A gluing construction of projective K3 surfaces

We construct a non-Kummer projective K3 surface $X$ which admits compact Levi-flats by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective plane at nine general points.


Introduction
In the paper [KU19], we gave a method, the so-called gluing method, for constructing a family of K3 surfaces, that is, we constructed such a K3 surface by holomorphically gluing two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at nine points. The family has complex dimension 19 and each K3 surface of the family admits compact Levi-flat hypersurfaces. In this paper, we will show that there are projective K3 surfaces among the family. One of the main results is given as follows: Theorem 1.1. There exists a deformation π : X → B of projective K3 surfaces over an 18 dimensional complex manifold B with injective Kodaira-Spencer map such that each fiber X b := π −1 (b) admits a holomorphic immersion F b : C → X b with the property that the Euclidean closure of the image F b (C) in X b is a compact real analytic hypersurface C ω -diffeomorphic to a real 3-dimensional torus S 1 × S 1 × S 1 which is Levi-flat. Especially, F b (C) is Zariski dense in X b whereas it is not Euclidean dense. Moreover, X b is non-Kummer for almost every b ∈ B in the sense of the Lebesgue measure.
In the construction of K3 surfaces given in the paper [KU19], we prepare two surfaces S + and S − obtained from the blow-ups of the projective plane P 2 at nine points {p ± 1 , . . . , p ± 9 } with smooth elliptic curves C ± ∈ |K −1 S ± |. Here we assume that (S ± , C ± ) satisfy the following two conditions: (a) there exists an isomorphism g : C + → C − such that g * N − N + , where N ± := N C ± /S ± are the normal bundles of C ± in S ± , and (b) the normal bundles N ± ∈ Pic 0 (C ± ) satisfy the Diophantine condition (see Definition 2.2).
In the present paper, we take an appropriate ξ ∈ C and consider g ξ := ξ • g, where ξ : C − C/ 1, τ is the translation induced from C z → z + ξ ∈ C. Note that g * ξ N − N + remains true since N ± ∈ Pic 0 (C ± ).
Then by identifying V + s and V − s via the biholomorphic map f s , we can patch M + s and M − s to define a compact complex surface X s . In the paper [KU19], we showed that X s is a K3 surface and that the nowhere vanishing holomorphic 2-form σ s on X s satisfies For each ξ, these K3 surfaces X s with s ∈ ∆ \ {0} are the fibers of a proper holomorphic map X → ∆ from a smooth complex manifold X (= X (ξ)) such that -each fiber over s ∈ ∆ \ {0} coincides with the K3 surface X s , -the fiber X 0 over 0 ∈ ∆ is a compact complex variety with normal crossing singularities whose irreducible components are S + and S − and whose singular part is the one obtained by identifying C + and C − via g ξ , and thus -X → ∆ is a type II degeneration of K3 surfaces (see Section 4.1).
We notice that V s ⊂ X s is biholomorphic to a topologically trivial annulus bundle over the elliptic curve C := C + C − , and hence homotopic to S 1 α × S 1 β × S 1 γ , where S 1 α and S 1 β are circles in V s such that S 1 α × S 1 β is a C ∞ section of the bundle, and S 1 γ is a circle in a fiber of the bundle which generates the fundamental group. Then we define the 2-cycles A αβ , A βγ , A γα by In addition to the 2-cycles A αβ , A βγ , A γα , each K3 surface X s admits a marking, which gives 22 generators of the second homology group H 2 (X s , Z) denoted by In §5, we will give the definitions of these generators. Now let L ± be holomorphic line bundles on S ± with (L + · C + ) = (L − · C − ). Assume that there exists ξ ∈ C such that g * ξ (L − | C − ) L + | C + . Note that such a ξ always exists when (L + · C + ) = (L − · C − ) 0. We fix such a ξ ∈ C, and consider the deformation family X → ∆.  (c) (L · A βγ ) = (L · A γα ) = 0.
In our arguments it is important to describe the line bundles on V ± s and on W ± , which is given in Section 3 after preliminary studies in Section 2. Then we will prove the main theorems in Section 4. Moreover, we will determine the Chern class c 1 (L s ) of the line bundle L s in terms of the marking (1.2) in Section 5.

Neighborhoods of elliptic curves
First we give the following definition.
Definition 2.1. Let (p, q) ∈ R 2 be a pair of real numbers.
(2) (p, q) is said to satisfy the Diophantine condition if there exist α > 0 and A > 0 such that Of course, if (p, q) satisfies the Diophantine condition, then (p, q) is a non-torsion pair. Let X be a complex manifold. Denote by Pic(X) the Picard group of X, the group of isomorphism classes of holomorphic line bundles on X, and by Pic 0 (X) the subgroup of Pic(X) consisting of (isomorphism classes of) topologically trivial line bundles. Note that L ∈ Pic(X) is topologically trivial if and only if L satisfies c 1 (L) = 0 ∈ H 2 (X, Z), where c 1 (L) stands for the first Chern class of L ∈ Pic(X). If X = C is a smooth elliptic curve, then any topologically trivial line bundle L ∈ Pic 0 (C) admits a structure of unitary flat line bundle (see [Ued83]). In particular, the monodromy of L ∈ Pic 0 (C) along any loop in C is expressed as a complex number with modulus 1.
Definition 2.2. For τ ∈ H, let C = C/ 1, τ be a smooth elliptic curve, and let α and β be the loops in C corresponding to the line segments [0, 1] and [0, τ], respectively. Then a topologically trivial line bundle L ∈ Pic 0 (C) on C is said to satisfy the Diophantine condition if so does the pair (p, q) ∈ R 2 , where (p, q) is defined from L, that is, exp(p · 2π √ −1) and exp(q · 2π √ −1) are the monodromies of L along the loops α and β, respectively. Now, assume C 0 = C/ 1, τ ⊂ P 2 is a smooth elliptic curve embedded in the projective plane P 2 . Let Z := {p 1 , . . . , p 9 } ⊂ C 0 be nine points on C 0 , and S := Bl Z P 2 be the blow-up of P 2 at Z with the strict transform C of C 0 . In this case, the normal bundle p 9 ) ∈ Pic 0 (C 0 ) Pic 0 (C), and the pair (p, q) ∈ R 2 defined from L = N C/S (see Definition 2.2) is given by where p 0 is an inflection point of C 0 . Moreover, if N C/S ∈ Pic 0 (C) satisfies the Diophantine condition, then Arnol' d's theorem [Arn77] guarantees that there exists a analytically linearizable neighborhood of C in S, namely, a tubular neighborhood of C in S which is biholomorphic to a neighborhood of the zero section in N C/S . In other words, there exists a neighborhood of C in S biholomorphic to With the neighborhood W at hand, we can construct a family of K3 surfaces as mentioned in the introduction.
The maps F b in Theorem 1.1 can be constructed in this manner.

Holomorphic line bundles on toroidal groups
The neighborhood W given in (2.1) is closely related to the toroidal group. For τ ∈ H and a non-torsion pair (p, q) ∈ R 2 , we consider It is seen that U becomes a toroidal group (see e.g. [AK01]). On the toroidal group U , an important class of line bundles is the theta line bundles, given as follows. Let be a Hermitian matrix satisfying the condition where H(x, y) = t xHy for x, y ∈ C 2 , and let ρ : Λ → U (1) be a semi-character of Im H, that is, it satisfies Then we define the holomorphic function α λ = α From (2.3), the function α λ (x) satisfies the cocycle condition defines a line bundle on U , which is called a theta line bundle on U . In our setting, note that λ 2 ∈ R for any t (λ 1 , λ 2 ) ∈ Λ. Hence a nowhere vanishing holomorphic function β : C 2 → C * , given by which means that L H,ρ is holomorphically isomorphic to L H 0 ,ρ . Hereafter, we assume c = 0 and put On the line bundle L H,ρ , there is a natural metric h = h H , given by In particular, the curvature form of h H is given by Moreover the following result holds (see [AK01]).
Proposition 2.4. Assume that (p, q) satisfies the Diophantine condition. Then any line bundle L on U τ,(p,q) is holomorphically isomorphic to L H,ρ for some (H, ρ).

Deformations of K3 surfaces and Picard numbers
The following results are taught by Dr. Takeru Fukuoka.
Proposition 2.5. Let P : X → T be a deformation family of K3 surfaces. Assume that the Kodaira-Spencer map ρ KS,P : Proof. Take a base point 0 ∈ T and denote by L := Π 3,19 the K3 lattice H 2 (X 0 , Z). Fix a marking where we are regarding P(L C ) as the set of hyperplanes of L C . It follows from Torelli's theorem that the map V • is a locally closed embedding of T into P(L C ). Therefore Image V • is a locally closed subvariety of P(L C ) of dimension dim(T ). Define r :

Line bundles on W and V
For τ ∈ H, let C = C z / 1, τ be a complex torus, and for a non-torsion pair (p, q) ∈ R 2 and 0 ≤ r < R ≤ ∞, let W = W R τ,(p,q) be defined in (2.1) and V = V r,R τ,(p,q) be defined by given by (2.2). We notice that V is isomorphic to an open submanifold of the toroidal group U = U τ,(p,q) = (C z × C η )/Λ, namely, (p,q) , and W is obtained from V 0,R τ,(p,q) by adding the complex torus C. Let π : W → C be the natural projection, given by π ([(z, w)]) = [z], and denote π| V : V → C by π : V → C for simplicity.
Proof. As the topologically trivial bundle L satisfies c 1 (L) = 0, L can be represented by some Moreover f jk can be expressed on W j as a convergent power series where (z j , w j ) are coordinates on W j which come from (z, w). Then it is enough to show that there are holomorphic functions g j : W j → C such that Note that there exists a multiplicative 1-cocycle {t jk } with t jk ∈ U (1) representing N C/W such that w k = t kj ·w j for any j, k. Since (U jk , f jk,n ) ∈ H 1 {U j }, N −n C/W and N C/W is non-torsion, the δ-equation −g j,n + t −n jk · g k,n = f jk,n has a unique solution g j,n : U j → C for each n > 0. Furthermore the power series In our setting, since N C/W satisfies the Diophantine condition, there exist A > 0 and α > 0 such that d(I C , N n C/W ) ≥ A · n −α holds for any n ≥ 1. Cauchy's inequality shows that for any ∈ (0, R), there exists M > 0 such that f jk,n (z j ) ≤ M/ n for any n ≥ 1 and z j ∈ U j ∩ U k . Hence we have which means that the power series (3.1) indeed converges because ∈ (0, R) is chosen arbitrarily. Therefore we have π * (α| C ) = α in H 1 (W , O W ).
Remark 3.2. The following can be proved in a similar manner by replacing a Taylor power series with a Laurent power one: for any L ∈ Pic 0 (V ), there exists an F ∈ Pic 0 (C) such that L = π * F, which is proved in [AK01] for the case where V = U is a toroidal group. Conversely, [AK01] also proves the statement that if a pair (p, q) does not satisfy the Diophantine condition, then there exists an L ∈ Pic 0 (U ) such that L π * F for any F ∈ Pic 0 (C).

Proposition 3.3.
Assume that (p, q) satisfies the Diophantine condition. Then L = π * (L| C ) holds for any L ∈ Pic(W ). In particular, the restriction map Pic(W ) → Pic(C) is an isomorphism.

Now let us recall the three 2-cycles
Proof. We will only prove the assertion (1) as the other cases can be treated in the same manner. Note that the class c 1 (L H,ρ ) can be represented as where w = exp(η · 2π √ −1). By the definition of A αβ , put z = α + τβ and η = pα + qβ. Since p, q, α, β ∈ R, we have where j αβ : A αβ → V is the embedding induced by i α and i β . In a similar manner, one has Therefore we have Proposition 3.5. Let L ∈ Pic(V ) be a holomorphic line bundle on V . Assume that (p, q) satisfies the Diophantine condition. Then the following are equivalent.
(1) There exists a holomorphic line bundle G ∈ Pic(W ) on W such that L = G| V .
(2) (L · A βγ ) = (L · A γα ) = 0. In order to describe the holomorphic line bundle L → X on X via the isomorphisms (1.1), we define manifolds M ± and V by where ∼ is the equivalence relation generated by Then M ± and V are glued together to yield the deformation family X via injective holomorphic maps f ± : M ± ⊃ V ± → V , where The restriction of X → ∆ on M ± is the natural projection M ± → ∆, while that on V is given by [(z + , w + , w − )] → w + · w − . Moreover, it should be noted that there are natural projections ϕ ± : M ± → S ± and ϕ : V → C + given by ϕ([(z + , w + , w − )]) = [z + ]. Then a holomorphic line bundle L → X is defined by the pullbacks ϕ * ± (L ± ) on M ± and ϕ * (L + | C + ) on V . We notice that the line bundle L → X is well-defined since the line bundles f * + ϕ * (L + | C + ) and f * − ϕ * (g * ξ (L − | C − )) are the same as the restrictions L| V + and L| V − respectively, by virtue of Proposition 3.3 and the assumption g * ξ (L − | C − ) L + | C + .

Idea of proof of Theorem 1.2 (ii)
Let X = X s be a K3 surface obtained by gluing M + = M + s and M − = M − s , and L ± be an ample line bundle on S ± . In order to show Theorem 1.2 (ii), we will construct a C ∞ -Hermitian metric on L := L s = L + ∨ L − with positive curvature in the following manner for fixed 0 < R 1 < R 2 < R: Step 1: Construct a C ∞ -Hermitian metric h ± on L ± such that: h ± can be glued to define a C ∞ -Hermitian metric h on L (if 0 < |s| < ε 0 ), -the Chern curvature of h ± is semi-positive: Step 2: Construct a C ∞ function ψ ± on S ± \ C ± such that: ψ ± can be glued to define a C ∞ function ψ on X, ψ ± is psh on M ± \ {R 2 ≤ |w ± | ≤ R}: √ −1∂∂ψ ± | M ± \{R 2 ≤|w ± |≤R} ≥ 0, ψ ± | W ± depends only on |w ± |, and - Step 3: For 0 < c 1, h·e −cψ is a desired metric on L with positive Chern curvature In our construction, h ± · e −cψ ± is a C ∞ -Hermitian metric on L ± | S ± \C ± with positive Chern curvature such that h ± · e −cψ ± ∼ (log |w ± |) 2 as w ± → 0. Moreover, ω ± := complete Kähler metric on S ± \ C ± , and on a neighborhood |w ± | < √ ε 0 R of C ± , the form ω ± is expressed as

Proof of Theorem 1.2 (ii)
Let S be the blow-up of P 2 at nine points, and C ⊂ S be an elliptic curve in |K −1 S | such that N C/S ∈ Pic 0 (C) satisfies the Diophantine condition. Then Arnol' d's theorem says that there is an analytically linearizable neighborhood W ⊂ S of C. By shrinking W if necessary, we may assume that W is isomorphic to W R τ, (p,q) for some R > 0, τ ∈ H and (p, q) ∈ R 2 satisfying the Diophantine condition, and let π : W → C be the projection given in Section 3.
Let L ∈ Pic(S) be an ample line bundle, which implies that there exists n ∈ N such that L n ⊗ [−C] is very ample, and let g 1 , g 2 , . . . , g N be a basis of H 0 (S, L n ⊗ [−C]), which are regarded as sections of L n with zeros along C. Then the singular Hermitian metric h L on L is defined by The metric h L has a pole along C and its restriction h L | S\C induces a C ∞ -metric on S \ C with positive curvature form Fix 0 < R 1 < R 2 < R. Then we define a metric h on L by where ε > 0 and RegularizedMax : R 2 → R is the regularized maximum function (see [Dem12,Chapter I,Lemma 5.18]). Note that, by choosing ε > 0 sufficiently small, one may assume that h = h L holds on {[(z, w)] ∈ W | R 1 < |w|}, which ensures the smoothness of h. Then where ϕ L and ϕ C are the local weight functions of h L and h C , respectively. By the construction of h, there exists a positive constant ε 0 such that h = ε −1 · π * h C holds on |w| < √ ε 0 R . By shrinking ε 0 if necessary, we may assume It is easy to see that ∂∂ψ = 0 outside {|w| ≤ R} and ∂∂ψ = 2 · dw ∧ dw/ |w| 2 on {0 < |w| < R 2 }. Finally, we choose c > 0 so that Here note that such a c > 0 exists since We consider the metric h · e −cψ on S \ C. Our assumption on c > 0 says that Moreover, h · e −cψ has positive curvature also on {0 < |w| < R 2 }, since it holds [Dem12, Chapter I, Lemma 5.18(e)]). Therefore the curvature of h · e −cψ is positive on S \ C. Now we consider two pairs (S ± , C ± ) of surfaces S ± and curves C ± ⊂ S ± given in the introduction, which admit analytically linearizable neighborhoods W ± ⊂ S ± of C ± , and assume that W ± are regarded as subspaces {[(z ± , w ± )] | |w ± | < R} of toroidal groups. Moreover let L ± be ample line bundles with (L + · C + ) = (L − · C − ) and g ξ : In what follows we abuse the notation to denote g ξ simply by g. Then the above argument shows that there exist C ∞ -metrics h ± · e −cψ ± on S ± \ C ± such that √ −1Θ h ± ·e −cψ ± > 0 on S ± \ C ± and h ± = ε −1 · π * ± h C ± , ψ ± (z ± , w ± ) = log |w ± | 2 |s| on 0 < |w ± | < √ |s|R (< √ ε 0 R < R 1 ) . As our K3 surface X s is given by gluing two surfaces M ± s = S ± \ w ± ≤ |s|/R via the map (z + , w + ) → (z − , w − ) = (g(z + ), s/w + ), it follows from Proposition 3.3 that h ± can be glued together and become a global C ∞ -Hermitian metric on L s = L + ∨ L − . Moreover, on √ |s|/R < |w + | < √ |s|R , we have ψ + (z + , w + ) = log |w + | 2 / |s| 2 and which means that ψ ± can be glued together and become a global C ∞ -function ψ on X s . Therefore h ± · e −cψ ± yield a C ∞ -metric on X s with positive definite curvature form.

Proof of Theorem 1.2 (iii)
The equivalence (a) ⇐⇒ (c) follows from Proposition 3.5, and the implications (a) =⇒ (b) follows from Theorem 1.2 (i). In what follows we show (b) =⇒ (c). Take a line bundle L → X as in (b) and consider the function h : ∆ → Z defined by h(t) := L| M + t · A βγ , where we are regarding A βγ as a cycle of M + t . As M + → ∆ is a submersion, h is continuous. Thus h is a constant function. Therefore, in order to show that (L · A βγ )(= h(s)) is equal to zero, it is sufficient to show that h(0) = 0, which follows from Proposition 3.5 since L| M + 0 coincides with the restriction of the line bundle (L| X 0 )| S + to M + 0 . The equation (L · A γα ) = 0 can be shown in the same manner.

Proof of Theorem 1.1
Our construction of K3 surfaces has 19 complex dimensional degrees of freedom if we allow the variation of ξ [KU19]. Indeed, for a fixed pair (p, q) ∈ R 2 satisfying the Diophantine condition, we have the following parameters: (I) 1 parameter τ ∈ H determining the elliptic curve C + C − , (II) 16 parameters {p ± 1 , . . . , p ± 8 } determining the centers of the blow-ups π ± (here p + 9 and p − 9 are fixed from the conditions (a) and (b) in the introduction), (III) 1 parameter ξ ∈ C determining the isomorphism g ξ : C + → C − , and (IV) 1 parameter s ∈ ∆ \ {0} determining the gluing function f s : V + s → V − s . Note that there always exist ample line bundles L ± → S ± with (L + ·C + ) = (L − ·C − ). If such ample line bundles L ± are fixed, then ξ is determined uniquely up to modulo 1, τ from the condition g * ξ (L − | C − ) L + | C + , and depends holomorphically on the parameters given in (I) and (II) (see also the relation (5.5)). Moreover, for any s ∈ ∆ \ {0} with sufficiently small |s| 1, the K3 surface X s admits an ample line bundle L s = L + ∨ L − by Theorem 1.2 (ii). Hence we have an 18 dimensional family of projective K3 surfaces, whose Kodaira-Spencer map is injective by [KU19, Theorem 1.1]. Moreover it follows from [KU19] that there exists a holomorphic immersion F b : C → X b mentioned in Theorem 1.1 (see also Remark 2.3). Finally among the family, almost every fiber is a non-Kummer K3 surface since if follows from Proposition 2.5 that almost every fiber X s has the Picard number ρ(X s ) ≤ 2.

Calculation of the Chern class c 1 (L)
Let S ± be surfaces obtained from the blow-ups π ± : S ± → P 2 of the projective plane P 2 at nine points {p ± 1 , . . . , p ± 9 } with smooth elliptic curves C ± ∈ K −1 S ± . In our assumption (S ± , C ± ) satisfy Conditions (a) and (b) given in the introduction. Moreover let L ± be holomorphic line bundles on S ± . In this section, we compute