujiki's class C.
Theorem 7.2 (Abundance theorem for Q-factorial log surfaces in Fujiki's class C). Let (X, ∆) be a Q-factorial log surface in Fujiki's class C. Assume that (K X + ∆) · C ≥ 0 for every curve C on X. Then K X + ∆ is semi-ample.
For the proof of Theorem 7.2, we prepare some easy lemmas.
Lemma 7.3. Let X be a compact normal analytic surface and let L be a line bundle on X such that L · C ≥ 0 for every curve C on X. Assume that κ(X, L) = 1. Then L is semi-ample.

.1) Theorem].
Lemma 7.4. Let S be a compact smooth analytic surface in Fujiki's class C with κ(S, K S ) = 0. Assume that the algebraic dimension a(S) of S is less than two. Then S is bimeromorphically equivalent to a K3 surface or a two-dimensional complex torus.
Proof. Since S is in Fujiki's class C, the first Betti number b 1 (S) of S is even. Then the Enriques-Kodaira classification (see [BHPV, Chapter VI]) and κ(S, K S ) = 0 give the desired statement.
Lemma 7.5. Let B be a non-zero effective divisor on a two-dimensional complex torus S. Then we have κ(S, B) ≥ 1.
Proof. Without loss of generality, we may assume that B is an irreducible curve on S. If B is not an elliptic curve, then we can see that S is an Abelian surface (see [U,Lemma 10.8]). In this case, it is well known that |2B| is basepoint-free. In particular, κ(S, B) ≥ 1. Therefore, from now on, we assume that B is an elliptic curve. By taking a suitable translation, we may further assume that B is a complex subtorus of S. We set A = S/B. Let p : S → A be the canonical quotient map. Then B = p * P holds for P = p(B) ∈ A. Therefore, we obtain κ(S, B) = κ(A, P ) = 1. Hence, we always have κ(S, B) ≥ 1.
Lemma 7.6. Let S be a K3 surface and let B be a non-zero effective divisor on S such that B 2 = 0. Then we have κ(S, B) ≥ 1. Before we prove Theorem 7.2, we explicitly state the abundance theorem for log canonical surfaces in Fujiki's class C.
Corollary 7.7 (Abundance theorem for log canonical surfaces in Fujiki's class C). Let (X, ∆) be a log canonical surface in Fujiki's class C. Assume that (K X + ∆) · C ≥ 0 for every curve C on X. Then K X + ∆ is semi-ample.
Proof. Let f : Y → X be the minimal resolution of X. We put K Y + ∆ Y := f * (K X + ∆). Then ∆ Y is effective by the negativity lemma and is a subboundary Q-divisor on Y since (X, ∆) is log canonical. Therefore, ∆ Y is a boundary Q-divisor on Y . We can easily see that (K Y + ∆ Y ) · C Y ≥ 0 for every curve C Y on Y . Thus, by Theorem 7.2, K Y + ∆ Y is semi-ample. This implies that K X + ∆ is also semi-ample.
Let us start the proof of Theorem 7.2.
Step 3 (κ = 0). In this step, we assume κ(X, K X + ∆) = 0. If X is projective, then K X + ∆ is semi-ample by [Fn4,Theorem 6.2]. Here, we will explain that the proof of [Fn4,Theorem 6.2] works with some minor modifications when X is not projective. From now on, we will freely use the notation of the proof of [Fn4, Theorem 6.2].
The first part of the proof of [Fn4, Theorem 6.2] works without any changes (see page 361 in [Fn4]). We note that Z is a member of |m(K S + ∆ S )| for some divisible positive integer m. We also note that Mumford's arguments on indecomposable curves of canonical type work on smooth analytic surfaces (see [Mu2,Definition,Lemma, and Corollary 1 in Section 2]). Therefore, [Fn4,Lemma 6.3] holds true. In particular, we obtain that Z 2 = 0. The compact smooth surface S constructed in the first part of [Fn4, Theorem 6.2] is not projective. Of course, S is in Fujiki's class C because S is bimeromorphically equivalent to X by construction. As in the proof of [Fn4, Theorem 6.2], we will derive a contradiction assuming Z = 0.
By Lemma 4.3, we have κ(S, K S ) ≥ 0 since S is not projective. Thus, all we have to do is to check that Step 1 in the proof of [Fn4, Theorem 6.2] works when S is not projective. In Step 1 in the proof of [Fn4, Theorem 6.2], S is a compact smooth analytic surface with κ(S, K S ) = 0 and there are no (−1)-curves on S. Since S is in Fujiki's class C, the first Betti number b 1 (S) of S is even. Therefore, by the Enriques-Kodaira classification, S is a K3 surface or a complex torus (see Lemma 7.4). Then, by Lemmas 7.5 and 7.6, we have κ(X, K X + ∆) = κ(S, K S + ∆ S ) = κ(S, Z) ≥ 1 and get a contradiction. This means that Step 1 in the proof of [Fn4, Theorem 6.2] works when S is not projective. Therefore, K X + ∆ is always semi-ample. This is what we wanted.

Contraction theorem for log canonical surfaces
In this section, we discuss a contraction theorem for log canonical surfaces. Note that compact log canonical surfaces are not necessarily Q-factorial. Therefore, we need Mumford's intersection theory (see [Mu1], [Ma,, and [S]).
Definition 8.1 (Mumford's intersection theory). Let X be a normal analytic surface and let π : Y → X be a resolution. Let Exc(π) = i E i be the irreducible decomposition of the exceptional curve of π. Let D be a Q-divisor on X. Then we can define the inverse image π * D as where D † is the strict transform of D by π and the rational numbers α i are uniquely determined by the following linear equations: for every j. We call π * D the pull-back of D in the sense of Mumford. Of course, π * D coincides with the usual one when D is Q-Cartier. From now on, we further assume that X is compact. The intersection number D · D ′ (in the sense of Mumford) is defined to be the rational number (π * D) · (π * D ′ ), where D and D ′ are Q-divisors on X. We can easily see that D · D ′ is well-defined. We note that it coincides with the usual one when D or D ′ is Q-Cartier.
Let us recall some definitions and basic properties of surface singularities for the reader's convenience.
Definition 8.2 (Numerically log canonical and numerically dlt, see [KM,Notation 4.1]). Let X be a normal analytic surface and let ∆ be a Q-divisor on X. Let f : Y → U ⊂ X be a proper bimeromorphic morphism from a smooth surface Y to an open set U of X.
Then we can define f * (K U + ∆| U ) in the sense of Mumford (see Definition 8.1) without assuming that K U + ∆| U is Q-Cartier. Thus we can always write If ∆ is effective and a(E i , X, ∆) ≥ −1 for every exceptional curve E i and f : Y → U ⊂ X, then we say that (X, ∆) is numerically log canonical. We say that (X, ∆) is numerically dlt if (X, ∆) is numerically log canonical and there exists a finite set Z ⊂ X such that X \ Z is smooth, Supp ∆| X\Z is a simple normal crossing divisor on X \ Z, and a(E, X, ∆) > −1 for every exceptional curve E which maps to Z. It is well known that if (X, ∆) is numerically log canonical then K X + ∆ is Q-Cartier (see [Fn4,Proposition 3.5] and [Ma,). Moreover, if (X, ∆) is numerically dlt then X has only rational singularities (see [KM,Theorem 4.12]).
Remark 8.3. In Definition 8.2, we only require that Supp ∆| X\Z is a simple normal crossing divisor on X \Z in the classical topology. So it permits some irreducible component of Supp ∆| X\Z to have nodal singularities. Therefore, our definition does not coincide with [KM,Notation 4.1] when X is an algebraic surface. However, since we are mainly interested in local analytic properties of singularities of pairs (X, ∆), this difference causes no subtle problems.
We need the following contraction theorem for log canonical surfaces in Sections 9 and 10.
Theorem 8.4 (Contraction theorem for log canonical surfaces, see [Fn6,Theorem 4.1]). Let (X, ∆) be a compact log canonical surface and let C be an irreducible curve on X such that −(K X + ∆) · C > 0 and C 2 < 0, where C 2 is the self-intersection number of C in the sense of Mumford (see Definition 8.1). Then we have a projective bimeromorphic morphism ϕ : X → Y onto a normal surface Y such that Exc(ϕ) = C ≃ P 1 and that C passes through no non-rational singular points of X, that is, X has only rational singularities in a neighborhood of C. In particular, C is Q-Cartier. Moreover, (Y, ∆ Y ) is log canonical with ∆ Y := ϕ * ∆.
Proof. By Sakai's contraction theorem (see [S,Theorem (1.2)]), we have a bimeromorphic morphism ϕ : X → Y which contracts C to a normal point P ∈ Y . Since −(K X +∆)·C > 0, (Y, ∆ Y ) is numerically dlt in a neighborhood of P by the negativity lemma. Therefore, K Y + ∆ Y is Q-Cartier and Y has only rational singularities in a neighborhood of P . Of course, (Y, ∆ Y ) is a compact log canonical surface. By Theorem 11.3 below, R i ϕ * O X = 0 for every i > 0. Thus, X has only rational singularities in a neighborhood of C by Lemma 2.8. In particular, C is Q-Cartier (see [Nk,Chapter II,2.12. Lemma]). Since R 1 ϕ * O X = 0, we can easily check that C ≃ P 1 as in Claim 1 of the proof of Theorem 5.1. We see that ϕ is projective, −(K X + ∆) and −C are ϕ-ample by construction.
We close this section with simple but very important remarks.
Remark 8.5 (Extremal rays). Theorem 8.4 says that X has only rational singularities in a neighborhood of the exceptional curve C and then C is automatically Q-Cartier. Therefore, if X is projective, then C spans a (K X + ∆)-negative extremal ray R of NE(X) in the usual sense (see Lemma 5.2). Thus the contraction ϕ in Theorem 8.4 is nothing but the usual contraction morphism associated to the extremal ray R. In particular, Y is also projective when so is X.

Log canonical surfaces in Fujiki's class C with negative Kodaira dimension
The main purpose of this section is to prove the following theorem.
Let us recall the following well-known lemma for the reader's convenience (see [Ma,).
Lemma 9.2. Let (X, ∆) be a log canonical surface. Assume that P ∈ X is not a rational singularity. Then P ∈ Supp ∆ and X is Gorenstein at P .
Sketch of Proof. If P ∈ Supp ∆, then (X, 0) is numerically dlt in a neighborhood of P . In particular, X has only rational singularities in a neighborhood of P . Therefore, we have P ∈ Supp ∆. By the classification of two-dimensional log canonical singularities (see [KM,Theorem 4.7] and [Ma,), P ∈ X is a simple elliptic singularity or a cusp singularity (see [KM,Note 4.8] and [Ma,). We can check that all the other two-dimensional log canonical singularities are rational singularities (see [Ma,). Therefore, X is Gorenstein at P .
Let us start the proof of Theorem 9.1.
Proof of Theorem 9.1. We divide the proof into several small steps.
Step 1. In this step, we will prove that X is Moishezon, that is, the algebraic dimension a(X) of X is two.
Let f : Y → X be the minimal resolution of X with K Y + ∆ Y := f * (K X + ∆). Then (Y, ∆ Y ) is log canonical since so is (X, ∆) by assumption. By applying Lemma 4.3 to (Y, ∆ Y ), we obtain that Y is a smooth projective surface. This implies that X is Moishezon, that is, the algebraic dimension a(X) of X is two.
Step 2. If X has only rational singularities, then X is Q-factorial (see Lemma 3.10). In this case, by Lemma 4.1, X is projective since we have already known that X is Moishezon in Step 1.
Therefore, from now on, we may assume that X has at least one non-rational singular point.
Step 3. By applying Theorem 8.4 finitely many times, we may assume that if C is an irreducible curve on X with −(K X + ∆) · C > 0 then C 2 ≥ 0 holds (see Remark 8.6).
Step 4. Let g : Z → X be the minimal resolution of non-rational singularities of X. Then we get the following commutative diagram.
Since Z has only rational singularities by construction, we see that Z is Q-factorial (see Lemma 3.10). Therefore, Z is projective by Lemma 4.1 since Z is Moishezon (see Step 1). We put K Z +∆ Z := g * (K X +∆). Then ∆ Z is effective by the negativity lemma. Of course, (Z, ∆ Z ) is log canonical. More precisely, by Lemma 9.2, we have ∆ Z = i E i +g −1 * ∆, where Exc(g) = i E i . Since K Z + ∆ Z never becomes nef by κ(Z, K Z + ∆ Z ) = κ(X, K X + ∆) = −∞ (see Theorem 7.1), there exists a (K Z + ∆ Z )-negative extremal ray R = R ≥0 [C] of NE(Z), whereC is an irreducible rational curve on Z.
Step 5. In this step, we will prove the following claim.
Claim 2. The self-intersection numberC 2 ofC is non-negative, whereC is a (K Z + ∆ Z )negative extremal rational curve as in Step 4.
Proof of Claim 2. We assume thatC 2 < 0 holds. Since −(K Z + ∆ Z ) ·C > 0,C is not g-exceptional. If C X := g * C is disjoint from non-rational singularities of X, then we have (C X ) 2 < 0 and −(K X + ∆) · C X > 0 because g is an isomorphism in a neighborhood ofC. This is a contradiction by Step 3. Therefore, C X passes though at least one nonrational singular point P of X. This implies that C X ⊂ Supp ∆ by Lemma 9.2. Let C Y be the strict transform ofC on Y . Then we can easily see that (C Y ) 2 < 0 holds and that On the other hand, ∆ Y ·C Y ≥ 1 since C X passes through a non-rational singular point P and the reduced part of ∆ Y contains f −1 (P ). Thus we have This is a contradiction. Therefore,C 2 ≥ 0 holds. Therefore, every (K Z + ∆ Z )-negative extremal ray is spanned by an irreducible rational curveC withC 2 ≥ 0.
Step 6. We assume that there exists a (K Z + ∆ Z )-negative extremal ray R of NE(Z) spanned by an irreducible rational curveC withC 2 > 0. In this case, the numerical equivalence class ofC is an interior point of NE(Z) (see [KM,Corollary 1.21]) and it also generates an extremal ray of NE(Z). Therefore, it is easy to see that ρ(Z) = 1 and −(K Z + ∆ Z ) is ample. This is a contradiction because −(K Z + ∆ Z ) · E = 0 for every g-exceptional curve E on Z. Thus, this case does not happen.
Step 7. Hence we may assume that there exists a (K Z + ∆ Z )-negative extremal ray R spanned by an irreducible rational curveC withC 2 = 0. Then there exists a surjective morphism ϕ R : Z → W onto a smooth projective curve W such that ρ(Z/W ) = 1 and −(K Z + ∆ Z ) is ϕ R -ample, that is, ϕ R : Z → W is a Mori fiber space. Without loss of generality, we may assume thatC is a general fiber of ϕ R . Since the self-intersection number of any irreducible component of Exc(g) is negative, every irreducible component of Exc(g) is dominant onto W by ϕ R . Since −K Z ·C = 2 and ∆ Z ≥ i E i = Exc(g), we can check that E := Exc(g) is an irreducible curve such that ϕ R | E : E → W is an isomorphism. By the classification of two-dimensional log canonical singularities (see [KM,Theorem 4.7] and [Ma,), E is an elliptic curve and P is a simple elliptic singularity.
Claim 3. −K X is ample, that is, X is Fano. In particular, X is projective.
Proof of Claim 3. Since ϕ R : Z → W is a Mori fiber space and −(K Z + E) ·C = E ·C = 1, we obtain that −( Thus, we obtain that −K X is ample by Nakai-Moishezon's ampleness criterion for algebraic spaces (see [P, (1.4

) Theorem]).
Thus we see that X is projective when X has at least one non-rational singular point.
Therefore, we obtain that X is always projective if (X, ∆) is a log canonical surface in Fujiki's class C with κ(X, K X + ∆) = −∞.
Remark 9.3. Let (X, ∆) be a log canonical surface in Fujiki's class C with κ(X, K X +∆) = −∞. The proof of Theorem 9.1 shows that if X has non-rational singularities then X has one simple elliptic singularity and no cusp singularities.
By Theorem 9.1, we can freely apply the minimal model theory of projective log canonical surfaces established in [Fn4] to log canonical surfaces in Fujiki's class C with negative Kodaira dimension.

Proof of Theorem 1.5
In this section, we prove Theorem 1.5, that is, the minimal model theory for log canonical surfaces in Fujiki's class C. We give a detailed proof for the reader's convenience, although it is essentially the same as that of Theorem 1.1.
Proof of Theorem 1.5. If (K X + ∆) · C ≥ 0 for every curve C on X, then K X + ∆ is semiample by Corollary 7.7. So (X, ∆) is itself a good minimal model of (X, ∆). Therefore, we may assume that (K X + ∆) · C < 0 for some curve C on X. If X is projective, then we can run the minimal model program for projective log canonical surfaces and finally get a good minimal model or a Mori fiber space (see [Fn4]). Thus we may assume that X is not projective. By Theorem 9.1, we obtain κ(X, K X + ∆) ≥ 0. Therefore, we have an effective Cartier divisor D ∈ |m(K X + ∆)| for some positive divisible integer m. Since (K X +∆)·C < 0 for some curve C on X, C is an irreducible component of Supp D such that the self-intersection number C 2 is negative. We apply the contraction theorem: Theorem 8.4. Since there are only finitely many irreducible components of Supp D, we finally get (X * , ∆ * ) such that K X * + ∆ * is semi-ample after finitely many contractions.
Since R 1 ϕ i * O X i = 0 by Theorem 11.3 below, X i has only rational singularities if and only if so does X i+1 by Lemma 2.8. Therefore, we have (2).
Since ϕ i is projective by construction, X i is projective when so is X i+1 . If X i is projective, then ϕ i is the usual contraction morphism associated to a (K X i + ∆ i )-negative extremal ray (see Remark 8.5). This implies that X i+1 is also projective. Thus we have (1).
Since Exc(ϕ i ) ≃ P 1 , C i = Exc(ϕ i ) is Q-Cartier, and −C i is ϕ i -ample, we can easily check (3) by Theorem 3.11.

Appendix: Vanishing theorems
In this section, we quickly explain some vanishing theorems for the reader's convenience. Fortunately, we do not need difficult analytic methods. They follow from elementary arguments.
Let us start with the following vanishing theorem. We learned it from [KK] (see [K2,Theorem 10.4]).
Theorem 11.1 (Relative vanishing theorem). Let ϕ : V → W be a proper bimeromorphic morphism from a smooth analytic surface to a normal analytic surface W . Assume that there exists a point P ∈ W such that ϕ is an isomorphism over W \P . Let Exc(ϕ) = i E i be the irreducible decomposition of the ϕ-exceptional locus Exc(ϕ). Let L be a line bundle on V , let N be a Q-divisor on V , and let E = i b i E i be an effective Q-divisor on V . Assume that N · E i ≥ 0 and L · E i = (K V + E + N) · E i hold for every i. We further assume that one of the following conditions holds.
(2) 0 < b i ≤ 1 for every i and there exists some j such that b j = 1.
(3) 0 < b i ≤ 1 for every i and there exists some j such that N · E j > 0. Then R i ϕ * L = 0 holds for every i > 0.
Theorem 11.1 (1) is a special case of the Kawamata-Viehweg vanishing theorem. Note that condition (1) is most useful.
Remark 11.2. In Theorem 11.1, it is sufficient to assume that N is a Q-line bundle on V , that is, a finite Q-linear combination of some line bundles on V such that N · E i ≥ 0 and L · E i = (K V + E) · E i + N · E i hold for every i.
Proof of Theorem 11.1. Let us consider Z = i r i E i , where r i is a non-negative integer for every i. Although [K2,Theorem 10.4] is formulated and proved for two-dimensional regular schemes, the proof works for two-dimensional complex manifolds. Therefore, by the proof of [K2,Theorem 10.4], we obtain H 1 (Z, L ⊗ O Z ) = 0 (see also [KK,2.2.1 Theorem]). Hence, by using the theorem on formal functions for proper morphisms between analytic spaces (see [BS,Chapter VI,Corollary 4.7]), we get R i ϕ * L = 0 for every i > 0 (see also [KK,2.2.5 Corollary]).
As an application of Theorem 11.1, we can prove the following vanishing theorem, which is a Kawamata-Viehweg vanishing theorem for proper bimeromorphic morphisms between surfaces.
Theorem 11.3. Let X be a normal analytic surface and let ∆ be an effective Q-divisor on X such that K X + ∆ is Q-Cartier. Let f : X → Y be a proper bimeromorphic morphism onto a normal analytic surface Y . Let L be a line bundle on X and let D be a Q-Cartier Weil divisor on X. Assume that one of the following conditions holds.
(1) L · C + (D − (K X + ∆)) · C > 0 for every f -exceptional curve C on X and the coefficients of ∆ are less than or equal to one.
(2) L · C + (D − (K X + ∆)) · C ≥ 0 for every f -exceptional curve C on X and the coefficients of ∆ are less than one. Then R i f * (L ⊗ O X (D)) = 0 holds for every i > 0.
Let us prove Theorem 11.3. The following proof is essentially the same as that of [FT,Theorem 6.2].
Proof of Theorem 11.3. We divide the proof into small steps.
Step 1. Without loss of generality, we can freely shrink Y and assume that Y is a small relatively compact Stein open subset of normal analytic surface. We may further assume that f is an isomorphism outside P ∈ Y and f −1 (P ) is one-dimensional.
Step 2. When ⌊∆⌋ = 0, we can take an f -ample Cartier divisor H (see (1)). Then we can find an effective Q-divisor ∆ ′ on X such that ⌊∆ ′ ⌋ = 0 and that ∆ ′ is Q-linearly equivalent to ∆ + εH for some 0 < ε ≪ 1. More precisely, we take a general member B of |⌊∆⌋ + mH| for some large positive integer m and put By replacing ∆ with ∆ ′ , we can always assume that ⌊∆⌋ = 0.
Step 3. Let ϕ : Z → X be the minimal resolution of X. We set K Z + ∆ Z := ϕ * (K X + ∆). We note that ∆ Z is effective by the negativity lemma. We note that We put Θ := ∆ Z + {−ϕ * D}. Then We note that we can write {Θ} = E + M where E is a ϕ-exceptional effective Q-divisor with ⌊E⌋ = 0 and M is an effective Q-divisor such that every irreducible component of M is not ϕ-exceptional. Let us consider By Theorem 11.1, we have Therefore, we get the following short exact sequence By construction, ⌊Θ⌋ is ϕ-exceptional. Therefore, L⊗ϕ * O ⌊Θ⌋ (⌈ϕ * D⌉) is a skyscraper sheaf on X. Thus we obtain the following surjection . By the Leray spectral sequence, we have As before, we can write {Θ} = E ′ + M ′ where E ′ is a f • ϕ-exceptional effective Q-divisor with ⌊E ′ ⌋ = 0 and M ′ is an effective Q-divisor such that every irreducible component of M ′ is not f • ϕ-exceptional. By Theorem 11.1, we know that This implies that by (11.2). By the surjection (11.1), we get R 1 f * (L ⊗ O X (D)) = 0.
Step 4. Since f −1 (P ) is one-dimensional, R i f * (L ⊗ O X (D)) = 0 for every i > 2 by the theorem on formal functions for proper morphisms between analytic spaces (see [BS,Chapter VI,Corollary 4.7]).
Therefore, we have R i f * (L ⊗ O X (D)) = 0 for every i > 0.
We close this section with an obvious remark.
Remark 11.4. Theorems 11.1 and 11.3 can be formulated and proved easily for proper birational morphisms between algebraic surfaces defined over any algebraically closed field. Therefore, we can formulate and prove Theorem 3.11 for projective birational morphisms between (not necessarily complete) algebraic surfaces defined over any algebraically closed field. This is because the proof of Theorem 3.11 only needs the vanishing theorem: Theorem 11.3.

Appendix: Complete non-projective algebraic surfaces
In this section, we construct some examples of complete non-projective log canonical algebraic surfaces. From Example 12.1 to Example 12.5, we will work over C, the complex number field.
Let us start with Kollár's example. Although it is not stated explicitly in [K1], it does not satisfy Kleiman's ampleness criterion. We note that the arguments in Example 12.3 below help the reader understand Example 12.1. Therefore, we do not explain the details of Kollár's example.
Example 12.1 ([K1, Aside 3.46]). In this example, we will freely use Kollár's notation in [K1,Aside 3.46]. In [K1,Aside 3.46], we assume that C is an elliptic curve. Then the surface S constructed in [K1,Aside 3.46] is a complete non-projective algebraic surface with two simple elliptic singularities. In particular, S is Gorenstein and log canonical. Let C ′ be the strict transform of {1} × C on S. Then we have Pic(S) = ZO S (C ′ ). We can directly check that π 1 (S) = {1}, that is, S is simply connected, K S ∼ 0, H 1 (S, O S ) = 0, and H 2 (S, O S ) = C. Therefore, S is a log canonical Calabi-Yau algebraic surface. Of course, we have κ(S, K S ) = 0. Let F be a general fiber of the second projection P 1 × C → C and let F ′ be the strict transform of F on S. Then NE(S) = R ≥0 [F ′ ] holds. We note that O S (C ′ ) is positive on NE(S) \ {0}. However, C ′ is nef but is not ample. This means that Kleiman's ampleness criterion does not hold for S.
Let us prepare an easy lemma.
Lemma 12.2. Let C be a smooth projective curve and let A be an ample line bundle on C. We consider π : X := P C (O C ⊕ A) → C. Then O X (1) ≃ O X (C + ) is semi-ample, where C + := P C (A) is the positive section of π, and the complete linear system |O X (m)| only contracts the negative section C − := P C (O C ) of π to a point for some sufficiently large positive integer m.
Proof. We can easily check that O X (1) is semi-ample (see [La,Lemma 2.3.2]). We note that C − · O X (1) = 0 and D · O X (1) > 0 for every irreducible curve D on X with D = C − . Therefore, the complete linear system |O X (m)| contracts C − only.
Let us construct complete non-projective normal algebraic surfaces S with Pic(S) = {0}. The following construction was suggested by Kento Fujita.
Example 12.3. Let C be a smooth projective curve of genus ≥ 1 and let L = O C (L) be a non-torsion element of Pic 0 (C). We considr π : V := P C (O C ⊕ L) → C. Let C 1 (resp. C 2 ) be the section of π corresponding to O C ⊕ L → O C → 0 (resp. O C ⊕ L → L → 0). We note that C 2 ∼ π * L + C 1 holds. We take an arbitrary point P ∈ C and blow up P 1 and P 2 , where P i := π −1 (P ) ∩ C i for i = 1, 2, to get p : W → V . Let C ′ i be the strict transform of C i on W for i = 1, 2 and let ℓ be the strict transform of π −1 (P ) on W . Let E i denote the (−1)-curve on W with p(E i ) = P i for i = 1, 2. We put . We note that C ′ i ⊂ U i for i = 1, 2 and ℓ ⊂ U 0 by construction. Then we can realize U i as a Zariski open subset of P C (O C ⊕ A i ) with deg A i = 1 such that C ′ i corresponds to the negative section C − on P C (O C ⊕ A i ) for i = 1, 2. By Lemma 12.2, we can construct a projective birational morphism U i → S i onto a normal quasi-projective surface S i such that the exceptional locus is C ′ i for i = 1, 2. Since ℓ is a (−2)-curve on a smooth projective surface W , we can construct a projective birational morphism U 0 → S 0 onto a normal quasi-projective surface S 0 such that the exceptional locus is ℓ and that ℓ is contracted to an A 1 singularity. Now S 0 , S 1 , and S 2 can be glued together to get a birational contraction morphism q : W → S onto a complete normal algebraic surface S which only contracts C ′ 1 , C ′ 2 , and ℓ. Claim 4. Pic(S) = {0} holds.
Proof of Claim 4. We take an arbitrary Cartier divisor D on S. We put D := q * D and D † := p * D. Then we can write D † ∼ π * M + kC 1 for some integer k and some Cartier divisor M on C because V is P C (O C ⊕ L). We note that we can write for some integers m 1 and m 2 . By construction, O W (D)| C ′ i is trivial for i = 1, 2. Therefore, we have . Since deg L = 0, we have m 1 = m 2 . By assumption, L = O C (L) is a non-torsion element of Pic 0 (C). Thus, we get k = 0. By construction again, we have D · ℓ = 0. Therefore, holds. Since we have already known that m 1 = m 2 holds, we obtain that m 1 = m 2 = 0.
i is trivial for i = 1, 2, we see that O C (M) is trivial. Therefore, we obtain D ∼ 0. So we get D ∼ 0. This means that Pic(S) = {0}.
By Claim 4, we see that S is obviously non-projective. By construction, we can see that π 1 (S) = {1}, that is, S is simply connected.
From now on, we assume that C is an elliptic curve. Then S has two simple elliptic singularities and one A 1 singularity. Moreover, S is Gorenstein and K S ∼ 0. We note that K V + C 1 + C 2 ∼ 0 and K W + C ′ 1 + C ′ 2 ∼ 0 by construction. We can easily check that H 1 (S, O S ) = 0 and H 2 (S, O S ) = C by using the Leray spectral sequence. Therefore, S is a complete non-projective log canonical Calabi-Yau algebraic surface with Pic(S) = {0}.
From now on, by taking blow-ups of S in Example 12.3, we construct complete nonprojective algebraic surfaces with large Picard number.
Example 12.4. Let S be the surface constructed in Example 12.3, which is birationally equivalent to P C (O C ⊕ L). As we saw above, Pic(S) = {0} holds. We take the minimal resolution µ : S → S of the unique A 1 singularity of S. Since S is simply connected, so is S. Let E denote the exceptional curve of µ. Of course, E is the strict transform of ℓ in Example 12.3 and is a (−2)-curve on S. Proof of Claim 5. By construction, we see that E ≃ P 1 and that −E and −(K S + E) are both µ-ample. Thus, by Theorem 3.11 (see also Remark 11.4), we have Pic( S) ⊗ Q = QO S (E) since Pic(S) = {0}. Let N be a torsion element of Pic( S). Then we have N · E = 0. Therefore, by Theorem 3.11 again, N is trivial. This means that Pic( S) is torsion-free. Thus, we can write Pic( S) = ZM for some line bundle M on S. Therefore, there exists some integer a such that O S (E) ≃ M ⊗a . Since E 2 = −2, we have aE·M = −2.
Note that E · M is an integer. If a = ±2, then −2 = E 2 = a 2 M 2 = 4M 2 . This is a contradiction because M 2 is an integer. Thus, we get a = ±1. This means that Pic( S) = ZO S (E).
By Claim 5, we have N 1 ( S) = R and NE( S) = N 1 ( S) = R. We note that E 2 = −2 and that there exists a curve G on S such that G · E > 0.
We further assume that C is an elliptic curve. Then K S ∼ 0 and S has only two simple elliptic singularities. Therefore, S is a complete non-projective log canonical Calabi-Yau algebraic surface with NE( S) = R.
Example 12.5. Let S be the surface constructed in Example 12.4. We take finitely many smooth points Q 1 , Q 2 , . . . , Q ρ−1 of S with ρ ≥ 2 such that Q i = Q j for i = j and Q i ∈ E for every i. We blow up Q 1 , Q 2 , . . . , Q ρ−1 to get σ : S → S. Let B i denote the (−1)-curve on S with σ(B i ) = Q i for every i. Then, by Theorem 3.11 (see also Remark 11.4), we can easily check that Pic(S) = ZO S (σ * E) ⊕ ZO S (B 1 ) ⊕ · · · ⊕ ZO S (B ρ−1 ).
Proof of Claim 6. Let D be a nef R-Cartier R-divisor on S. It is sufficient to prove that D is numerically trivial. By the above description of Pic(S), we can assume that D = b 0 σ * E + ρ−1 i=1 b i B i , where b i ∈ R for every i. Since D · B i ≥ 0 for every i and D · σ * E ≥ 0, b i ≤ 0 holds for every i. We assume that D = 0. Then we can take an irreducible curve G on S such that G ⊂ Supp D and that G ∩ Supp D = ∅. This is because the smooth locus of S is a quasi-projective open subset of S and contains σ * E and B i for every i. Thus we get D · G < 0. This is a contradiction. This means that D = 0. Hence we have NE(S) = N 1 (S) = R ρ .
If C is an elliptic curve, then S is a complete non-projective log canonical algebraic surface with NE(S) = N 1 (S) = R ρ . By construction, we see that K S = ρ−1 i=1 B i . We can apply the minimal model program established in Theorem 1.5 to a complete non-projective log canonical algebraic surface S. Then every B i is contracted to a smooth point and we finally get a good minimal model S, which is a surface with K S ∼ 0 constructed in Example 12.4.
The following commutative diagram may help the reader understand the constructions in Examples 12.3, 12.4, and 12.5.
Finally, we make a remark on complete non-projective algebraic surfaces defined over an algebraically closed filed k with k = C.
Remark 12.6. We note that every complete algebraic surface defined over F p is always Qfactorial (see [T,Theorem 4.5]). So it automatically becomes projective (see [Fn4,Lemma 2.2]). By the following lemma (see Lemma 12.7 below), the constructions in this section and Kollár's construction in [K1, Aside 3.46] can work for algebraic surfaces defined over an algebraically closed field k such k = F p for every prime number p with some suitable modifications. We note that Theorem 3.11 holds true for algebraic surfaces defined over any algebraically closed field (see Remark 11.4). Thus, we can construct complete nonprojective algebraic surfaces over an algebraically closed field k such that k = F p for every prime number p.
Lemma 12.7. Let C be a smooth projective curve defined over an algebraically closed field k whose genus g(C) is positive. Let P be an arbitrary closed point of C. Assume that k = F p for every prime number p. Then there exists Q ∈ C such that O C (Q − P ) is a non-torsion element of Pic 0 (C).
Proof. By g(C) ≥ 1 and k = F p , we can take a non-torsion element L of Pic 0 (C) (see [T,Fact 2.3]). We take a large positive integer m. Then L ⊗ O C (mP ) is very ample. We consider a general member Q 1 + · · · + Q m of |L ⊗ O C (mP )|. Then we have L ≃ O C ((Q 1 − P ) + · · · + (Q m − P )). Therefore, there exists some i 0 such that O C (Q i 0 − P ) is a non-torsion element of Pic 0 (C).