On Kummer-like surfaces attached to singularity and modular forms
aa r X i v : . [ m a t h . AG ] D ec On Kummer-like surfaces attached to singularity andmodular forms
Atsuhira Nagano and Hironori ShigaDecember 23, 2020
Abstract
The aim of this paper is to obtain a natural counterpart of the family of Kummer surfaces fromthe viewpoint of lattice polarized K K Introduction
A Kummer surface Kum( A ) attached to an Abelian surface A is a K A/ h κ i by the Kummer involution κ . It is a traditionalresearch object, firstly studied by Kummer in the 19th century. It is defined by a quartic equation inthe projective space P ( C ) with remarkable geometric properties (for example, see [H]). Also, the familyof Kummer surfaces plays a significant role in the proof of the Torelli type theorem for K K E × E ) derived from the product of two elliptic curves E and E , he showed that there exists adouble covering ϕ Shio : Kum( E × E ) S Shio . (0.1)Here, the K E × E ) and S Shio are explicitly defined by the following equationsKum( E × E ) : Z = Y − αY + (cid:16) U + 1 U − β (cid:17) , (0.2) S Shio : Z = Y − αY + (cid:16) X + 1 X − β (cid:17) , (0.3)where α and β are complex parameters, and ϕ Shio has an absolutely simple expression: (
U, Y, Z ) ( X, Y, Z ) = ( U , Y, Z ) . After that, Ma [Ma] studied the lattice structure for general Kummer surfacesKum( A ) in detail, and he proved that there is a rational double covering ϕ Kum : Kum( A ) S CD . (0.4) Keywords: Kummer surfaces ; K K S CD , which we call the partner surface of Kum( A ), is given by a member of thefamily of K S CD is very closely related to theSiegel modular forms of degree 2 (see Section 5.1). In this paper, let U be the unimodular hyperboliclattice of rank 2. Let A j ( −
1) and E j ( −
1) be the root lattices with negative eigenvalues. Also, if a lattice L with the intersection matrix ( c jk ) j,k are given, L ( n ) for n ∈ N means the lattice given by the intersectionmatrix ( nc jk ) j,k . Then, the K L K is isometric to U ⊕ U ⊕ U ⊕ E ( − ⊕ E ( − . The N´eron-Severi lattice of a K L K . It is proved that the transcendental lattice of S CD is generically givenby U ⊕ U ⊕ A ( − A ) is U (2) ⊕ U (2) ⊕ A ( − . By the way, it is very interesting to study K K ,
2) introduced by Ishiiand Watanabe [IW]. The exceptional set of a minimal resolution of such a singularity is given by a normal K K F ( ζ , ζ , ζ , ζ ) = 0. We canregard this hypersurface as a K K K K K K ( t ) be the corresponding K K t = ( t , t , t , t , t ). Wewill see that there exists a double covering ϕ : K ( t ) S ( t ) , (0.5)which gives a natural extension of (0.4). Here, the partner surface S ( t ) is the lattice polarized K S ( t ) naturally contains the family of the Clingher-Doran surfaces S CD .The family of S ( t ) has interesting properties from various viewpoints. For example, we have an explicitexpression of the moduli space and the period mapping for S ( t ) in [Na]. Also, the transcendental latticeof S ( t ) is given by U ⊕ U ⊕ A ( − K t , t , t , t and t give generators of the ring of theHermitian modular forms for the imaginary quadratic field of the minimal discriminant (for detail, seeSection 2). Like the explicit forms (0.2) and (0.3), the K K ( t ) and S ( t ) are defined by explicitequations K ( t ) : Z = Y + t Y + t + (cid:16) U + t Y + t U (cid:17) + t U , (0.6) S ( t ) : Z = Y + t Y + t + (cid:16) X + t Y + t X (cid:17) + t X . (0.7)In our argument, the surfaces K ( t ) and S ( t ) will give natural extensions of the results of [Sh] and [Ma].We will apply solid results of the partner surface S ( t ) to the research on K ( t ). In this way, we will seethat our K ( t ) gives a natural counterpart of the Kummer surface as follows. • The family of K ( t ) naturally contains that of Kum( A ). • The double covering (0.5) is given by a very clear form (
U, Y, Z ) ( X, Y, Z ) = ( U , Y, Z ) , like(0.1). 2 The transcendental lattice Tr( K ( t )) of K ( t ) is isometric to Tr( S ( t ))(2), as Tr(Kum( A )) ≃ Tr( S CD )(2)holds. • The inverse of the period mapping for the partner S ( t ) of K ( t ) gives Hermitian modular forms, asthat for the partner S CD of Kum( A ) gives Siegel modular forms.Here, we need to remark that there is an error in the list of [B] (for detail, see Remark 4.1). Namely,as we will prove in Section 4, the transcendental lattice of K ( t ) is U (2) ⊕ U (2) ⊕ A ( − U ⊕ U ⊕ A ( −
1) in [B].The authors expect that our results provide a steady step for arithmetic researches of K K ( t ) beyonds that ofAbelian surfaces. So, non-trivial arithmetic results of K ( t ) must be inherent in K U (2 , K ( t ) naturally contains the family of the classical Kummer surfaces, which hasfruitful arithmetic properties as we remarked at the begging of the introduction.At the end of the introduction, we mention counterparts of the Kummer sandwich. According to[Ma], there is a rational double covering ψ Kum : S CD Kum( A ) :Kum( A ) ϕ Kum S CD ψ
Kum
Kum( A ) . (0.8)They call such a phenomenon the Kummer sandwich. Since special members of the family of S ( t ) areequal to S CD , the sandwich (0.8) has a very clear expression by using (0.6) and (0.7) (Theorem 5.1).However, general members S ( t ) do not have a rational double covering S ( t ) K ( t ) (Theorem 5.2).Namely, there exists only one side of the sandwich for our surfaces K ( t ) and S ( t ). Now, let us remarkthe result of Clingher, Malmendier and Shaska [CMS]. They obtain another family of K S CMS which naturally contains the family of S CD (see Table 1). We will see that there is a natural extension K MSY ϕ
MSY S CMS ψ
CMS K MSY (0.9)of the Kummer sandwich (0.8), where K MSY is a K P ( C )blanched along 6 lines and precisely studied by Matsumoto, Sasaki and Yoshida [MSY]. There are veryclear expressions of ϕ MSY and ψ CMS , like the result of (0.2) and (0.3) in [Sh] (Theorem 5.3).Kummer and [CD] This paper and [Na] [MSY] and [CMS]Kummer-like surface Kum( A ) K ( t ) K MSY U (2) ⊕ ⊕ A ( − U (2) ⊕ ⊕ A ( − U (2) ⊕ ⊕ A ( − ⊕ A ( − S CD S ( t ) S CMS U ⊕ ⊕ A ( − U ⊕ ⊕ A ( − U ⊕ ⊕ A ( − ⊕ A ( − , , ,
12 4 , , , ,
18 4 , , , , K K K K ( t ). In Section 4, we will provide a direct andgeometric construction of 2-cycles on K ( t ) in order to calculate the lattice structure of K ( t ) precisely.The main result of this section shows that our surface K ( t ) is a counterpart of the Kummer surface. Also,3t corrects an error in the list of [B]. In Section 5, we will consider natural counterparts of the Kummersandwich. K surfaces from singularities K singularities We survey the results of simple K X , x ) be a normal isolated singularity in an analytic space X of 3-dimension. Let ρ : ( e X , E ) → ( X , x ) be a good resolution, where E is the exceptional set. For any positive integer m , the plurigenera δ m ( X , x ) is defined as dim C (Γ( X − { x } , O ( mK )) /L m ( X − { x } )). Here, K is the canonical bundle on X − { x } and L m ( X − { x } ) is the set of holomorphic m -ple 3-forms on X − { x } which are L m -integrableat x . A singularity ( X , x ) is said to be purely elliptic if it holds δ m ( X , x ) = 1 for any m .Suppose that ( X , x ) is quasi-Gorenstein. Letting E = ∪ i E i be the irreducible decomposition, we havean expression in the form K e X = π ∗ K X + P i ∈ I m i E i − P j ∈ J m j E j , where m i , m j ∈ Z > . If ( X , x ) ispurely elliptic, we can prove that m j = 1 holds for any j ∈ J . Now, setting E J = P j ∈ J E j , there is anunique j ∈ { , , } such that H ( E J , O J ) ≃ H ,j ( E J ) ≃ C . Such a singularity ( X , x ) is said to be of(0 , j )-type. Now, we have the following result. Proposition 1.1. ([IW] Section III) For a -dimensional isolated singularity ( X , x ) , the following twoconditions are equivalent.(i) ( X , x ) is Gorenstein, purely elliptic and of (0 , -type,(ii) ( X , x ) is quasi-Gorenstein and the exceptional divisor E for any minimal resolution ρ : ( e X , E ) → ( X , x ) is a normal K surface. If ( X , x ) satisfies the conditions in the above proposition, it is called a simple K K X , x ) is given by a hypersurface singularity.Let X be a hypersurface X = { ( ζ , ζ , ζ , ζ ) ∈ C | F ( ζ , ζ , ζ , ζ ) = 0 } where F is a non-degeneratepolynomial given by F ( ζ , ζ , ζ , ζ ) = X ( p ,p ,p ,p ) λ p ,p ,p ,p ζ p ζ p ζ p ζ p = X p λ p ζ p , (1.1)and x = (0 , , , . It is known that p ∈ Z ≥ such that λ p = 0 are integral points of a 3-dimensionalNewton polytope ∆. Such a polytope ∆ is determined by a weighted vector a = ( a , a , a , a ) ∈ Q > asfollows. Let ∆ ′ be the convex hull of n(cid:16) a , , , (cid:17) , (cid:16) , a , , (cid:17) , (cid:16) , , a , (cid:17) , (cid:16) , , , a (cid:17)o . Then, ∆ is the convex hull of all integral points of ∆ ′ . We can prove that ( X , x ) is a simple K , , ,
1) in the relativeinterior (see [W]). We note that such a weighted vector a satisfies P i =1 a i p i = 1 for any integral point p ∈ ∆. Especially, it holds that P i =1 a i = 1. Yonemura [Y] classifies such weight vectors into 95 types.By the way, we can obtain minimal resolutions of the above simple K a is written as ( w w , w w , w w , w w ) such that gcd( w i , w j , w k ) = 1. Then,from the 4 unit vectors in Z ≥ and the integral vector ( w , w , w , w ) , we can obtain a 4-dimensional fanΥ ⊂ R ≥ . Then, we have a 4-dimensional toric variety V Υ . There is a canonical morphism ˜ ρ : V Υ → C such that V Υ − ˜ ρ − (0) is isomorphic to C − { } and ˜ ρ − (0) is a 3-dimensional weighted projective space P ( w , w , w , w ) . Letting e X be the proper transform of X by ˜ ρ , set ρ = ˜ ρ | e X . Then, we have the followingresult. 4 roposition 1.2. ([Y] Section 3, see also [IW] Section IV) The morphism ρ : ( e X , E ) → ( X , x ) givesa minimal resolution of ( X , x ) . Here, the exceptional divisor is given by a -dimensional set ρ − (0) =˜ ρ − (0) ∩ e X , which is a hypersurface in P ( w , w , w , w ) corresponding to (1.1). Let K λ be the hypersurface in P ( w , w , w , w ) in the above proposition. So, we obtain a family { K λ } λ of K P ( w , w , w , w ) with complex parameters λ = λ p ,p ,p ,p from a given weightedvector for a simple K K K K surfaces corresponding to No.88 vector In this paper, we focus on the weight vector a = (cid:16) , , , (cid:17) . This is the No.88 in the list of the weight vectors of [Y]. The purpose of this paper is to show that thefamily of No.88 gives a natural extension of the family of Kummer surfaces.In this case, the polynomial F ( ζ ) of (1.1) is given by a linear combination of 11 monomials ζ ζ , ζ ζ , ζ , ζ ζ , ζ ζ , ζ ζ , ζ ζ ζ ζ , ζ ζ ζ , ζ ζ ζ , ζ ζ ζ , ζ ζ over C . We can regard F ( ζ ) as a quasi homogeneous polynomial of weight 27, where the weight of ζ ( ζ , ζ , ζ , resp.) is 11 (9 , ,
2, resp.). A K K λ is corresponding to the hypersurface { F = 0 } in P (11 , , , . So, K λ is given by a hypersurface in C , which is defined by a linear combination ofmonomials x z , x , y , y , z , z , x y z , y z , x z , y z , z . A generic member of this family of complex surfaces is a K K (88) ( t ) = K ( t ) : x = y + ( t z + t z ) y + ( z + t z + t z + t z ) . (1.2)We note that the equation (1.2) defines an elliptic surface with 5 complex parameters t , t , t , t and t . In order to investigate the family of K K (88) ( t ), let us consider the family of the partner surfaces(2.1). We can consider the moduli of the partner surfaces, since they have clear structures of markedlattice polarized K t = ( t , t , t , t , t ) ∈ C − { } . Let us consider the elliptic surfaces z = y + ( t x + t x ) y + ( x + t x + t x + t x ) . (2.1)Suppose the weight of x ( y , z , t j , resp.) is 6 (14 , , j , resp.), the equation of (2.1) is homoge-neous of weight 42. So, we have the family of the complex surfaces (2.1) over P (4 , , , ,
18) =Proj( C [ t , t , t , t , t ]) . The point of P (4 , , , ,
18) corresponding to t = ( t , t , t , t , t ) ∈ C − { } is denoted by [ t ] = ( t : t : t : t : t ). Set T = P (4 , , , , − { [ t ] ∈ P (4 , , , , | t = t = t = 0 } . (2.2)5f [ t ] ∈ T , then the complex surface (2.1) is a K K L K , which is isomorphic to the 2-homology group of a K II , = U ⊕ U ⊕ U ⊕ E ( − ⊕ E ( − . Also, the lattice A = U ⊕ U ⊕ A ( −
1) (2.3)of signature (2 ,
4) is necessary for our study.
Proposition 2.1. ([Na] Corollary 1.1) For a generic point [ t ] of T , the transcendental lattice (the N´eron-Severi lattice, resp.) of the K surface (2.1) is given by the intersection matrix A of (2.3) ( U ⊕ E ( − ⊕ E ( − , resp.). We remark that A is the simplest lattice satisfying the Kneser conditions, which are arithmeticconditions for quadratic forms. Due to the Kneser conditions, we can obtain a discrete group derivedfrom A with very good properties as follows. First, we have a 4-dimensional space D M = { [ ξ ] ∈ P ( A ⊗ C ) | ( ξ, ξ ) = 0 , ( ξ, ξ ) > } from A . Let D be a connected component of D M . We have a subgroup O + ( A ) = { γ ∈ O ( A ) | γ ( D ) = D} . Also, set ˜ O ( A ) = Ker( O ( A ) → Aut( A ∨ /A )) , where A ∨ = Hom( A, Z ).Set Γ = ˜ O + ( A ) = O + ( A ) ∩ ˜ O ( A ) . (2.4)Due to the Kneser conditions of A , we can prove that Γ = ˜ O + ( A ) is generated by the reflections of( − , C × ) is equal to { id , det } .We can define the multivalued period mapping Φ : T → D defined by[ t ] (cid:16) Z ∆ ,t ω S [ t ] : · · · : Z ∆ ,t ω S [ t ] (cid:17) , (2.5)where ω S [ t ] is the unique holomorphic 2-form up to a constant factor and ∆ ,t , . . . , ∆ ,t are appropriate2-cycles on the K ,t , . . . , ∆ ,t are constructed geometricallyand explicitly in [NS] Section 2. We can prove that this mapping induces the biholomorphic isomorphism¯Φ : T ≃ D / Γ (2.6)by a detailed argument of the period mappings for lattice polarized K T of (2.2) gives the moduli space of marked pseudo-ample ( U ⊕ E ( − ⊕ E ( − K D ∗ bethe C × -bundle of D . If a holomorphic function f : D ∗ → C given by Z f ( Z ) satisfies the conditions(i) f ( λZ ) = λ − k f ( Z ) (for all λ ∈ C ∗ ),(ii) f ( γZ ) = χ ( γ ) f ( Z ) (for all γ ∈ Γ) , where k ∈ Z and χ ∈ Char(Γ), then f is called a modular form of weight k and character χ for the groupΓ. Theorem 2.1. ([Na], Theorem 5.1) (1) The ring A (Γ , id) of modular forms of character id is isomor-phic to the ring C [ t , t , t , t , t ] . Namely, via the inverse of the period mapping ¯Φ of (2.6), thecorrespondence Z t k ( Z ) gives a modular form of weight k and character id .(2) There is a modular form s of weight and character det . Here, s is given by s s , where s and s are holomorphic functions on D ∗ such that s = t , s = ( a polynomial in t , t , t , t , t of weight in [Na] Section 1 ) . These relations determine the structure of the ring A (Γ) of modular forms with characters. D and the 4-dimensional complex bounded symmetricdomain H I of type I . Via this biholomorphic mapping, the modular forms in Theorem 2.1 are identifiedwith the Hermitian modular forms for the imaginary quadratic field of the smallest discriminant (see[NS] Section 3). Moreover, we have an explicit expression of the modular forms in Theorem 2.1 by thetafunctions introduced by Dern-Krieg [DK] (see [NS] Section 4).Thus, the family of K K surfaces Let us summarize results of Nikulin involutions. For detail, see [Ni1], [Ni2] and [Ma].Let X be an algebraic K ω be the unique holomorphic 2-form on X up to a constantfactor. An involution ι ∈ Aut( X ) is called a Nikulin involution (or symplectic involution) if it holds ι ∗ ω = ω . If ι is a Nikulin involution on X , then the minimal resolution Y = ^ X / h ι i is also an algebraic K X Y . Conversely, any given rational doublecovering X Y of K ι be a Nikulin involution on X and set T ι = H ( X , Z ) ι ∗ . Then, we can see that the transcendentallattice Tr( X ) is a primitive sublattice of T ι . We note that the orthogonal complement ( T ι ) ⊥ of T ι in L K has remarkable properties. The lattice ( T ι ) ⊥ is an even negative definite lattice and has no ( − . Moreover, we have(( T ι ) ⊥ ) ∨ / ( T ι ) ⊥ ≃ ( Z / Z ) (see [Ni1] Proposition 10.1). According to these properties, we can see that( T ι ) ⊥ is isometric to E ( −
2) and T ι ≃ U ⊕ U ⊕ U ⊕ E ( − . (3.1)Thus, the lattice E ( −
2) is important to study Nikulin involutions. We will use the following result.
Lemma 3.1. ([Ni2] Section 2, see also [Ma] Proposition 2.1) Let X be an algebraic K surface.(1) If X admits a Nikulin involution ι , there is a primitive embedding Tr( X ) ֒ → U ⊕ U ⊕ U ⊕ E ( − . (3.2) Here, the transcendental lattice of Y = g X /ι is given as follows: Tr( Y ) ≃ (cid:16) (Tr( X ) ⊗ Q ) ∩ (cid:16) U ⊕ U ⊕ U ⊕ E ( − (cid:17)(cid:17) (2) (3.3) (2) Conversely, if the transcendental lattice of X admits a primitive embedding (3.2), there is a Nikulininvolution ι . In this section, let us give a relation between the K K ( t ) = K (88) ( t ) of (1.2), which is comingfrom the simple K A of (2.3) with the Kneser conditions. Theorem 3.1. (1) The K surface K ( t ) of (1.2) is birationally equivalent to the surface K ( t ) : Z = Y + t Y + t + (cid:16) U + t Y + t U (cid:17) + t U . (3.4)7
2) The K surface of (2.1) is birationally equivalent to the surface S ( t ) : Z = Y + t Y + t + (cid:16) X + t Y + t X (cid:17) + t X (3.5) (3) One has an explicit double covering K ( t ) S ( t ) given by ( U, Y, Z ) ( X, Y, Z ) = ( U , Y, Z ) ,which is coming from a Nikulin involution ι on K ( t ) .Proof. (1) By putting x = U Z, y = U Y, z = U, the equation (1.2) is transformed to (3.4).(2) By putting x = X, y = X Y, z = X Z, the equation (2.1) is transformed to (3.5).(3) Since the unique holomorphic 2-form on K ( t ) is given by dY ∧ dUUZ up to a constant factor, an explicitinvolution given by ( U, Y, Z ) ( − U, Y, Z ) is a Nikulin involution on K ( t ). Remark 3.1.
The authors found the very simple expressions (3.4) and (3.5) of K surfaces duringour research in order to describe our period mapping via solutions of a system of GKZ hypergeometricdifferential equations. This investigation will be published elsewhere. Proposition 3.1.
For a generic point [ t ] = ( t : t : t : t : t ) ∈ T , the Picard number of the K surface K ( t ) of (3.4) is equal to .Proof. By Proposition 2.1 and Lemma 3.1 (especially (3.3)), the rank of Tr( K ( t )) is equal to that ofTr( S ( t )). From Proposition 2.1, we have the assertion. In this section, we will determine the lattice structure of the surface K ( t ) of (3.4).According to Proposition 3.1, the rank of the transcendental lattice of K ( t ) is generically equal to6. If we had a double covering S ( t ) K ( t ) , we could determine the structure of the transcendentallattice of K ( t ) by applying Lemma 3.1 to Proposition 2.1. However, in practice, S ( t ) does not have anyNikulin involutions, as we will see in Section 5. Therefore, in this section, we will construct 2-cycles on K ( t ) geometrically, in order to determine the lattice structure of K ( t ) directly. Eventually, we will provethe following theorem. Theorem 4.1.
For a generic point [ t ] = ( t : t : t : t : t ) ∈ T , the transcendental lattice of the K surface K (88) ( t ) , which is birationally equivalent to K ( t ) of (3.4), is given by the intersection matrix A (2) = U (2) ⊕ U (2) ⊕ A ( − . This is a primitive sublattice of U ⊕ U ⊕ U ⊕ E ( − . Remark 4.1.
Belcastro [B] determines the structures of the lattices for the K surfaces which are comingfrom resolutions of simple K singularities. Especially, in the list of [B], it is stated that the transcendentallattice corresponding to the K surfaces for the singularity of No.88 is isomorphic to A of (2.3). If thatis true, the family of K ( t ) should coincide with that of S ( t ) .We tried to follow such a story, but it was not successful. Contrally, we found that the transcendentallattice of the family of K ( t ) should be A (2) and this family naturally contains the family of the classicalKummer surfaces.Thus, at least for the case of No.88, the statement of [B] is not correct. Our Theorem 4.1 gives acorrection of this defect.
8n order to calculate the transcendental lattice of K ( t ), we consider the elliptic K K ( t ) of(1.2), which is birationally equivalent to K ( t ). There is an involution : ( x , y , z ) ( x , y , − z ) . (4.1)Also, on K ( t ), there is the unique holomorphic 2-form ω t = ω = dy ∧ dzx (4.2)up to a constant factor. Then, we have ∗ ( ω ) = − ω. Namely, is not a Nikulin involution on the K t ) : x = y + ( t w + t w ) y + ( w + t w + t w + t w ) (4.3)by the involution is a rational surface.By putting ( t , t , t , t , t ) = (0 , , , − , − K : x = y + 3 z y + z − z − z (4.4)of the family of the surfaces of (1.2). Namely, we regard K as a reference surface of the family. By virtueof a consideration of local period mapping, as in the argument in [Na] Section 1.3, we can see that thelattice structure for the reference surface K is valid for generic members of the family of K ( t ) of (3.4).We obtain the corresponding quotient surface Σ of K by :Σ : x = y + 3 wy + w − w − w. (4.5)We start an investigation on the simpler surface Σ . Properties of Σ will be useful to construct 2-cycleson K . Σ Let us observe the elliptic surface (Σ , π, P ) over w -sphere P . Let τ , τ , τ , τ , τ and τ be the so-lutions of 1 + 6 w + w − w − w + w = 0 such that approximate values are ( τ , τ , τ , τ , τ , τ ) ; ( − . , − . , − . . i, − . − . i, . . i, . − . i ) . We have singular fibres of Σ of type I over these points. Moreover, there is a singular fiber of type II ( IV , resp.) over w = τ = 0 ( w = τ ∞ = ∞ , resp.). Set τ b = − . w -space. Wehave a regular fiber π − ( τ ) : x = y − . y + 0 . . It has four ramification points on y -plane: ( v , v , v , v m ) ; ( − . , . , . , ∞ ) . On π − ( τ b ),set an oriented closed arc γ ( γ , resp.) such that its projection goes around v and v ( v and v , resp.)in the positive direction. We define their branch and orientation so that the intersection number ( γ · γ )is equal to 1 (see Figure 1).We make an oriented closed circuits δ i ( i = 0 , , . . . , , ∞ ) on the w -plane which goes around τ i in thepositive direction with the starting point τ b (see Figure 2).Let M i be the matrix which represent the monodromy transformation of the homology basis t ( γ , γ )by the continuation of the fiber π − ( τ b ) along δ i . We call it the circuit matrix. They are given in Table2. Note that they are assumed to be left actions. Here, “ V. cycle” means the 1-cycle which vanishes at τ i . We have M M M M M M M M ∞ = id , since δ δ δ δ δ δ δ δ ∞ is homotopic to zero.9 v m v v v ¬ y ¬Γ Γ Figure 1: Cycles γ , γ on π − ( τ b ) Τ Τ Τ Τ Τ Τ Τ Τ ¥ Τ b ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ¥ y zyz yzyy Figure 2: The circuits δ i ( i = 0 , , . . . , , ∞ ) K We have the elliptic K K , π z , P ) of (4.4), where the base space P is the z -sphere and π z isthe natural projection. There is the double covering K → Σ given by the involution (4.1). Recallingthe singular fibers over Σ , we find singular fibers of the elliptic K K over the points ζ j = √ τ j , ζ ′ j = −√ τ j ( j ∈ { , . . . , , b } ) , ζ = 0 , ζ ∞ = ∞ . Remark 4.2.
We have a singular fiber of type IV ( IV ∗ , resp.) over ζ ( ζ ∞ , resp.). So, besides thegeneral fiber and the global section, we have two (six, resp.) components of π − z (0) ( π − z ( ∞ ) , resp.).Hence, we obtain a sublattice of NS( K ) of rank 10 which is generated by these divisors. We denote it by L B . We lift up the cut lines ℓ , . . . , ℓ , ℓ , ℓ ∞ in the w -sphere to m , . . . , m , m ′ , . . . , m ′ , m , m ∞ . Thoseare indicated in Figure 3. We take an oriented arc α from ζ b to ζ ′ b in the simply connected region P ′ = P − ( S i m i ∪ S i m ′ i ∪ ˜ BB ′ ), where B ( B ′ , resp.) is the initial points of the cut lines m , m , m , m ′ , m ′ , m ′ and m ( m , m , m , m ′ , m ′ , m ′ and m ∞ , resp.) and ˜ BB ′ is an arc connecting them indicated in Figure3. We make the liftings δ zi ( i = 1 , . . . ,
6) of δ i indicated in Figure 4. In any case, we take ζ b as theirstarting point. By a similar manner, we make circuits δ ′ zi starting from ζ ′ b as indicated in Figure 4. Also,we take a closed circle δ z ( δ z ∞ , resp.) around z = 0 ( z = ∞ , resp.).Then, π − z ( ζ b ) and π − z ( ζ ′ b ) are the same elliptic curve which is identified with π − ( τ b ) on Σ . So wecan define γ and γ on them by this identification. On the other hand, the involution in (4.1) inducesan isomorphism π − z ( ζ b ) ≃ π − z ( ζ ′ b ) on K . So, we obtain the 1-cycles γ ′ = ( γ ) and γ ′ = ( γ ) on π − z ( ζ ′ b ) .For every δ zi , we have the local monodromy of the system { γ , γ } along these circuits. We denote10 i τ τ τ τ τ τ matrix (cid:18) −
10 1 (cid:19) (cid:18) −
10 1 (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) −
10 1 (cid:19) (cid:18) (cid:19)
V. cycle γ γ γ γ γ γ τ i τ τ ∞ matrix (cid:18) −
11 0 (cid:19) (cid:18) − −
11 0 (cid:19)
V. cycle any cycle any cycle
Table 2: Circuit matrix of δ i O Ζ Ζ Ζ Ζ Ζ Ζ Ζ ¥ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¥ B B ¢ m m m m m m m 'm 'm ' m 'm 'm 'm ¥ m ¯Α Ζ b Ζ b ¢ Figure 3: A simply connected region P ′ them by M zi . By observing the covering structure K → Σ and Figure 2, we have M zi = M i ( i = 1 , . . . , M z = M . δ zi δ z δ z δ z δ z δ z δ z matrix M zi (cid:18) −
10 1 (cid:19) (cid:18) −
10 1 (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) −
10 1 (cid:19) (cid:18) (cid:19)
V. cycle γ γ γ γ γ γ δ zi δ z δ z ∞ matrix M zi (cid:18) − − (cid:19) (cid:18) − − (cid:19) V. cycle any cycle any cycle
Table 3: Circuit matrix M zi For the local monodromy induced from δ ′ zi on the system { γ ′ , γ ′ } is just the copy of those in Table3, we have the same matrix for i ∈ { , . . . , } . We have the transformation between two systems { γ , γ } and { γ ′ , γ ′ } on π − z ( ζ ′ b ) induced from the arc α : (cid:18) γ ′ γ ′ (cid:19) = M α (cid:18) γ γ (cid:19) , where M α = (cid:18) −
11 0 (cid:19) . (4.6)Then, the local monodromy M ′ zi induced from δ ′ zi on the system { γ , γ } is given by M ′ zi = M − α M zi M α . Hence, we obtain Table 4 for them.
Remark 4.3.
We can determine M α by the facts that M ′ z = M z and that the fiber π − z (0) is a singularfiber of type IV. By the relation M z M ′ z M ′ z M ′ z M ′ z M ′ z M ′ z M z M z M z M z ∞ M z M z M z = id , thematrix M z ∞ is determined as in Table 3. Let us construct a 2-cycle Γ . We take the points Q , Q , Q and Q on the z -plane indicated inFigure 5. Then, we make a “8 shaped” closed arc ρ connecting them in this order returning to the initial11 Ζ Ζ Ζ Ζ Ζ Ζ Ζ ¥ Ζ b Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¥ Ζ b ¢ B B ¢ ∆ z1 ∆ z2 ∆ z3 ∆ z4 ∆ z5 ∆ z6 ∆ z0 ∆ z ¥ yz yy yyyz O Ζ Ζ Ζ Ζ Ζ Ζ Ζ ¥ Ζ b Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¥ Ζ b ¢ B B ¢ Figure 4: Circuits δ zi and δ ′ zi δ ′ zi δ ′ z δ ′ z δ ′ z δ ′ z δ ′ z δ ′ z matrix M ′ zi (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) −
11 0 (cid:19) (cid:18) −
11 0 (cid:19) (cid:18) (cid:19) (cid:18) −
11 0 (cid:19)
V. cycle γ γ γ − γ γ − γ γ γ − γ Table 4: Circuit matrix M ′ zi point Q (see Figure 5). We take a 1-cycle γ on the fiber π − z ( Q ). Then, by making its continuationalong ρ , we obtain a 2-cycle Γ = U ( ρ , γ ). According to Table 2, we can see that the continuationreturns back to the original γ . Namely, the cycle γ changes to γ − γ after crossing the cut line m ,and it returns back to γ after crossing m . So Γ is a 2-cycle on K . Q Q Q Q y Γ O Ζ Ζ Ζ Ζ Ζ Ζ Ζ ¥ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¥ B B ¢ y Γ -Γ G m m Figure 5: Detail of 2-cycle Γ on K By the same way, according to the indication in Figure 6, we can obtain 2-cycles Γ , Γ and Γ .Next, we construct a 2-cycle Γ . We take the points P and P illustrated in Figure 7. Take a smallcircle c ( c , resp.), which starts from P and goes around ζ ( ζ , resp.) in the positive (negative, resp.)direction. Let β be an oriented arc from P to P crossing the cut line m . So, we have 2-chains U ( c , γ ) , U ( β , γ ) and U ( c , γ ) on K . Here, we use the circuit matrices in Table 3. For example, γ on P is transformed to γ + γ after crossing the cut line m . Thus, we obtain a 2-cycleΓ = U ( c , γ ) ∪ U ( β , γ ) ∪ U ( c , γ ) , that is illustrated by Figure 7.Also, let us construct a 2-cycle Γ Take the points P , P , P and P indicated in Figure 8. Let c be as above. Let c ( c , resp.) be a small circle which starts from P ( P , resp.) and goes around ζ ( ζ ′ , resp.) in the negative (positive, resp.) direction. For ( l, m ) ∈ { (4 , , (4 , , (5 , } , let β lm be an arc12 Ζ Ζ Ζ Ζ Ζ Ζ Ζ ¥ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¥ B B ¢ y ¯ ¯Γ -Γ Γ -Γ Γ Γ +Γ G G G G Figure 6: Cycles Γ , . . . , Γ on K Ζ Ζ Ζ Ζ Ζ Ζ Ζ ¥ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¥ B B ¢ G c c Β Γ +Γ Γ -Γ -Γ Γ Γ Γ y zyz O P P m m m G Figure 7: Cycle Γ on K from P l to P m . We have the 2-chains U ( c , γ − γ ) , U ( c , γ + γ ) , U ( c , γ ′ ) , U ( β , − γ ) , U ( β , γ ) and U ( β , γ ′ ). Here, according to (4.6), we have γ ′ = γ and γ ′ = γ − γ . As with the case of Γ , we obtainthe 2-cycleΓ = U ( c , γ − γ ) ∪ U ( c , γ + γ ) ∪ U ( c , γ ′ ) ∪ U ( β , − γ ) ∪ U ( β , γ ) ∪ U ( β , γ ′ ) , which is illustrated in Figure 8. We put Γ ′ i = (Γ i ) ( i ∈ { , , , , , } ) . Set L GG ′ = h Γ , · · · , Γ , Γ ′ , · · · , Γ ′ i . (4.7) Proposition 4.1.
The rank of L GG ′ is equal to . It is orthogonal to the system L B in Remark 4.2.Hence, it holds h L GG ′ , L B i ⊗ Q = H ( K , Q ) . Proof.
We have rank( H ( K , Q )) = 22 and dim Q ( L B ⊗ Q ) = 10. By the construction, any member of L GG ′ is orthogonal to L B . So, it is enough to check that the system { Γ , · · · , Γ , Γ ′ , · · · , Γ ′ } are independent.Since we have Proposition 4.2 below, it follows that the intersection matrix to be nonsingular. Hence wehave the assertion. 13 Ζ Ζ Ζ Ζ Ζ Ζ ¥ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¥ B B ¢ c z Γ -Γ P c z Γ +Γ P Ζ Ζ ' Ζ ' ‰ -Γ P ®Γ y Γ 'P Γ ' +Γ ' ®Γ ' =Γ -Γ G c Figure 8: Cycle Γ on K Proposition 4.2.
The intersection matrix M GG ′ of the system (4.7) is given by M GG ′ = (cid:18) C G C GG ′ t C GG ′ C G (cid:19) , where C G = ((Γ i · Γ j )) ≤ i,j ≤ = ((Γ ′ i · Γ ′ j )) ≤ i,j ≤ = − − − − −
11 1 − − − − − − − − − − − ,C GG ′ = ((Γ i · Γ ′ j )) ≤ i,j ≤ − −
10 0 0 1 − . Especially, M GG ′ is nonsingular.Proof. Let us calculate the intersection number (Γ · Γ ). We have two geometric intersections R and R on the z -plane as described in Figure 9. We observe both local intersections (Γ · Γ ) R and (Γ · Γ ) R .At R , Γ ∩ π − z ( R ) is the 1-cycle γ − γ and Γ ∩ π − z ( R ) is the same 1-cycle γ − γ . So, it holds(Γ · Γ ) R = 0. At R , Γ ∩ π − z ( R ) is the 1-cycle γ and Γ ∩ π − z ( R ) is the 1-cycle γ − γ . Hence,we have (Γ · Γ ) R = ( − · ( γ · ( γ − γ )) · ( ρ · ρ ) R = − . So (Γ · Γ ) = − Z Γ ′ i ω = − Z Γ i ω ( i = 1 , . . . , . It means that every Γ i + Γ ′ i ( i = 1 , . . . ,
6) is an algebraic cycle.
Proposition 4.3.
Every Γ i − Γ ′ i ( i = 1 , . . . , is an element of the orthogonal complement of the N´eron-Severi lattice NS( K ) .Proof. By Proposition 3.1, the rank of NS( K ) is 16. By the construction, it is apparent that Γ i − Γ ′ i isorthogonal to the lattice L B . Also, we obtain ((Γ i − Γ ′ i ) · (Γ j + Γ ′ j )) = 0 from Proposition 4.2. Hence,the assertion is proved. 14 R y Γ Γ Ζ O Ζ Ζ Ζ Ζ Ζ Ζ ¥ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¥ B B ¢ y ¯Γ -Γ Γ -Γ G G m m Figure 9: Calculation of the intersection number (Γ · Γ ) Ζ Ζ Ζ Ζ Ζ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ Ζ ¢ B B ¢ m m m m m 'm 'm ' m 'm 'm ' m m ¥ m y h m y h Ζ h h ¯ h Ζ ¥ O h Figure 10: The h -arcs for dual 2-cyclesWe make oriented arcs h , h , . . . , h on the z -plane given by Figure 10. Each arc starts from ζ i withthe terminal ζ ∞ . Over these arcs, we make a system of 2-cycles: ( C = U ( h , γ ) , C = U ( h , − γ ) , C = U ( h , − γ ) ,C = U ( h , γ ) , C = U ( h , − γ ) , C = U ( h , − γ ) . (4.8)Note that on C i ( i = 1 , . . . ,
6) every 1-cycle of the fiber on h i is a vanishing cycle at the starting point ζ i ( i = 0 , , . . . , C i determines a 2-cycle on K . By a direct calculation as in the proof ofProposition 4.2, we obtain the following result. Proposition 4.4.
The intersection matrix M CG = (( C i · (Γ j − Γ ′ j ))) i,i ∈{ ,..., } is given by M CG = . It holds det( M CG ) = 1 . By this proposition, it is guaranteed that the system { Γ − Γ ′ , . . . , Γ − Γ ′ } becomes to be a systemof Z -basis of H ( K , Z ) / NS( K ). Set G j = Γ j − Γ ′ j ( j ∈ { , . . . , } ). Also, setting t ( C ′ , · · · , C ′ ) =15 − CGt ( C , · · · , C ) , it holds ( C ′ i · G j ) = δ ij ( i, j ∈ { , . . . , } ) . We set M G = ( G i · G j ) ≤ i,j ≤ . By a direct calculation, we have the following proposition.
Proposition 4.5. M G = 2 − − − − −
11 1 − − − − − −
10 0 − − − − . Theorem 4.2.
The system { G , · · · , G } gives a basis of the transcendental lattice Tr( K ) of the referencesurface with the intersection matrix A (2) .Proof. Set M J = − − − −
10 1 0 1 0 −
10 0 0 1 − − . This is a unimodular matrix. By a direct calculation, itholds M J M Gt M J = (cid:16) U (2) ⊕ U (2) ⊕ (cid:18) − − (cid:19) (cid:17) . Theorem 4.1 immediately follows from Theorem 4.2 and Lemma 3.1. K ( t ) As in [Na] Section 1.3, we can define a marking H ( K ( t ) , Z ) → L K for t ∈ T using a basis of NS( K ( t )) andthe analytic continuation in the parameter space T of (2.2). Here, the 2-cycles on the reference surface K give the initial marking. Via such a marking, we can define the period mapping for the family of K ( t ), also.We can obtain an expression of the period mapping for K ( t ) under the notation of this section as follows.Set ( H , · · · , H ) = ( G , · · · , G ) t M J . We can expand the system { H , · · · , H } to { H , · · · , H } ,which gives a basis of H ( K , Z ) . Let { D , · · · , D } be the dual basis of { H , · · · , H } with respect tothe unimodular lattice L K . By using the above mentioned marking, we can naturally obtain 2-cycles D ,t , . . . , D ,t ∈ H ( K ( t ) , Z ). Now, let us recall the double covering K ( t ) S ( t ) in Theorem 3.1 andthe period mapping (2.5) for the family of the partner surface S ( t ). Since Tr( K ( t )) = Tr( S ( t ))(2), theRiemann-Hodge relations for K ( t ) is equal to that for S ( t ). Hence, we obtain the following theorem. Theorem 4.3.
The multivalued period mapping for the family of K ( t ) , which is birationally equivalentto K ( t ) of (3.4), coincides with (2.5): T ∋ [ t ] (cid:16) Z D ,t ω t : · · · : Z D ,t ω t (cid:17) = (cid:16) Z ∆ ,t ω S [ t ] : · · · : Z ∆ ,t ω S [ t ] (cid:17) ∈ D , where ω t is the holomorphic -form of (4.2) on K ( t ) . By virtue of the argument at the end of Section 2, we have the following corollary.
Corollary 4.1.
One has an explicit expression of the inverse correspondence of the period mapping forthe family of K ( t ) by the Dern-Krieg theta functions. The theta expression is the same as that of [NS]Theorem 4.1. K ( t ) and the parameters t j via the theta functions.At the end of this section, we note that the family of K ( t ) does not coincide with the set of theequivalent classes of marked lattice polarized K K ( t )) in the sense of [D].The moduli space of such polarized K T of(2.2). Recall that T gives the moduli space of marked ( U ⊕ E ( − ⊕ E ( − K Shioda [Sh] studies the following explicit defining equations of the Kummer surface Kum( E × E ) forthe product of two elliptic curves and a K S Shio :Kum( E × E ) : Z = Y − αY + (cid:16) U + 1 U − β (cid:17) , (5.1) S Shio : Z = Y − αY + (cid:16) X + 1 X − β (cid:17) , (5.2)where these equations define elliptic surfaces and α and β are complex parameters. He shows that thereare double coverings ϕ Shio and ψ Shio such thatKum( E × E ) ϕ Shio S Shio ψ
Shio
Kum( E × E ) . Explicitly, ϕ Shio and ψ Shio are derived from the involutions(
U, Y, Z ) ( − U, Y, Z ) , ( X, Y, Z ) (cid:16) X , Y, − Z (cid:17) , (5.3)respectively. He calls this phenomenon the Kummer sandwich.The surface K ( t ) of (3.4) ( S ( t ) of (3.5), resp.) can be regarded as a natural extension of Kum( E × E )( S Shio , resp.). In this section, we will study natural and explicit counterparts of the Kummer sandwichphenomenon.
Following the work [Sh], Ma [Ma] proves that an arbitrary Kummer surface Kum( A ) of general type,where A is a principally polarized Abelian surface, admits the Kummer sandwich. Namely, there aredouble coverings ϕ Kum and ψ Kum such thatKum( A ) ϕ Kum S CD ψ
Kum
Kum( A ) (5.4)(see [Ma] Theorem 2.5). Here, S CD is a lattice polarized K U ⊕ U ⊕ A ( − K S CD . The proof of[Ma] is based on a lattice theoretic argument, but he does not give explicit forms of the defining equationsand the double coverings like (5.1) (5.2) and (5.3).As an application of Theorem 3.1, let us give a simple expression of ϕ Kum and ψ Kum in (5.4). Thisexplicit result gives a natural extension of (5.3) of [Sh].
Theorem 5.1.
The Kummer surface
Kum( A ) is given by Z = Y + t Y + t + (cid:16) U + t Y + t U (cid:17) . (5.5)17 lso, the K surface S CD is given by Z = Y + t Y + t + (cid:16) X + t Y + t X (cid:17) . (5.6) The mappings ϕ Kum and ψ Kum , which give the Kummer sandwich (5.4), are explicitly given by theNikulin involutions ι ϕ,Kum : ( U, Y, Z ) ( − U, Y, Z ) , ι ψ,Kum : ( X, Y, Z ) (cid:16) t Y + t X , Y, − Z (cid:17) , respectively.Proof. In [Na], the K U ⊕ U ⊕ A ( −
1) is introduced as anextension of the K S CD with the transcendental lattice U ⊕ U ⊕ A ( − t = 0, then(2.1) is degenerated to S CD . So, together with Theorem 3.1, the surface defined by (5.6) is birationallyequivalent to S CD .The involution ι ψ,Kum defines a Nikulin involution on (5.6). Since S CD admits a Shioda-Inose struc-ture in the sense of [Mo], the minimal resolution of the quotient S CD / h ι ψ,Kum i coincides with Kum( A ).This image is defined by the equation U V − U + 4 t + t V + 4 t Y + t V Y + V Y = 0 , (5.7)where U = X + t Y + t X and V = Z ( X − t Y + t X ). By the birational transformation ( U , V ) =(2 U ( U + Z ) , U ), (5.7) is transformed to (5.5) with the transcendental lattice U (2) ⊕ U (2) ⊕ A ( −
2) by[Mo] Theorem 5.7. Therefore, the equation (5.5) defines the Kummer surface of general type.The involution ι ϕ,Kum is just a special case of the involution of Theorem 3.1. Remark 5.1.
The involution ι ψ,Kum is equal to the van Geemen-Sarti involution (see [GS]) for thefamily of S CD studied in [CD]. Theorem 5.1 also gives a natural visualization of the Kummer sandwich phenomenon and Siegelmodular forms of degree 2. Recall that the ring of Siegel modular forms on the 3-dimensional Siegelupper half plane S of the trivial character is generated by the modular forms of weight 4 , , ,
12 and35. In fact, each of the parameters t j ( j ∈ { , , , } ) appeared in Theorem 5.1 gives a member of asystem of generators of the ring of Siegel modular forms via the period mapping for the family of S CD .Also, the generator of weight 35 and modular forms of the non-trivial character can be calculated byconsidering the degeneration of S CD (see [CD], see also [Na] Proposition 2.3). K ( t ) and S ( t ) Let us recall that K ( t ) ( S ( t ), resp.) is a natural extension of Kum( A ) ( S CD , resp.). As we proved in thelast subsection, we have the sandwich if t = 0. However, if t = 0, we have only one side K ( t ) S ( t )of the sandwich, because Theorem 3.1 and the following theorem hold. Theorem 5.2.
The parameter t gives an obstruction to the existence of a double covering S ( t ) K ( t ) for the K surfaces (3.4) and (3.5). Namely, if t = 0 , there are no Nikulin involutions on S ( t ) .Proof. In order to apply the results of Section 3.1, we assume that there is a primitive embedding i : U ⊕ U ⊕ A ( − ֒ → U ⊕ U ⊕ U ⊕ E ( − . (5.8)Now, this direct summand E ( −
2) should be a primitive sublattice of E ( − ⊕ E ( − ⊂ L K ): E ( − ֒ → E ( − ⊕ E ( − . (5.9)18o, we have a primitive embedding i ′ : U ⊕ U ⊕ A ( − ֒ → L K = II , as a composition of (5.8) and (5.9). By virtue of the results of [Se] Chapter V, we can regard the directsummand U ⊕ U of L K as the image of i ′ | U ⊕ U . Thus, we can suppose that the embedding i satisfies i ( A ( − ⊂ U ⊕ E ( − . (5.10)Now, let { e, f } be a system of basis of U such that ( e · e ) = ( f · f ) = 0 and ( e · f ) = 1. Also, let { ν , ν } be a system of basis of A ( −
1) satisfying ( ν · ν ) = ( ν · ν ) = − ν · ν ) = 1. If (5.10) holds, wehave an expression i ( ν j ) = l j e + m j f + µ j for j = 1 ,
2, where l j , m j ∈ Z and µ j ∈ E ( − − i ( ν ) · i ( ν )) = 2 l m + ( µ · µ ) . Since self-intersection numbers of E ( −
2) are in 4 Z , it follows that both l and m are odd numbers.Similarly, both of l and m are odd numbers. On the other hand, we obtain1 = ( i ( ν ) · i ( ν )) = l m + l m + ( µ · µ ) . Here, ( µ · µ ) ∈ Z , because µ j ∈ E ( − l m + l m is an odd number, but this is acontradiction. Therefore, there are no primitive embedding (5.8). So, by Lemma 3.1, there are noNikulin involutions on the K As a by-product of our explicit expression of Nikulin involutions, we can obtain a very simple expressionof a sandwich phenomenon for another families of Kummer-like surfaces. In this subsection, we will seethat.Clingher, Malmendier and Shaska [CMS] obtains another natural extension of the story for the K A ) and S CD . They consider the family of lattice polarized K S CMS with thetranscendental lattice U ⊕ U ⊕ A ( − ⊕ A ( − . Also, they study the relation between S CMS and K MSY ,where K MSY is the K U (2) ⊕ U (2) ⊕ A ( − ⊕ A ( − K MSY is precisely studied by Matsumoto, Sasaki and Yoshida [MSY] based on the viewpoint ofthe double covering of P ( C ) branched along 6 lines and hypergeometric differential equations. In [CMS],they show that there is a van Geemen-Sarti involution on S CMS and the corresponding double covering S CMS K MSY . They call the K S CMS the van Geemen-Sarti partner of K MSY .In this section, we use the Weierstrass equation of the K S CMS appeared in [NU]: z = y + ( u , x + u , x + u , x ) y + ( u , x + u , x + u , x ) , (5.11)Set t = u , , t = u , , t = u , u , , t = u , u , + u , u , , t = u , u , . Via the inverse of the period mapping for the family of (5.11), t , t , t , t and t give modular formson the 4-dimensional symmetric domain D of type IV for the orthogonal group O (2 , Z ). Also, thereare 3 modular forms for O (2 , Z ) of the 3 non-trivial characters (see [NU] Theorem 1.1). Now, remarkthat the K S CMS given by (5.11) is degenerated to the K S CD if u , = 0 and u , = 1.Thus, the family of S CMS , which is characterized by the connection with the configuration of 6 lines in P ( C ) and modular forms for O (2 , Z ), gives another natural extension of the family of S CD .19y the birational transformation ( x, y, z ) = ( X, X Y, X Z ) , the equation (5.11) is transformed to S CMS : Z = Y + u , Y + u , + (cid:16) X + u , u , Y + ( u , u , + u , u , ) Y + u , u , X (cid:17) . (5.12)There is a Nikulin involution ι ψ,CMS : ( X, Y, Z ) (cid:16) u , u , Y + ( u , u , + u , u , ) Y + u , u , X , Y, − Z (cid:17) on the K ι ψ,CMS induces a double covering ψ CMS : S CMS K MSY , where K MSY is a K K MSY : Z = Y + u , Y + u , + (cid:16) U + u , u , Y + ( u , u , + u , u , ) Y + u , u , U (cid:17) . (5.13)The transcendental lattice of K MSY is given by U (2) ⊕ U (2) ⊕ A ( − ⊕ A ( − . If u , = 0 and u , = 1, K MSY is degenerated to Kum( A ) of (5.5). So, we can regard K MSY as another Kummer-like surface.There is a Nikulin involution ι ϕ,MSY : ( U, Y, Z ) ( − U, Y, Z )which induces a double covering ϕ MSY : K MSY S CMS . Summarizing this argument, we have thefollowing theorem.
Theorem 5.3.
The notation being as above, one has the Kummer-like sandwich for S MSY of (5.12) and S CMS of (5.13). K MSY ϕ
MSY S CMS ψ
CMS K MSY . (5.14)This explicit result (5.14) with clear expressions will help one to understand the results of [CMS] and[MSY] from the viewpoint of lattice polarized K Acknowledgement
The first author is supported by JSPS Grant-in-Aid for Scientific Research (18K13383) and MEXTLEADER. The second author is supported by JSPS Grant-in-Aid for Scientific Research (19K03396).The authors are thankful to Professor Jiro Sekiguchi for valuable discussions about the surface Σ( t ) inSection 4 from the view point of complex reflection groups. References [B] S. M. Belcastro,
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