Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle

Let $X$ be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if $X$ is $d$-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus of $X$, and the first author's existence result of holomorphic volume forms on global smoothings of $X$. In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples of $d$-semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces and $K3$ surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable to $K3$ surfaces.


INTRODUCTION
This is a sequel to the first author's paper [D09], where he obtained a construction of compact complex surface with trivial canonical bundle by gluing together compact complex surfaces with an anticanonical divisor. As an application, he proved that if a simple normal crossing (SNC) complex surface X is d-semistable and has at most double intersections, then there exists a family of smoothings of X in a differential geometric sense. More precisely, there exist a smooth 6-manifold X and a smooth surjective map : X → ∆ ⊂ C such that −1 (0) = X, −1 (ζ) for each ζ ∈ ∆ \ {0} is a smooth compact complex surface with trivial canonical bundle, and the complex structure on −1 (ζ) depends continuously on ζ outside the singular locus of X. One purpose of this article is to extend the smoothability result in [D09] to cover the cases where X has triple intersections. For another purpose, we provide many examples of SNC complex surfaces with trivial canonical bundle including double and triple intersections, to which we can apply our smoothability result.
Throughout this article, X = N i=1 X i (or possibly X = N −1 i=0 X i ) denotes a compact connected complex surface with normal crossings of N irreducible components with dim C X i = 2 for each i, unless otherwise specified. Before stating the main result, we give some definitions.
In particular, the above X has no fourfold intersections: Q ijk = X i ∩ X j ∩ X k ∩ X = ∅ for all i, j, k, . Let D ij = X i ∩ X j be a set of double curves and T ijk = X i ∩ X j ∩ X k be a set of triple points. We define index sets I i and I ij ⊂ I i by Then D i = j∈I i D ij and T ij = k∈I ij T ijk are divisors on X i and D ij respectively, and T i = j∈I i T ij is the set of triple points on X i . Note that we admit self-intersections, that is, it may happen that X i ∩ X i = ∅. An example of an SNC complex surface with a self-intersection will be given in Example 3.7 for N = 1.
where N ij denotes the holomorphic normal bundle N D ij /X i to D ij in X i , and [T ij ] denotes the associated holomorphic line bundle of T ij (see, e.g., [GH], p. 145 and p. 134 for the respective definitions). This is a differential geometric interpretation of Friedman's original complex analytic (and also algebro-geometric) definition that a SNC complex surface X is said to be d- for the singular locus D on X, where I X i and I D are the ideal sheaves of X i and D in X respectively. Now the main theorem of this article is described as follows.
Theorem 1.3. Let X = N i=1 X i be an SNC complex surface. Assume the following conditions: (i) X is d-semistable; (ii) each D i = j∈I i D ij is an anticanonical divisor on X i ; and (iii) there exists a meromorphic volume form Ω i on each X i with a single pole along D i such that the Poincaré residue res D ij Ω i of Ω i on D ij is minus the Poincaré residue res D ij Ω j of Ω j on D ij for all i, j. (For the definition of Poincaré residues, see [GH], pp. 147-148).
(a) X is a smooth real 6-dimensional manifold and is a smooth map.
(d) The complex structure on X ζ depends continuously on ζ outside the singular locus D = N i=1 D i ⊂ X 0 . More precisely, for any point p ∈ X \ D there exist a neighborhood U of p and a diffeomorphism U V × W with W ⊂ ∆, such that the induced complex structures on V depend continuously on ζ ∈ W .
Remark 1.4. Conditions (ii) and (iii) of Theorem 1.3 are equivalent to the condition that the canonical bundle K X of the SNC complex surface is trivial (see Section 2.3). Also, if • the normal crossing complex surface X = i X i is not simple, or • we are given {X i , D i , Ω i } i satisfying conditions (i)-(iii) and gluing isomorphisms between double curves, but not all local embeddings of X i into C 3 , we obtain an alternative SNC complex surface X as follows: If we glue together the irreducible components X i along the double curves using the isomorphisms and local embeddings into C 3 , we obtain a desired SNC complex surface X to which we can apply Theorem 1.3.
Let us compare Theorem 1.3 with Friedman's smoothability result. According to [Ku77], Theorem II, the central fiber of a semistable degeneration of K3 surfaces is classified into either Type I, II, or III, where the central fiber corresponding to a degeneration of Type ν contains a ν-ple intersection but no (ν + 1)-ple intersection. Conversely, Friedman proved in [Fr83], Theorem 5.10 that if X is a d-semistable K3 surface (see Section 2.4 for the definition), then there exists a family of smoothings : X → ∆ ⊂ C of X with K X = O X , where X is a 3-dimensional complex manifold, ∆ is a domain in C, and is a holomorphic map. See Section 2.4 for more details. On the other hand, our smoothability result holds even when H 1 (X, O X ) = 0 or not all irreducible components of X are Kählerian. In exchange for a broader scope of application, our smoothings : X → ∆ are not holomorphic but only smooth, that is, both and X are smooth, although each fiber admits a complex structure which depends continuously on ζ ∈ ∆. In Section 3.5, we give some examples of SNC complex surfaces X with trivial canonical bundle including double curves and satisfying H 1 (X, O X ) = 0, which are smoothable to complex tori and primary Kodaira surfaces, in addition to some examples of d-semistable K3 surfaces of Type II. Also, in Sections 4 and 5, we construct some explicit examples of d-semistable K3 surfaces of Type III which are smoothable to K3 surfaces. However, it remains an interesting problem whether there exists an example of a d-semistable SNC complex surface with trivial canonical bundle including triple points which is smoothable to a complex torus or a primary Kodaira surface.
From the modern viewpoint of logarithmic geometry which is a central tool for smoothings, Kawamata and Namikawa generalized Friedman's smoothability result in higher complex dimensions n 3 [KN94]. In the paper, they required that H n−1 (X, O X ) = 0, H n−2 (X i , O X i ) = 0 for all i, and all irreducible components X i are Kählerian for proving the existence of smoothings of a compact Kähler normal crossing variety X. Note that the first requirement H n−1 (X, O X ) = 0 comes from the use of the T 1 -lifting property. Kawamata-Namikawa's smoothability result is particularly effective in constructing many examples of Calabi-Yau manifolds. Indeed, they obtained new examples of Calabi-Yau threefolds by smoothing d-semistable SNC Calabi-Yau threefolds X = X 1 ∪ X 2 with a double intersection D = X 1 ∩ X 2 . Also, N.-H. Lee [L19] used [KN94] to obtain further new examples of Calabi-Yau threefolds by smoothing d-semistable SNC Calabi-Yau threefolds of Type III (i.e., with triple intersections). Meanwhile, Hashimoto and Sano [HS19] noticed that in [KN94] it is not necessary for X as a whole to be Kählerian although its irreducible components have to be (see [HS19], Remark 2.7), and constructed an infinite number of examples of non-Kähler threefolds with trivial canonical bundle by smoothing d-semistable SNC non-Kähler threefolds with trivial canonical bundle including a double intersection.
A major breakthrough in algebraic extensions of Friedman's smoothability result has been achieved by Felten, Filip and Ruddat in [FFR19]. Conceptually, T 1 X = Ext 1 O X (Ω 1 X , O X ) measures the failure of the normal sequence and the notion of d-semistability defined in (1.2) is then given by T 1 X ∼ = O D . It was proved in [FFR19], Theorem 1.1 that an SNC variety with trivial canonical bundle is smoothable if T 1 X is generated by global sections and the singular locus of X is projective. Moreover, their smoothability theorem holds even when not all of the irreducible components X i of an SNC variety X are Kählerian, or when not all of the cohomology groups of X and X i vanish. We also mention that Chan, Leung, and Ma proved that the existence of smoothings of d-semistable log smooth Calabi-Yau varieties [CLM19]. Keeping these modern logarithmic viewpoints in mind [RS20], one can interpret Theorem 1.3 as a differential geometric counterpart of algebraic extensions of Friedman's smoothability result (e.g., Theorem 1.1 in [FFR19]).
Meanwhile, our result of differential geometric smoothings using the gluing technique brings some insight into both differential and algebraic geometry. As mentioned above, the smoothing technique in complex analytic and algebraic geometry is particularly effective for constructing special smooth complex manifolds such as Calabi-Yau manifolds. For this purpose, our construction of differential geometric smoothings is also useful because we need a complex structure not on the total space X of the smoothings : X → ∆ but only on a smooth fiber X ζ = −1 (ζ). Meanwhile in differential geometry, the gluing technique is effectively used for constructing many compact manifolds with a special geometric structure such as Calabi-Yau, G 2 -and Spin(7)structures and complex structures with trivial canonical bundle by Joyce [J], Kovalev [Ko03], Clancy [C11], and Doi and Yotsutani [DY14, DY15, DY19, D09]. Our result here not only enables us to reconstruct such a manifold as a smooth fiber of global differential geometric smoothings of an SNC manifold with a special geometric structure including only double intersections, but also opens up the possibilities of treating global smoothings of such SNC manifolds admitting triple intersections to obtain new examples.
The proof of Theorem 1.3 is based on an explicit construction of global smoothings of X by gluing together local smoothings around double curves and triple points. For this purpose, we note that the holomorphic coordinates on a neighborhood of D ij in X i are approximated by those on a neighborhood of D ij in N ij with a Taylor expansion in terms of the fiber coordinate of N ij via an exponential map. Then we realize differential geometric local smoothings ij : V ij → ∆ ⊂ C of X i ∪ X j around each double curve D ij as a complex hypersurface of N ij ⊕ N ji . In particular, we see from the simplicity of the normal crossing complex surface X = i X i that the space V ij of local smoothings around each triple point in D ij includes a local model written as By gluing together (X i \D i )×∆ and V ij for all i, j, we obtain differential geometric global smoothings : V → ∆. For each ζ ∈ ∆, we can consistently define an SL(2, C)-structure Ω ζ on the fiber −1 (ζ) using condition (iii) of Theorem 1.3 such that we have d Ω ζ → 0 as ζ → 0 in an appropriate sense. Finally, we prove that if |ζ| is sufficiently small, then we can deform Ω ζ to a d-closed SL(2, C)-structure Ω ζ using the main result of [D09], so that −1 (ζ) is a compact complex surface with trivial canonical bundle. This article is organized as follows. In Section 2, we briefly state the results in [D09] which will be used in the proof of Theorem 1.3. We introduce the notions of SL(2, C)and SU(2)-structures in Section 2.1, state the existence theorem of complex structures with trivial canonical bundle in Section 2.2, define the canonical bundles of SNC complex surfaces in Section 2.3, and review the semistable degenerations of K3 surfaces in Section 2.4. Before constructing explicit local smoothings in Sections 3.2 and 3.3, in Section 3.1 we will introduce local holomorphic coordinates suited to the smoothing problem, and give an example which provides a local model around a triple point. The proof of Theorem 1.3 is given in Section 3.4. Also, in Section 3.5 we give several examples of d-semistable SNC complex surfaces with trivial canonical bundle including only double curves, which are smoothable to complex tori, primary Kodaira surfaces, and K3 surfaces. In the last two sections, we construct examples of SNC complex surfaces with triple points which are smoothable to K3 surfaces. In Section 4.1, we see that the blow-up of an SNC complex surface with trivial canonical bundle at finite points in the double curves excluding the triple points inherits good properties from the original one. Then in Section 4.2, we produce examples of d-semistable K3 surfaces with four points in Example 4.2. Section 5 is devoted to considering a more technical example. After fixing our notation in Section 5.1, we consider in Section 5.2 the mismatch problem which we encounter when we try to glue all components together along their intersections. In order to handle this kind of mismatch issue, we shall take the order of blow-ups carefully. Consequently, we will show that one can still glue all components together after taking appropriate blow-ups in Section 5.3.
The first author is mainly responsible for Sections 1, 2.1-2.3, and 3, and the second author mainly for Sections 1, 2.4, 4, and 5.
Acknowledgements. The authors would like to thank Professor Kento Fujita for allowing us to use his example which is the source of Example 3.9. Naoto Yotsutani also thank to Professors Nam-Hoon Lee, Taro Sano, and Yuji Odaka for fruitful discussions through e-mails. Finally, we are grateful to the referee for valuable comments which improved the presentation of our manuscript. This work was partially supported by JSPS KAKENHI Grant Number 18K13406 and Young Scientists Fund of Kagawa University Research Promotion Program 2021 (KURPP).

A BRIEF REVIEW OF COMPLEX SURFACES WITH TRIVIAL CANONICAL BUNDLE
In dealing with complex surfaces with trivial canonical bundles in differential geometry, it is crucial to note that a complex structure of such a surface is characterized by a dclosed SL(2, C)-structure, which becomes a holomorphic volume form with respect to the resulting complex structure (see Proposition 2.4). Then with the help of a Hermitian form which forms an SU(2)-structure together with an SL(2, C)-structure, we can reduce the problem whether a given SL(2, C)-structure ψ with small dψ can be deformed into a dclosed SL(2, C)-structure, to the solvability of a partial differential equation given by (2.1). The first two subsections provide more details. We introduce the notions of SL(2, C)-and SU(2)-structures in Section 2.1, and state in Section 2.2 the existence result of an d-closed SL(2, C) structure as a solution of the above differential equation (2.1) below. Meanwhile, Section 2.3 describes the canonical bundle of an SNC complex surface according to [Fr83], and Section 2.4 reviews the classification of semistable degenerations of K3 surfaces and smoothability result of d-semistable K3 surfaces from the algebro-geometric viewpoint.
2.1. SL(2, C)-and SU(2)-structures. In this subsection, we briefly review the notions and results in [D09] without proofs. (See also [Go04] for reference.) An SL(2, C)-structure ψ 0 on V gives a decomposition of V ⊗ C: and ι ζ is the interior multiplication by ζ. Thus if v ∈ V , then v is uniquely written as v = v 1,0 + v 1,0 , and v → v 1,0 gives an isomorphism between real vector spaces. Then the composition Thus ψ 0 defines a complex structure I ψ 0 on V such that ψ 0 is a complex differential form of type (2, 0) with respect to I ψ 0 .
Let A SL(2,C) (V ) be the set of SL(2, C)-structures on V . Then A SL(2,C) (V ) is an orbit space under the action of the orientation-preserving general linear group GL + (V ). Since each ψ ∈ A SL(2,C) (V ) has isotropy group SL(2, C), there is a one-to-one correspondence from the orbit A SL(2,C) (V ) to the homogeneous space GL + (V )/SL(2, C).
We define A SL(2,C) (M ) to be the fiber bundle which has fiber A SL(2,C) (T x M ) over x ∈ M . Then an SL(2, C)-structure can be regarded as a smooth section of A SL(2,C) (M ).
Since an SL(2, C)-structure ψ on M induces an SL(2, C)-structure on each tangent space, ψ defines an almost complex structure I ψ on M such that ψ is a (2, 0)-form with respect to I ψ .
Lemma 2.3 (Grauert, Goto [Go04]). Let ψ be an SL(2, C)-structure on an oriented 4manifold M . If ψ is d-closed, then I ψ is an integrable complex structure on M with trivial canonical bundle and ψ is a holomorphic volume form on M with respect to I ψ .
The above lemma gives the following characterization of complex surfaces with trivial canonical bundle by d-closed SL(2, C)-structures. Thus, if we say that X be a complex surface with canonical trivial bundle, then we understand that X consists of an underlying oriented 4-manifold M and a d-closed SL(2, C)structure ψ on M such that ψ induces a complex structure I ψ on M and becomes a holomorphic volume form on X = (M, I ψ ).
Let X be a compact complex surface with trivial canonical bundle. If X is simply connected or H 1 (X, O X ) = 0, then X is called a K3 surface. According to the Enriques-Kodaira classification, it is known that a compact complex surface with trivial canonical bundle is either a complex torus, a primary Kodaira surface, or a K3 surface (see [BHPV], Chapter VI).
We also have the orthogonal decomposition is the orthogonal complement to T (ψ 0 ,κ 0 ) A SU(2) (V ) with respect to g (ψ 0 ,κ 0 ) . The next lemma is crucial in solving the partial differential equation in the proof of Theorem 2.11.
Definition 2.9. Let M be an oriented 4-manifold.
Define A SU(2) (M ) to be the fiber bundle whose fiber over x ∈ M is A SU(2) (T x M ). Then an SU(2)-structure can be regarded as a smooth section of A SU(2) (M ).
If ψ and κ are both d-closed, then X = (M, I ψ , κ) is a Kähler surface with trivial canonical bundle. Moreover, the Ricci curvature of the Kähler metric g vanishes due to condition (iv) of Definition 2.5.
Definition 2.10. Let M be an oriented 4-manifold. Choose ρ < ρ * so that the projection Θ is well-defined. We define T SU(2) (M ) to be the fiber bundle whose fiber over x ∈ M is T SU(2) (T x M ), and denote by Θ the projection from T SU(2) (M ) to A SU(2) (M ).

2.2.
Existence theorem of d-closed SL(2, C)-structures. We are now in a position to state the following existence theorem of a complex structure with trivial canonical bundle.
Note that the Hermitian form κ in Theorem 2.11, which forms an SU(2)-structure together with ψ, only plays an auxiliary role for obtaining a d-closed SL(2, C)-structure. Since we only require a mild estimate for κ, it is not difficult to find such a κ.
The proof of Theorem 2.11 is summarized as follows. The equation dΘ 1 (ψ + η, κ) = 0 is rewritten as We note that F (η) is quadratic in η ∈ C ∞ (∧ 2 − T * M ⊗ C) due to Lemma 2.6. To solve (2.1) we consider the recurrence equations with j > 0 and η 0 = 0. According to the Hodge theory, there exists a unique η j ∈ Then one can show that the sequence {η j } converges to a unique η in the Sobolev space The hypothesis on the injectivity radius and Riemann curvature in Theorem 2.11 is a technical assumption to evaluate ∇χ L 8 for χ ∈ C ∞ (∧ 2 − T * M ) in terms of dχ L 8 and χ L 2 , and then χ C 0 in terms of ∇χ L 8 and χ L 2 . Regularity of η follows from the ellipticity of (2.1) when it is considered as an equation on L 8 (V ) with V = 4 i=0 T * M . Then using the Sobolev embedding L 8 1 → C 0,1/2 and the standard bootstrapping method, we prove that η is smooth. For further details, see [D09], Section 4 (see also [J], Chapter 13).

2.3.
Canonical bundle of an SNC complex surface. According to [Fr83], Remark 2.11, the canonical bundle K X of an SNC complex surface X is described as follows.
Definition 2.12. Let X = N i=1 X i be an SNC complex surface with irreducible components X i , given by gluing isomorphisms f ij : D ij → D ji for all i, j with i j and j ∈ I i , where we distinguish D ij and D ji by regarding D ij ⊂ X i and D ji ⊂ X j . Also, we understand that if D ij has more than one irreducible component, then f ij is a union of the corresponding isomorphisms, and if D ii = ∅, then we divide the irreducible components of D ii into two as D ii = D ii ∪ D ii and consider f ii as an isomorphism from D ii to D ii . Define line bundles L i on X i and L ij on D ij by By the adjunction formula, we calculate L ij as Also, the restriction of L i to D ij is given by the Poincaré residue map. Then the canonical bundle of X is given by the collection of the line bundles L i on X i , together with the gluing isomorphisms −f * ij : L ji → L ij , that is, a set {s i } i of local sections s i ∈ H 0 (X i , L i ) together define a global section s ∈ H 0 (X, K X ) if and only if res D ij The minus sign in the gluing isomorphisms −f * ij : L ji → L ij naturally arises when we consider a local model as follows. Consider a local embedding {ζ 1 ζ 2 = 0} (resp. {ζ 1 ζ 2 ζ 3 = 0}) of X 1 and X 2 around p ∈ D 12 \ T 12 (resp. X 1 , X 2 , and X 3 around p ∈ T 123 ) into C 3 with local representations X i = {ζ i = 0}. Let Ω 0 be a meromorphic volume form on C 3 given by Then Ω 0 induces local meromorphic volume forms Ω i,0 on X i for i = 1, 2 given by which explains the minus sign in the gluing isomorphisms −f * ij : L ji → L ij of K X . We immediately obtain the following result from Definition 2.12 because a section of H 0 (X i , L i ) is given by a meromorphic volume form on X i with a single pole along D i .
Lemma 2.13. Let X = N i=1 X i be an SNC complex surface given by gluing isomorphisms f ij : D ij → D ji for all i, j with i j and j ∈ I i . Then the canonical bundle of K X is trivial if and only if the following conditions hold: • each D i = j∈I i D ij is an anticanonical divisor on X i ; and • there exists a meromorphic volume form Ω i on each X i with a single pole along D i such that This lemma implies that if we have D ij = D ji for all i, j, then conditions (ii) and (iii) of Theorem 1.3 are necessary and sufficient for the canonical bundle K X of X to be trivial. Also, Definition 2.12 leads to the following result, which is useful for computing H 0 (X, K X ).
Proposition 2.14. Let X = N i=1 X i be an SNC complex surface given by gluing isomorphisms f ij : D ij → D ji for all i, j with i j and j ∈ I i . Then we have an exact sequence where L i , L ij , and ρ ij are given by Hence, H 0 (X, K X ) is given by the kernel of the linear map ρ. In particular, if D i = j∈I i D ij is an anticanonical divisor for all i, then we have L i ∼ = C and L ij ∼ = C for all i, j, so that ρ defines a linear map from C N to C M , where M is the number of double curves D ij with i j.
We will use Proposition 2.14 in Example 3.9.
2.4. Semistable degenerations of K3 surfaces. In this subsection, we give a summary of the classification of degenerations of K3 surfaces. Let : X → ∆ be a proper surjective holomorphic map from a compact complex 3-dimensional manifold X to a domain ∆ in C such that (1) X is smooth outside the central fiber X = −1 (0), and (2) for each ζ ∈ ∆ * = ∆ \ {0}, the general fiber X ζ = −1 (ζ) is a smooth compact complex surface. We call a degeneration of compact complex surfaces. Furthermore, a degeneration is said to be semistable if (3) the total space X is smooth, and (4) the central fiber X is a normal crossing complex surface whose irreducible components are all Kählerian. Let : X → ∆ be a semistable degeneration of K3 surfaces, that is, the general fiber X ζ is a K3 surface. A degeneration : X → ∆ with the smooth total space X is said to be a modification of : X → ∆ if there exists a birational map ρ : X X which is compatible with the projections and . In particular, ρ is an isomorphism over ∆ * . Mumford's semistable reduction theorem states that after a base change and a birational modification, the central fiber of a degeneration is a reduced divisor with normal crossings. Furthermore, if : X → ∆ is a semistable degeneration of K3 surfaces, then there is a modification : X → ∆ such that the total space X has trivial canonical bundle, according to the results of Kulikov [Ku77] and Persson-Pinkham (see [H], Chapter 6, Theorem 5.1).
The new family : X → ∆ is said to be a Kulikov degeneration of the original degeneration : X → ∆, and the central fibers of Kulikov degenerations are classified into the following three cases, due to Kulikov [Ku77] and Persson [Pe77].
Theorem 2.15 ([Ku77], Theorem II. See also [H], Chapter 6, Theorem 5.2). Let : X → ∆ be a Kulikov degeneration, that is, a semistable degeneration of K3 surfaces with K X = O X as above. Then the central fiber X = −1 (0) is one of the following three types: Type I: X is a smooth K3 surface. Type II: X = X 1 ∪· · ·∪X N is a chain of surfaces, where X 1 and X N are rational surfaces, X 2 , . . . , X N −1 are elliptic ruled surfaces, and X i ∩ X i+1 , i = 1, . . . N − 1 are smooth elliptic curves.
where each X i is a rational surface and the double curves D ij = X i ∩ X j ⊆ X i are cycles of rational curves.
If one omits the assumption that all irreducible components of X are Kählerian in Theorem 2.15, other types of surfaces may arise as irreducible components of the central fiber of semistable degenerations [N88]. Meanwhile, it is known that d-semistability defined in (1.2) is a necessary condition for an SNC complex manifold X to be the central fiber of a semistable degeneration. Hence, it is natural to consider the converse problem: For what d-semistable SNC complex manifolds (or surfaces) X does there exist a family of (global) smoothings : X → ∆ of X, i.e., a semistable degeneration : X → ∆ such that X is the central fiber of ? Friedman investigated this problem for K3 surfaces [Fr83]. In order to state his result more precisely, we need the following.
When we posted a preprint of this article on arXiv, we did not know whether there exists an SNC complex surface X which satisfies conditions (i), (ii) of Definition 2.16 and H 1 (X, O X ) = 0, but not condition (iii). However, it was known that such surfaces actually exist and the following example was kindly mentioned to us by the referee.
Starting from a d-semistable K3 surface of Type II, we consider the total space of a family of smoothings : X → ∆. Assume the rational surface X 1 of the central fiber X = N i=1 X i contains a (−1)-curve E. We further assume that E is a (−1, −1)-curve on X , i.e., E is an algebraic curve on X such that E ∼ = CP 1 and N C/X ∼ = O CP 1 (−1) ⊕ O CP 1 (−1). Then the elliptic curve X 1 ∩ X 2 intersects to E at the point P , which is an ordinary double point in X 2 ⊂ X . Taking the Atiyah flop ϕ : X X (which is called the elementary modifications in [FM]), we consider the proper transform X 2 of X 2 under ϕ. Then the restriction of ϕ on X 2 gives the blow-up which yields that X 2 is no longer a ruled surface. Thus, we conclude that the central fiber X of X does not belong to any of the types in Theorem 2.15.

EXPLICIT CONSTRUCTION OF DIFFERENTIAL GEOMETRIC GLOBAL SMOOTHINGS
In this section, we explicitly construct differential geometric global smoothings of a given SNC complex surface X satisfying conditions (i)-(iii) of Theorem 1.3. For this purpose, we introduce in Section 3.1 local holomorphic coordinates on X suited to the smoothing problem. Then we construct local smoothings around double curves D ij without a triple point, and triple points T ijk in Sections 3.2 and 3.3, respectively. In Section 3.4, we construct global smoothings : X → ∆ of X by gluing together the above local smoothings, and then use Theorem 2.11 to prove Theorem 1.3 which states that each fiber X ζ = −1 (ζ) admits a complex structure with trivial canonical bundle depending continuously on ζ ∈ ∆. Section 3.5 provides several explicit examples of d-semistable SNC complex surfaces with trivial canonical bundle including at most double curves, which we see are smoothable to complex tori, primary Kodaira surfaces, and K3 surfaces by Theorem 1.3. In particular, we construct in Example 3.8 d-semistable K3 surfaces of Type II with any number N 2 of irreducible components.

Local coordinates on an
We can find a local holomorphic coordinate system In particular, we see from condition (E) that Λ i j k = Λ ijk for any permutation (i , j , k ) of (i, j, k). By condition (ii) of Theorem 1.3, we can choose the above coordinate system so that (F) the meromorphic volume form Ω i in (iii) of Theorem 1.3 can be locally represented on U i,α as (2) i takes either 1 or −1. In terms of the above coordinate system, we define a new one {U i,α , (z ij,α , w ij,α )} α∈Λ ij around D ij as follows: ij , then between z 1 i,α and z 2 i,α , we choose as w ij,α the coordinate which is a defining function of D ij on U i,α , and z ij,α as the remainder, so that In particular, we have Condition (E) is rephrased as We can further choose the coordinate system so that the following condition holds.
where ij = (j − i)/ |j − i|, and σ ijk ∈ {1, −1} does not depend on α ∈ Λ ijk and satisfies Also, the Poincaré residue res D ij Ω i gives a meromorphic volume form ψ D ij on D ij , which is locally represented on U ij,α for α ∈ Λ ij as Indeed, condition (iii) of Theorem 1.3 and equation (3.1), respectively, give that which is locally represented on U ij,α for α ∈ Λ ij as Example 3.1. Here we suppose indices i, j, k, and will take 0, 1, 2, or 3. Let us consider We will see that X is an SNC complex surface satisfying conditions (ii) and (iii) of Theorem 1.3, but not condition (i). We will also calculate all σ ijk ∈ {1, −1} which appear in the local representation (3.2) of the meromorphic volume form Ω i on X i .
(1) X is an SNC complex surface.
where the hatted component is meant to be omitted. Note that in this example, we are using the notation U i,j above and U ij,k below in place of U i,α and U ij,α , respectively. Then we see that ζ j k for j, k = i is a local defining function of D ij on U i,k . Thus, D i is an anticanonical divisor on X i because we have where j, k, = i are all distinct.
(3) Meromorphic volume forms Ω i on X i , which give all σ ijk , satisfy condition (iii) of Theorem 1.3. Using (3.6), we obtain well-defined meromorphic volume forms Ω i on X i locally represented as by the Poincaré residue map as with ζ k k = 1 on U ij,k for k = i, j in D ij . Then the Poincaré residues res D ij Ω i and res D ij Ω j are locally represented on U ij,k as where k, ∈ {0, 1, 2, 3} \ {i, j} with k = , so that X is not d-semistable and does not satisfy condition (i).
Although the above X is not d-semistable, this example will provide a local model of the neighborhood of a triple point in Section 3.3. Also, we will see in Section 4 that if we blow up X at appropriate points, then the resulting SNC complex surface becomes a d-semistable K3 surface of Type III satisfying conditions (i)-(iii) of Theorem 1.3, so that we can apply Theorem 1.3.
Then from condition (i) of Theorem 1.3, we may further assume that Thus we see for all i, j with i = j that Now consider a Hermitian metric around D ij (which is temporary and different from what is considered later) such that the associated 2-form coincides with that of a flat metric ij . Then by the tubular neighborhood theorem, we have a diffeomorphism Thus, it follows from (3.1) and (3.9) that (3.10) In the rest of this subsection, we assume i < j unless otherwise mentioned. Let · ij be a bundle norm on N ij such that ij , which makes sense because N ij does not include the fibers over the triple points T ij . Then we define a cylindrical parameter t ij on N ij \ D ij by In particular, using (3.11) we have For later convenience, we further extend t ij so as to take ∞ on D ij , so that Thus, t ij takes values in (0, ∞] on V ij , and accordingly t ij takes values in (0, ∞] on Then we see from (3.7) that (3.13) and (3.14) still hold if we interchange i and j. As with t ij , we will regard t ji as a function on N ji ∪ T ji , taking ∞ on D ji . Now let us define Then by (3.8), (3.14) and (3.18), we have for all i, j with i = j and ζ ∈ C * that In the same way as above, we can define a neighborhood V ji (resp. V ji ) of D ij (resp. D ij ) in N ji (resp. N ji ), a neighborhood W ji of D ij in X j , and a cylindrical parameter t ji on 3.2. Local smoothings of X i ∪ X j around D ij without a triple point. Here we suppose D 12 = ∅ is a double curve without a triple point, so that T 12 = ∅. Also, indices i, j will take 1 or 2, and the pair (i, j) will take (1, 2) or (2, 1). For general D n 1 n 2 = X n 1 ∩ X n 2 with n 1 < n 2 , we replace subscripts 1, 2 and i, j with n 1 , n 2 and n i , n j , respectively. In Section 3.1, we have chosen the coordinate system α is a defining function of D ij on U i,α and the holomorphic volume form Ω i is locally represented as ij dz ij,α ∧ dw ij,α /w ij,α on U i,α for α ∈ Λ ij . Also, we have a holomorphic volume form ψ D ij = res D ij Ω i given by (3.4) and a Hermitian form ω D ij given by (3.5) on D ij .

Local smoothings of
Here we suppose T 123 = ∅ and consider local smoothings of X 1 ∪ X 2 ∪ X 3 around D 12 ∪ D 23 ∪ D 31 . Indices i, j, k will take 1, 2, or 3, while will take all possible values besides 1, 2, 3. For local smoothings of X n 1 ∪ X n 2 ∪ X n 3 around D n 1 n 2 ∪ D n 2 n 3 ∪ D n 3 n 1 for general n 1 , n 2 , n 3 with n 1 < n 2 < n 3 , we will be done if we replace subscripts 1, 2, 3 and i, j, k with n 1 , n 2 , n 3 and n i , n j , n k , respectively. For later convenience, we will use ij = (j − i)/ |j − i| as before, and the Levi-Civita symbol ijk = ij ik jk , so that we have n i n j = ij and n i n j n k = ijk .
as in Section 3.1. Recall that we have a holomorphic volume form Ω i on X i \D i with a local representation around D ij in (3.2), and a holomorphic volume form ψ D ij = res D ij Ω i in (3.4) and a smooth Hermitian form ω D ij in (3.5) on D ij . We define a smooth complex volume form Ω ∞ ij and a smooth Hermitian form ω ∞ ij on N ij \ D ij by (3.31) Then as in Lemma 3.2, we see that (Ω ∞ ij , ω ∞ ij ) defines an SU(2)-structure on N ij \ D ij such that Ω ∞ ij is holomorphic and the associated metric is cylindrical. In particular, Ω ∞ ij and ω ∞ ij are locally represented on where we used in (3.31) ij , which follows from (3.4) and (3.13) respectively. As in Section 3.2, we can also regard (Ω ∞ ij , ω ∞ ij ) as an SU(2)-structure on W ij \ D i ⊂ X i via Φ ij . We see from (3.1), (3.9), (3.10) and (3.32) that (1) ij , and (3.33) where ω ij is the (1, 1)-part of ω ∞ ij , normalized so that Ω i ∧ Ω i = 2ω ij ∧ ω ij , and |·| is measured by the cylindrical metric g ∞ ij associated with ω ∞ ij . Letting c > 1 and shrinking ij by the coordinate transformation w ij,α → cw ij,α in conditions (B), (G) and (C), (H) if necessary, we may assume that (Ω i , ω ij ) is an SU(2)-structure on W ij \ D i . Thus, we have the associated Hermitian metric g ij on W ij \ D i such that ij . We also see from (3.20) and (3.31) that Now we construct a family of local smoothings of X 1 ∪X 2 ∪X 3 around D 12 ∪D 23 ∪D 31 . The construction consists of the following steps.
Step 1. Fix e −1 and let ∆ = ∆ = { ζ ∈ C | |ζ| < } be a domain in C. Following Section 3.2, we construct a family of local smoothings ij : This gives a local model of a family of smoothings of X i ∪ X j around D ij ⊂ X i ∩ X j . We see that −1 ij (ζ) for ζ ∈ ∆ * is obtained by gluing together t −1 ij (0, 2T ζ ) ⊂ V ij and t −1 ji (0, 2T ζ ) ⊂ V ji using the diffeomorphism h ij,ζ given by (3.7), where T ζ is defined by (3.19). Also, −1 ij (ζ) has an SU(2)-structure which is induced from N ij and N ji .
Step 5. We define an injective diffeomorphism Φ ij : × {ζ} be as defined in (3.29), where W T ij and X T i are defined in (3.21). Then we can define injective diffeomorphisms Thus by gluing together X 1 , X 2 , X 3 and V 123 along W ij ∪ W ik ⊂ X i for all triples (i, j, k) with ijk = 1 using the injective diffeomorphisms Φ −1 ij ∪ Φ −1 ik , we obtain the desired family of local smoothings of X 1 ∪ X 2 ∪ X 3 around D 12 ∪ D 23 ∪ D 31 .
Step 1. Fix e −1 and let ∆ = ∆ = { ζ ∈ C | |ζ| < } be a domain in C. Let V ij be the neighborhood of D ij in N ij defined in (3.16). Then a family of local smoothings Note that the projection ij is well-defined according to condition (J) in Section 3.1. Let p ij : V ij → N ij be the projection. Then following the argument in Section 3.2, we see that p ij,ζ is a diffeomorphism on −1 ij (ζ) for ζ ∈ ∆ * , while on −1 ij (0) = V ij ∪ V ji we have (3.37) p ij,0 is an identity map on V ij , the projection map to D ij on V ji .
Thus, we can extend V ij,β smoothly to V ij,β defined by where V ij,β is defined in (3.15). Hence, replacing V ij,β with V ij,β for all β ∈ Λ (2) ij in V ij and extending the projection ij correspondingly, we obtain a family of local smoothings This turns out to be the same as Since V ij and V ji only differ by the order of V ij and V ji in the definition, we will identify V ij and V ji .
Then V 123,β is an open neighborhood of the triple point 0 in C 3 and −1 123,β (0) = (H 1 ∪ H 2 ∪ H 3 ) ∩ V 123,β . To define an SU(2)-structure on each fiber −1 123 (ζ) over ζ ∈ ∆, let η i = du i /u i be a meromorphic 1-form on C 3 , and consider a meromorphic volume form Ω H i ,β and a singular Hermitian form ω H i ,β on H i defined by where j and k are chosen so that ijk = 0. Then it is easy to check that ( where To define an SU(2)-structure (Ω 123,β,ζ , ω 123,β,ζ ) on each fiber −1 123,β (ζ) over ζ ∈ ∆, we define projections p i : C 3 → H i by where j and k are determined so that j < k and ijk = 0. Also, we define p i,ζ = p i | −1 123,β (ζ) . Suppose ζ ∈ ∆ * . Then we see that Meanwhile, if ζ = 0, then on each irreducible component of −1 123,β (0) we have that p i,0 is an identity map on H i ∩ V 123,β , the projection map to L i ⊂ H i on H ∩ V 123,β for = i.
Hence, regarding V ij (= V ji ) as an open submanifold of V 123 for all pairs (i, j) with ijν ij = 1, we can glue together X 1 , X 2 , X 3 , and V 123 along W ij ∪ W ik ⊂ X i for all i and j, k with ijk = 1 using the injective diffeomorphisms , to obtain the desired family of local smoothings of X 1 ∪ X 2 ∪ X 3 around D 12 ∪ D 23 ∪ D 31 . This gluing procedure for i is diagrammed as follows: where V ij is constructed by diagram (3.30) with subscripts 1, 2 replaced with i, j using Steps 1, 2, and V 123 is constructed by diagram (3.42). Also, the last line of (3.44) for all i yields the fiber −1 123 (ζ) of the local smoothings 123 : V 123 → ∆ over ζ ∈ ∆. Note that at this point we have only constructed differential geometric smoothings, and thus each fiber over ζ ∈ ∆ is only given as a smooth manifold without a complex structure. In Section 3.4, we shall construct on each fiber over ζ ∈ ∆ a complex structure which depends continuously on ζ.
3.4. Existence of holomorphic volume forms on global smoothings. Here we shall prove Theorem 1.3.
Proof of Theorem 1.3. Let ∆ = ∆ = { ζ ∈ C | |ζ| < } for e −3 and let T ζ for ζ ∈ ∆ be as in (3.19), so that we have T ζ ∈ (T , ∞] with T 3. Let X i and V ij , V ijk be as defined in (3.29) and Sections 3.2 and 3.3 respectively. Then for all pairs (n 1 , n 2 ) with n 1 < n 2 , Λ n 1 n 2 = ∅ and Λ (2) n 1 n 2 = ∅, we glue together X n 1 , X n 2 and V n 1 n 2 according to diagram (3.30), in which subscripts 1, 2 and i, j are replaced with n 1 , n 2 and n i , n j , respectively. At the same time, for all triples (n 1 , n 2 , n 3 ) with n 1 < n 2 < n 3 and Λ n 1 n 2 n 3 = ∅, we glue together X n 1 , X n 2 , X n 3 and V n 1 n 2 n 3 according to (3.44), in which subscripts i, j, k are replaced with n i , n i , n k . As a result, we obtain a family of global smoothings : X → ∆ of X, which satisfies parts (a) and (b) of Theorem 1.3.
For each double curve D ij ⊂ X i , we obtained the Hermitian form ω ij on W ij \ D i which defines an SU(2)-structure together with Ω i and satisfies ω ij = ω ik on W ij ∩ W ik \ D i = β∈Λ ijk U i,β . We also obtained the Hermitian metrics g ij on W ij \ D i associated with the SU(2)-structure (Ω i , ω ij ) such that g ij = g ik on W ij ∩W ik \D i . Then we have the following two results.
Lemma 3.3. There exists a Hermitian form ω i on X i \ D i such that (Ω i , ω i ) defines an SU(2)-structure on X i \ D i and we have Proof. We shall follow the argument in [D09], Section 3.3. Let ω 1 i be a Hermitian form on the compact submanifold X 1 i of X i \ D i normalized so that 2ω 1 i ∧ ω 1 i = Ω i ∧ Ω i , and g 1 i be the associated Hermitian metric. Then gluing together g 1 i and g ij for all j ∈ I i using a cut-off function which takes 1 on X 0 i = X i \ j∈I i W ij and 0 outside X 1 i , we have a Hermitian metric g i on X i \ D i such that Letting ω i be the associated Hermitian form, we have 2λ i ω i ∧ ω i = Ω i ∧Ω i for some positive function λ i on X i \ D i such that λ i ≡ 1 on X 0 i and outside X 1 i . Then ω i = λ 1/2 i ω i gives the desired Hermitian form.
Lemma 3.4. There exists a smooth complex 1-form ξ ij on W ij such that In particular, we have ξ ij = 0 on U i,β for β ∈ Λ (2) ij .
Hence for ζ = 0, we can define a pair (Ω i,ζ , ω i,ζ ) of a smooth complex and a real 2-form on X i \ D i by where ρ T (x) = ρ(x − T + 1) is a translation of the cut-off function ρ : R → [0, 1] with Then Ω i,ζ is d-closed, and under the decomposition (3.17) of W ij , we have (3.45) ij . By (3.23) and (3.33), we have an estimate ij , where the norm is measured by the associated metric g ∞ ij , and C i is a constant which is independent of T ζ . Now recall that X ζ is constructed as a differentiable manifold by the gluing procedures according to the last lines of diagrams (3.30) and (3.44) around all double lines and triple points. Then we see from (3.45) that the pairs (Ω i,ζ , ω i,ζ ) of 2-forms on X T ζ +1 i for all i extend to a pair ( Ω ζ , ω ζ ) on all of X ζ so that Ω ζ is d-closed, and Ω ζ , ω ζ ) coincides with the SU(2)-structure (Ω ij,ζ , ω ij,ζ ) on the image of . Now set T ρ * = 2 log(max i {C i }/ρ * ) and assume T ζ > T ρ * hereafter. Then we have C i e −T ζ /2 < ρ * for all i, and thus by (3.46), Lemma 2.8, and Definition 2.10, we can define an SU(2)-structure (ψ ζ , κ ζ ) = Θ( Ω ζ , ω ζ ) on X ζ . Let φ ζ = Ω ζ − ψ ζ , so that we have dψ ζ + dφ ζ = 0.
Lemma 3.5. We have estimates φ ζ L p Ce −T ζ /2 , dφ ζ L p Ce −T ζ /2 , and dκ ζ C 0 C for some positive constants C which are independent of T ζ , where the norms are measured by the metric g ζ on X ζ associated with the SU(2)-structure (ψ ζ , κ ζ ).
Since one can see that as ζ → ζ ∈ ∆ * , η ζ converges to η ζ in L 8 1 (∧ 2 − T * M, g ζ ) → C 0,1/2 (∧ 2 − T * M, g ζ ), the resulting family { Ω ζ | ζ ∈ ∆ * } of d-closed SL(2, C)-structures on M is continuous with respect to ζ. Also, for T ζ > T * (ρ) we have an estimate on X for some positive constants C and C independent of ρ, where the C 0 -norms · C 0 i and · C 0 ζ are measured on X T ζ +1 i by the metrics g i and g ζ associated with the SU(2)-structures (Ω i , ω i ) and (ψ ζ , κ ζ ), respectively. Then redefining T * (ρ) so that C e −T * (ρ)/2 ρ in (3.47), we have for all i and ζ ∈ ∆ * with T ζ > T * (ρ), which implies that the complex structure I ζ on X ζ induced by the SL(2, C)-structure Ω ζ converges uniformly as ζ → 0 to the original complex structure on the central fiber X = −1 (0) outside the singular locus D = i D i . Hence, the continuity in part (d) is proved. This completes the proof of Theorem 1.3.

3.5.
Examples of d-semistable SNC complex surfaces with trivial canonical bundle without triple points. Here we shall give some examples of d-semistable SNC complex surfaces with trivial canonical bundle without triple points, which are smoothable to complex tori, primary Kodaira surfaces, or K3 surfaces due to Theorem 1.3. Our examples are based on those given in [D09], Examples 5.1 and 5.3. It is worth mentioning that although the classical smoothability result of Friedman cannot be applied to the SNC complex surfaces X given in Example 3.7 because we have H 1 (X i , O X i ) = 0 for some irreducible component X i of X, the modern techniques for smoothings ([FFR19], Theorem 1.1 and [CLM19], Corollary 5.15) are applicable. A typical example to see this generalization is given as follows. Let X = X 1 ∪ X 2 be a 3-dimensional SNC complex manifold such that X 1 and X 2 are two copies of CP 3 , and D = X 1 ∩ X 2 is a quartic surface. Then X is not d-semistable, but T 1 X ∼ = N D/X 1 ⊗ N D/X 2 is generated by global sections (see [FFR19], Example 1.3). Consequently, X is smoothable to Calabi-Yau threefolds due to [FFR19], Theorem 1.1. A similar argument works for SNC complex surfaces, and so with Examples 3.6-3.8. Thus, it seems that some of the examples given here are already known to the experts. However, it is still valuable to list these examples in the rest of this article because they are helpful to illustrate properties and technical features of conditions (i)-(iii) in Theorem 1.3. Particularly, Example 3.9 provides a nice example to see that condition (ii) in Theorem 1.3 is only a necessary condition for the canonical bundle of an SNC complex surface to be trivial.
Example 3.6. For d ∈ {0, ±1, ±2, ±3}, let C and C d be smooth curves of degree 3 and 3− d in CP 2 , respectively, such that C d intersects C transversely, which we regard as trivially satisfied for C 3 = ∅. Then C is an anticanonical divisor on CP 2 . Let π d : X C (d) → CP 2 be the blow-up of CP 2 at the points C ∩ C d , and D be the proper transform of C in X C (d), so π d maps D isomorphically to C. Also, let E d be an exceptional divisor π −1 d (C ∩ C d ).
Then since we have [GH], p. 608), we calculate the canonical bundle K X C (d) of X C (d) as Meanwhile, using the adjunction formula and (3.48), we have mean that we consider these as divisors on D and C, respectively. Similarly, letting C be a cubic curve which intersects C transversely, so that C is linearly equivalent to C, we find that . Thus, putting (3.51) and (3.52) into (3.50) gives that Now for the above cubic curve C and d = 0, 1, 2, 3, let X 1 = X C (−d), X 2 = X C (d) and D 1 , D 2 be the corresponding anticanonical divisors isomorphic to C. Consider an SNC complex surface X = X 1 ∪ X 2 obtained by the gluing isomorphism D 1 and local embeddings into C 3 . Then (3.53) and (3.49) lead to conditions (i) and (ii) of Theorem 1.3, respectively, while (iii) is obvious. Thus by Theorem 1.3, we obtain a family : X → ∆ of global smoothings of X with −1 (0) = X. Calculating the Euler characteristics of X 1 , X 2 , and D as χ(X 1 ) = χ(CP 2 ) + #(C ∩ C −d ) · (χ(CP 1 ) − χ(point)) = 12 + 3d, χ(X 2 ) = 12 − 3d and χ(D) = 0, we see that the Euler characteristic of the general fiber X ζ = −1 (ζ) of over ζ ∈ ∆ * is given by χ(X ζ ) = χ(X) = χ(X 1 ) + χ(X 2 ) − χ(D) = 24, where we used the invariance of homology under continuous deformations. Hence, we see from the Enriques-Kodaira classification of compact complex surfaces with trivial canonical bundle [BHPV] that X ζ is a K3 surface. ])} be the zero and the infinity section of Y CP 2 (d), respectively, where z = (z 0 , z 1 , z 2 ) ∈ C 3 \ (0, 0, 0). Then we have under which isomorphisms we have Thus by the adjunction formula, we can calculate N D CP 2 ,0 /Y CP 2 (d) and N D CP 2 ,∞ /Y CP 2 (d) as by considering the transition functions. Also, for a cubic curve C in , and let D 0 = D C,0 and D ∞ = D C,∞ be the zero and the infinity section of Y C (d), respectively. Then in the same way as above, we calculate Now consider a local coordinate system {U i , ζ i } on CP 2 given by and ξ 0,i ξ ∞,i = 1. Letting ψ C be a holomorphic volume form on C, we can consistently define a meromorphic volume form Ω on Y C (d) with a single pole along D 0 ∪ D ∞ by Thus, we see that Ω gives a trivialization (3.56) so that Y C (d) has an anticanonical divisor D = D 0 + D ∞ . Also, Ω satisfies the relation For N ∈ N and i = 1, . . . , N , let X i be a copy of Y C (d) with an anticanonical divisor D i = D i,0 + D i,∞ . We construct an SNC complex surface X = N i=1 X i by gluing together D i,∞ and D i+1,0 for all i = 1, . . . , N using the isomorphisms together with local embeddings into C 3 . Then we see that (3.54), (3.56) and (3.57) give conditions (i), (ii) and (iii) of Theorem 1.3, respectively. Thus by Theorem 1.3, we obtain a family : X → ∆ of global smoothings of X with −1 (0) = X. One can show that the general fiber X ζ = −1 (ζ) for ζ ∈ ∆ * is topologically S d × S 1 , where S d is the U(1)-bundle associated with the complex line bundle O C (d), which has Betti numbers Consequently, the general fiber X ζ has Betti numbers Hence by the classification of compact complex surfaces with trivial canonical bundle [BHPV], we see that the general fiber X ζ is a complex torus for d = 0 and a primary Kodaira surface for d = 0. In particular, the central fiber X for d = 0 cannot be Kählerian because b 1 (X) = b 1 (X ζ ) = 3 is odd. We remark that we can construct Y C (d) from a CP 1 -bundle Y CP n over CP n of any complex dimension n 2 and an elliptic curve C embedded in CP n . See also Example 3.9, in which we take N = 2 and use a gluing map D 2,∞ → D 1,0 not being an identity isomorphism.
Example 3.8. Let C be a cubic curve in CP 2 , and let X C (d) and Y C (d) be as in Examples 3.6 and 3.7, respectively. For N 2 and d = 0, 1, 2, 3, let X 1 = X C (−d), X N = X C (d), and X 2 , . . . , X N −1 be copies of Y C (d). Let us denote by D i the anticanonical divisor on X i constructed in the above examples, where D i = D i,0 + D i,∞ for i = 2, . . . , N − 1. Then we obtain an SNC complex surface X = N i=1 X i using gluing isomorphisms D 1 → D 2,0 , D i,∞ → D i+1,0 for i = 2, . . . , N − 1, and D N −1,∞ → D N . In almost the same way as in Examples 3.6 and 3.7, we can apply Theorem 1.3 to obtain a family : X → ∆ of global smoothings of X with −1 (0) = X. Since we have χ(X ζ ) = χ(X) = χ(X 1 ) + χ(X N ) = 24 as in Example 3.6 using χ(Y C (d)) = 0 and χ(C) = 0, X is smoothable to K3 surfaces. Note that this example gives an explicit construction of d-semistable K3 surfaces of Type II with any number N of irreducible components.
Here we shall give some examples of SNC complex surfaces for which condition (ii) holds but the canonical bundle is not trivial. The following example for the case of d = 0 and k = 1 is due to K. Fujita [Fu21], which we extend to all integers d and k = 1, 2, 3. The proof given here is more explicit and elementary than the original one.
Example 3.9. Let C be an elliptic curve embedded in CP n for some n, so K C is trivial. We take C so that C is isomorphic to C/Λ with the standard coordinate z, where Λ is the lattice in C generated by 1 and √ −1. We will identify C with C/Λ below and use z ∈ C/Λ as a coordinate on C. For d ∈ Z, let Y C (d) be a CP 1 -bundle over C obtained as in Example 3.7. Then Y C (d) has an anticanonical divisor D = D 0 + D ∞ with D 0 , D ∞ ∼ = C, and a meromorphic volume form Ω given by (3.55) with a single pole along D 0 and D ∞ which Now let X 1 and X 2 be two copies of Y C (d), and define an SNC complex surface X = X 1 ∪ X 2 using gluing isomorphisms τ 1 : D 1,∞ → D 2,0 and τ 2 : D 2,∞ → D 1,0 , together with local embeddings into C 3 , where τ 1 and τ 2 are given by (3.58) τ 1 : z → z and τ 2 : z → ( √ −1) k z for k = 0, 1, 2, 3.
Note that if k = 0 in (3.58), the resulting SNC complex surface X is the same as that obtained in Example 3.7.
As we saw in Example 3.7, X satisfies condition (ii) of Theorem 1.3. However, the following result shows that the canonical bundle K X of X is not trivial for k = 1, 2, 3, while K X is trivial for k = 0 as desired.
Proof. Applying Proposition 2.14 to our example, we see that H 0 (X, K X ) is given by the kernel of the linear map where ρ i is given by s i for i = 1, 2 and i = 3 − i.
We can modify the above example as follows. For a modification, we can take any number N ∈ N of components X i = Y C (d) and take τ i : D i,∞ → D i+1,0 as z → ( √ −1) k i z with k i = 0, 1, 2, 3 for i = 1, . . . , N as in Example 3.7. Then one can see that H 0 (X, K X ) ∼ = C if i k i ≡ 0 mod 4 and H 0 (X, K X ) = 0 otherwise. For a further modification, we can take the lattice Λ as general, which still satisfies Λ = −Λ. In this case, we can take τ i as z → (−1) k i z with k i = 0, 1. Then similarly, one finds that H 0 (X, K X ) ∼ = C if i k i is even, and H 0 (X, K X ) = 0 if i k i is odd.

EXAMPLES OF d-SEMISTABLE K3 SURFACES OF TYPE III
In this section, we provide several examples of d-semistable SNC complex surfaces X with triple points to which we can apply Theorem 1.3. Furthermore, we show that all of our examples are d-semistable K3 surfaces of Type III by computing the Euler characteristic of the general fiber X ζ of the resulting smoothings, and using the result of the Enriques-Kodaira classification as well.
Let us recall Example 3.1, where we considered an SNC complex surface X in CP 3 with four hyperplanes X 0 , . . . , X 3 as irreducible components. We saw that in the special case where X i are given by X i = {[z] ∈ CP 3 |z i = 0}, X satisfies conditions (ii) and (iii) of Theorem 1.3, but does not satisfy (i), i.e., X is not d-semistable. If we change the configuration of X 0 , . . . , X 3 so that they do not have fourfold intersections, then the number of triple points may change, but we see that X still satisfies conditions (ii) and (iii). To obtain a d-semistable SNC complex surface from X, we will use N.-H. Lee's criterion given in [L19] (see also Lemma 4.3) and blow up X at appropriate points in the double lines.
We will rename X, X i , D ij in Example 3.1 as X , H i , L ij , respectively, because we want to let X = 3 i=0 X i with X i ∩ X j = D ij be the desired d-semistable SNC complex surface. We will otherwise use the same notation. Now for an arbitrary configuration of four hyperplanes H i in CP 3 , we newly define triple points p i by p i = H 0 ∩· · ·∩ H i ∩· · ·∩H 3 , where we omitted H i . 4.1. Blow-up of an SNC complex surface at finite points in double curves excluding triple points. Here we shall prove Proposition 4.1, which will be used in constructing examples of d-semistable K3 surfaces of Type III in later sections.
Consider an SNC complex surface X = i X i with double curves D i = j∈I i D ij and triple points T ij = k∈I ij T ijk on each X i . Suppose that X satisfies conditions (ii) and (iii) of Theorem 1.3. For a point p ∈ D ij \ T ij , we blow up X i ∪ X j at p as follows.
Meanwhile, the meromorphic volume form Ω i on U i,α lifts to Ω i on U i,α and U i,α , which is locally represented as for i = 1, 2 and j = i .
Thus, we see that Ω i has a single pole along D 12 and yields a trivialization which leads to condition (ii) of Theorem 1.3. Also, we have which leads to condition (iii) of Theorem 1.3. Defining U 3 0 ⊂ C 3 by we see that ι i ( U i ) is given by {ξ i = 0} in U 3 0 , and thus U 1,α ∪ U 2,α embeds into U 3 0 ∼ = C 3 as {ξ 12,α ξ 21,α = 0}. Hence, the blow-up of the SNC complex surface X 1 ∪ X 2 at p is again an SNC complex surface around π −1 p (p) ∩ D 12 corresponding to ((0, 0, 0), [1, 0, 0]) ∈ C 3 . Extending the above local argument to the whole of the blow-up of X , we finally have the following result.
Proposition 4.1. Let X = i X i be an SNC complex surface satisfying conditions (ii) and (iii) of Theorem 1.3. Let X = i X i be the blow-up of X at finite points in the double curves excluding the triple points. Then X is also an SNC complex surface satisfying conditions (ii) and (iii) of Theorem 1.3.

4.2.
A d-semistable K3 surface of Type III. Here and hereafter, we will denote tensor products of line bundles by their sums. Also, we will denote the divisor class [D] simply by D. Throughout this section, indices i, j, and k will take 0, 1, 2, or 3, and if i and j are placed together as subscripts, then we will understand as i < j unless otherwise stated.
Example 4.2. As we mentioned above, we rewrite the SNC complex surface with triple points in Example 3.1 as Then the triple points in X are given by To make X d-semistable, we consider the blow-up X i of H i at two points except for the triple points in each L ij . According to Remark 2.11 of [Fr83], we define N X (L ij ) in Pic(L ij ) by for the SNC variety X . See also equation (4.1) of [L19]. Then let us define the collective normal class of X by (N X (L ij )) 0 i<j 3 = (N X (L 01 ), N X (L 02 ), . . . , N X (L 23 )) ∈ 0 i<j 3 Pic(L ij ).
The following description of d-semistability is a consequence of (1.1) in Definition 1.2. Now we return to our example. In Example 3.1, (4), we saw that , and we see that the collective normal class of X is a divisor class We choose two points p ij and p ij in each L ij \ T ij as shown in Figure 1 with symbol ×. Setting P ij = {p ij , p ij } and P = 0 i<j 3 P ij , we take the simultaneous blow-up π of X at P : π : X = Bl P (X ) X ⊂ CP 3 . Let X i , D ij , and q i be the proper transforms of irreducible components H i , double lines L ij , and triple points p i under the blow-up π, respectively. Let E be the exceptional divisor, which is a disjoint union of 2#I i copies of CP 1 . Then X can be obtained as another SNC complex surface X = 3 i=0 X i . Hence, we have the following. Claim 4.4. The SNC complex surface X = 3 i=0 X i is d-semistable. Proof. We use the same notation as above. Then we see that N X (D ij ) is linearly equivalent to the divisor class of P . Hence (4.2) yields that where k and are chosen so that {i, j, k, } = {0, 1, 2, 3}. Hence, a straightforward computation shows that for all 0 i < j 3, which implies d-semistability of X by Lemma 4.3.
By Proposition 4.1 and Claim 4.4, we can apply Theorem 1.3 to obtain a family of smoothings : X → ∆. Since topology is invariant under continuous deformations, we can find the Betti numbers of the general fiber X ζ = −1 (ζ) for ζ ∈ ∆ * from those of the central fiber X. In particular, the Euler characteristic of the general fiber X ζ is calculated as where E denotes the exceptional divisor of the blow-up π. Moreover, one can compute the integral cohomology group of the general fiber from those of the components of the central SNC fiber. See [L06], Chapter IV and [L19], Proposition 3.2 for further details. According to the Enriques-Kodaira classification of compact complex surfaces with trivial canonical bundle, the resulting compact complex surface X ζ with trivial canonical bundle is a K3 surface.

TWO HYPERPLANES AND A QUADRIC SURFACE IN CP 3
We apply the argument in the previous section to more general SNC complex surfaces and will provide a more technical example. In this case, we encounter the issue that one cannot glue all components together along their intersections because the intersection parts are not isomorphic in general (see Section 5.2). This kind of problem does not happen in the case of the doubling construction [D09,DY14]. However, we will see that there is a good way to handle this sort of mismatch problem by choosing carefully where and in what order we take the blow-ups. 5.1. Notation. Let H 1 , H 2 be two hyperplanes and H 3 be a quadric surface in CP 3 . Assume the union Y = H 1 ∪ H 2 ∪ H 3 is an SNC surface. For an SNC complex surface Y = H i ∪ H j ∪ H k , let us denote L ij = H i ∩ H j and T ijk = H i ∩ H j ∩ H k , respectively. For later use, we denote the set of double curves L ij by C k with k = ν ij , where ν ij ∈ {1, 2, 3} is the unique number satisfying ijν ij = 0 as in Section 3.3. Then we find that the collective normal class of Y is a divisor class (O C 1 (4), O C 2 (4), O C 3 (4)) ∈ Pic(C 1 ) ⊕ Pic(C 2 ) ⊕ Pic(C 3 ).
For i = 1, 2, we choose nonsingular points P i in |O C i (4)| consisting of eight distinct points: P i = {p i,1 , p i,2 , . . . , p i,8 }. Also we choose P 3 in the linear system |O C 3 (4)| which consists of four nonsingular points. Furthermore, we may assume that all P i 's are distinct. Let τ be the set of triple points T 123 = H 1 ∩ H 2 ∩ H 3 where each H i intersects the rest transversely. In order to avoid the following mismatch problem, we may choose P i ∈ |O C i (4)| satisfying (5.1) P i ∩ τ = ∅ for all i ∈ {1, 2, 3}, that is, P i and τ are distinct sets of points for all i.

5.2.
The mismatch problem. We denote the blow-up of X at P by Bl P (X) as in Section 4. Suppose that we symmetrically set X 1 = Bl P 3 (H 1 ), X 2 = Bl P 1 (H 2 ), and X 3 = Bl P 2 (H 3 ) and take the proper transforms D ij of L ij = C k in X j . Then we have to glue together X i and X j along their intersections in a suitable way. Otherwise, if we mistakenly choose D 21 in X 1 and D 12 in X 2 , the mismatch problem may occur, that is, the intersections D 12 , D 21 which we want to glue together are not isomorphic. For instance, let D 21 be the proper transform of L 21 under the blow-up X 1 = Bl P 3 (H 1 ) H 1 . Now we assume that P i ∩ τ = ∅. Since the blow-up locus P 3 lies on L 21 , we see that D 21 ∼ = L 21 = C 3 . However, P 1 intersects C 3 at P 1 ∩τ (1 or 2 points). This implies that D 12 is obtained as the blow-up of L 12 along P 1 ∩ τ , namely D 12 = Bl P 1 ∩τ (L 12 ) L 12 . Thus, D 12 cannot be identified with D 21 . Consequently, we cannot glue together D 21 in X 1 and D 12 in X 2 . In the following subsection, we shall see how to deal with this sort of technical issue.

5.3.
A d-semistable SNC complex surface. As we saw in the previous section, if we choose the blow-up locus in a symmetric way, the mismatch problem may occur. Hence, we shall take the blow-up of each component H i not to be symmetric, whereas the proper transforms D ij and D ji to be isomorphic. More precisely, we will construct a d-semistable SNC complex surface X in three steps.
Step 1. For {i, j} = {1, 2}, we take the blow-up π i of H i at P j with P j ∈ |O C j (4)|, and consider the proper transform L ji of L ji under the blow-up π i . Then we show that L ji ∼ = L ij in Claim 5.1.
Step 2. We take the blow-up of H 1 at P 3 where P 3 is the proper transform of P 3 under π 1 .
Then we obtain H 1 = Bl P 3 (H 1 ) which will be a component of an SNC complex surface in the next step.
We introduce our setting in this subsection. In accordance with the previous argument in Section 5.2, we choose P i ∈ |O C i (4)| for i = 1, 2, 3 satisfying (5.1). Note that the triple locus τ consists of two points. Furthermore, we regard L ij as a divisor of H j , whereas we treat L ji as a divisor of H i , although L ij and L ji are isomorphic to each other.
Step 1. For {i, j} = {1, 2}, we consider the blow-up π i : H i = Bl P j (H i ) H i and take the proper transforms L 3i (resp. L ji ) of L 3i (resp. L ji ) under π i . Let E j = π −1 i (P j ) be the exceptional divisor in H i . Then we have isomorphisms Moreover, we claim the following isomorphism between L 21 ⊂ H 1 and L 12 ⊂ H 2 .
Step 2. Next we take the blow-up of H 1 at P 3 π 1 : H 1 = Bl P 3 (H 1 ) . The SNC complex surface X and consider the proper transform E 2 of E 2 under π 1 . Let E 3 = π −1 1 (P 3 ) be the exceptional divisor. Since P 3 / ∈ L 31 = C 2 , the blow-up π 1 does not change L 31 . Hence, the proper transform L 31 of L 31 in H 1 is isomorphic to L 31 , namely Step 3. Now we construct an SNC complex surface by gluing H 1 , H 2 and H 3 together along their intersections. As a consequence of (5.2), (5.3) and (5.4) we see that (5.5) L 21 ∼ = L 12 , L 32 ∼ = L 23 , and L 31 ∼ = L 31 ∼ = L 13 .
We use (5.5) and (5.6) for gluing all components H 1 , H 2 and H 3 together. For example, we can glue H 1 and H 2 together by using L 21 ∼ = L 12 , and further we need to consider the isomorphism L 21 ∩ L 31 ∼ = τ in H 1 because there are three components. Then one can construct an SNC complex surface X = X 1 ∪ X 2 ∪ X 3 with a normalization ψ : H 1 ∪ H 2 ∪ H 3 → X such that ψ(H 1 ) = X 1 , ψ(H 2 ) = X 2 and ψ(H 3 ) = X 3 . Setting D (i) = D jk = X j ∩ X k for {i, j, k} = {1, 2, 3}, we show the following result: Proposition 5.2. X is d-semistable.
The rest of this subsection is devoted to prove Proposition 5.2. In the light of Lemma 4.3, Proposition 5.2 is an immediate consequence of the following.
Claim 5.3. Let P i ∈ |O C i (4)| be nonsingular points as in Section 5.1. Then {P 1 , P 2 , P 3 } determines a collective normal class. Moreover, X has trivial collective normal class.
Proof of Claim 5.3. For the first part of the statement, we saw already in Section 5.1. For the proof of the second part, it suffices to show that (i) N X (D (1) ) = 0, (ii) N X (D (2) ) = 0, (iii) N X (D (3) ) = 0 for our purpose. Recalling π i : Step 1, and the definition of the normal bundle for the SNC complex surface Y , we see that N Y (L i3 ) is linearly equivalent to P j : N Y (L i3 ) = L i3 | L i3 + L 3i | L i3 + T ji3 ∼ P j .
This completes the proof of the claim.

5.4.
Computation of the Euler characteristic. Applying Theorem 1.3 to X, we obtain a family of smoothings : X → ∆ of X whose general fibers X ζ = −1 (ζ) are compact complex surfaces with trivial canonical bundle. In fact, we will show the following.
Proposition 5.4. X is a d-semistable K3 surface of Type III, that is, the Euler characteristic of X ζ is 24.
Proof. We remark that the Euler characteristic of X ζ is given by χ(X i ) − 2 i<j χ(D ij ) + 3χ(X 123 ).
Hence, the assertion is verified.