$ L $-Functions of Higher Dimensional Calabi-Yau Manifolds of Borcea

L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE

DOMINIK BUREK

Abstract. We construct a series of examples of Calabi-Yau manifolds in arbitrary dimension and compute the main invariants. In particular, we give higher dimensional generalisation of Borcea-Voisin Calabi-Yau threefolds. We compute Hodge numbers of constructed examples using orbifold Chen-Ruan cohomology. We also give a method to compute a local zeta function using the Frobenius morphism for orbifold cohomology introduced by Rose.

1. Introduction The first examples of mirror Calabi-Yau threefolds which are neither toric or complete intersection were constructed by C. Borcea ([Bor97]) and C. Voisin ([Voi93]). This construction involves a non- symplectic involutions γS : S → S, αE : E → E of a K3 surface S and an elliptic curve E. The quotient (S ×E)/(γS ×αE) has a resolution of singularities which is a Calabi-Yau threefold, the mirror Calabi-Yau threefold is given by the same construction using the mirror K3 surface. C. Vafa and E. Witten studied similar constructions in [VW95] as a resolution of singularities of (E × E × E)/(Z2 ⊕ Z2) and (E × E × E)/(Z3 ⊕ Z3), where E is an elliptic curve and Z2 (resp. Z3) acts as involution (resp. automorphism of order 3). This approach leads to abstract physical models studied by L. Dixon, J. Harvey, C. Vafa, E. Witten in [Dix+85; Dix+86]. More generally, quotients of products of tori by a finite group were classified by J. Dillies, R. Donagi, A. E. Faraggi an K. Wendland in [Dil07; DF04; DW09]. In this paper we shall study Calabi-Yau manifolds constructed as a resolution of singularities of n−1 (X1 × X2 × ... × Xn)/Zd ,

where Xi denotes a variety of Calabi-Yau type with purely non-symplectic automorphism φi,d : Xi → Xi k of order d, where d = 2, 3, 4 or 6. If Fix(φi,d) for k | d and i = 2, . . . , n is a smooth divisor and φ1,d satisﬁes assumption Ad then there exists a crepant resolution Xd,n which is a Calabi-Yau manifold. In the special case when Xi are elliptic curves we obtain the construction given in [CH07], [Bur20]. We shall give a formula for Hodge numbers of Xd,n using Chen-Ruan orbifold cohomology theory ([CR04]). In [Bur20] we have computed Hodge numbers in the special case of Xi = Ed by a careful study k of the action of φi,d on Fix(φi,d) for k | d and i = 1, . . . , n. This approach does not generalize to the current setup, in the present paper we use characteristic polynomials of the eigenspaces of the action of φi,d on the cohomology groups of Xi. We shall study in more details the case when X1 is a K3 surface and X2,X3,...,Xn are elliptic curves. In the case of d = 2 this is a higher dimensional generalization of Borcea-Voisin construction, for d = 3, 4, 6 we get a higher dimensional generalization of the construction given by Cattaneo and Garbagnati in [CG16]. arXiv:2107.04104v1 [math.AG] 8 Jul 2021 Comparing the Euler characteristics of constructed manifolds computed from formulas for Hodge numbers with the stringy Euler number we obtained new relations among invariants of K3 surfaces with purely automorphism of order 3, 4 and 6. We also propose a method to compute the local zeta function of Calabi-Yau manifolds. As the zeta function is of arithmetic nature we do not get a general formula, but we shall give a method to compute the zeta function in explicit examples. Computations of the zeta function uses description of the Frobenius action on the orbifold cohomology ([Ros07]). Acknowledgments. This paper is a part of author’s PhD thesis. I am deeply grateful to my advisor Slawomir Cynk for his enormous help. The author is supported by the National Science Center of Poland grant no. 2019/33/N/ST1/01502.

2010 Mathematics Subject Classiﬁcation. Primary 14J32; Secondary 14J40, 14E15. Key words and phrases. Calabi–Yau manifolds, Borcea-Voisin construction, crepant resolution, Chen-Ruan cohomology. 1 2 D. BUREK

2. Kummer type Calabi-Yau manifolds 2.1. Borcea-Voisin construction. The ﬁrst example of family of Calaby-Yau threefolds which is sym- metric with respect to Mirror Symmetry was constructed independently by C. Borcea ([Bor97]) and C. Voisin ([Voi93]). Their construction involves an elliptic curve E and K3 surface S with non-symplectic involutions αE : E → E, γS : S → S. They also computed Hodge numbers of constructed Calabi-Yau threefold.

2.1. Theorem ([Bor97; Voi93]). The quotient (S ×E)/(γS ×αE) has a crepant resolution of singularities X which is a Calabi-Yau threefold. Moreover h1,1(X) = 11 + 5N − N 0 and h2,1(X) = 11 + 5N 0 − N, 0 where N is a number of curves in Fix(γS) and N is a sum of their genera. The classiﬁcation of K3 surfaces with non-symplectic involution was given by Nikulin ([Nik87]). From Nikulin’s classiﬁcation it follows that a mirror of Borcea-Voisin Calabi-Yau threefold is a Borcea-Voisin Calabi-Yau threefold associated to the mirror K3 surfaces S. In [CG16] A. Cattaneo and A. Garbagnati generalized the Borcea-Voisin construction allowing a non-symplectic¬ automorphisms of a K3 surfaces of higher degrees i.e. 3, 4 and 6.

2.2. Theorem ([CG16]). Let Sd be a K3 surface admitting a purely non-symplectic automorphism γd of order d = 3, 4, 6. Let Ed be an elliptic curve admitting an automorphism αd of order d. Then the n−1 quotient (Sd × Ed)/(γd × αd ) is a singular variety which admits a crepant resolution of singularities (S × E )/(γ × αd−1). In particular (S × E )/(γ × αd−1) is a Calabi-Yau threefold. ¬d d d d ¬d d d d The authors gave a detailed crepant resolution and computed the Hodge numbers of the resulting algebraic varieties. For all possible orders they computed the Hodge numbers of these varieties and constructed elliptic ﬁbrations on them, in [Bur18] we gave much simpler derivations of formulas for Hodge numbers using Chen-Ruan cohomology 3.1. 2.2. Generalisation. Let X be a complex smooth projective manifold with trivial canonical bundle (we shall call them of Calabi-Yau type) with a purely non-symplectic automorphism η : X → X of order d i.e. satisfying ∗ η (ωX ) = ζdωX , n,0 where ωX ∈ H (X) denotes a non-zero canonical form and ζd is the ﬁxed d-th root of unity. Consider the following assumptions: Condition A3 (1) Fix(η) is a disjoint union of submanifolds of codimension at most 2. In particular η has linearisation of the form 2 • (ζ3 , 1, 1,..., 1) near a component of codimension one of Fix(η), • (ζ3, ζ3, 1, 1,..., 1) near a component of codimension two of Fix(η). Condition A4 (1) Fix(η) is a disjoint union of submanifolds of codimension at most 3. In particular η has linearisation of the form 3 • (ζ4 , 1, 1,..., 1) near a component of codimension one of Fix(η), 2 • (ζ4, ζ4 , 1, 1,..., 1) near a component of codimension two of Fix(η), • (ζ4, ζ4, ζ4, 1, 1,..., 1) near a component of codimension three of Fix(η). Condition A6 (1) Fix(η) is a disjoint union of submanifolds of codimension at most 3. In particular η has linearisation of the form 5 • (ζ6 , 1, 1,..., 1) near a component of codimension one of Fix(η), 4 3 2 • (ζ6 , ζ6, 1, 1,..., 1) or (ζ6 , ζ6 , 1, 1,..., 1) near a component of codimension two of Fix(η), (2) Fix(η2) \ Fix(η) is a disjoint union of smooth submanifolds of codimension at most 2, so η2 has 2 2 linearisation of the form (ζ3 , 1, 1,..., 1) or (ζ3, ζ3, 1, 1,..., 1) along any component of Fix(η ) \ Fix(η), (3) Fix(η3) \ Fix(η) is a disjoint union of smooth divisors, so η3 has linearisation of the form (−1, 1, 1,..., 1) along any component of Fix(η3) \ Fix(η), 2 2 (4) the automorphism η has a local linearisation of the form (ζ6 , ζ6 , ζ6, 1, 1,..., 1) along any codi- mensional 3 component of Fix(η). We have the following: L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 3

2.3. Proposition ([Bur20], [CH07]). Let X1 and X2 be projective manifolds of Calabi-Yau type and a purely non-symplectic automorphisms ηi : Xi → Xi of order d, where d ∈ {2, 3, 4, 6}. Assume moreover, k that automorphism η1 satisﬁes condition Ad and Fix(η2 ) for k | d is a smooth divisor in X2. Then the d−1 quotient (X1×X2)/(η1×η2 ) admits a crepant resolution of singularities X. Moreover the automorphism id ×η2 lifts to a purely non-symplectic automorphism of X satisfying Ad.

Let X1,X2,...,Xn be projective manifolds of Calabi-Yau type and let φi,d : Xi → Xi be a purely non-symplectic automorphism of order d, where d ∈ {2, 3, 4, 6}. Consider the following group n n−1 Gd,n := {(m1, m2, . . . , mn) ∈ Zd : m1 + m2 + ... + mn = 0}' Zd mi which acts symplecticly on X1 × X2 × ... × Xn by φi,d on the i-th factor. Using the above Proposition 2.3 we can prove by an easy induction the following theorem:

2.4. Theorem. Let X1,X2,...,Xn be projective manifolds of Calabi-Yau type and a purely non- symplectic automorphism φi,d : Xi → Xi of order d, where d ∈ {2, 3, 4, 6}. Assume moreover, that auto- k morphism φ1,d satisﬁes condition Ad and Fix(φi,d) for k | d is a smooth for i = 2, 3, . . . , n. Then the quotient (X1 × X2 × ... × Xn)/Gd,n

admits a crepant resolution Xd,n. Moreover the automorphism φ1,d × id lifts to a purely non-symplectic automorphism of Xd,n.

The manifold Xd,n has a trivial canonical bundle and hence it is a Calabi-Yau manifold if the outer Hodge numbers vanishes i,0 H (Xd,n) = 0, 0 < i < dim Xd,n.

As we shall in Corollary 3.7 this conditions immediately satisfy when X1,X2,...,Xn are Calabi-Yau manifolds. In [Bur20], [CH07] the special case of the above theorem, when Xi = Ed was considered. 2.5. Theorem ([CH07], [Bur20]). If d = 2, 3, 4, 6 then there exists a crepant resolution

n n E /Gd,n → E /Gd,n. ¬d d n Consequently, Xd,n := E /Gd,n is an n-dimensional Calabi-Yau manifold. ¬d In [Bur20] we have computed Hodge numbers of the resulting varieties by using orbifold formula 3.1 k and a careful study of the action of φi,d on Fix(φi,d). In the present paper we shall propose new approach which is much more general and allows for new applications. It is easy to see that if X1 equals Sd – a K3 surface with purely non-symplectic automorphism of order d, then the condition Ad is automatically satisﬁed. Therefore as a special case of Theorem 2.4 we have the following:

2.6. Theorem. Let Ed be an elliptic curve with automorphism of order d and Sd be a K3 surface with n−1 purely non-symplectic automorphism of order d ∈ {2, 3, 4, 6}. Then the quotient variety (Sd ×Ed )/Gd,n admits crepant resolution of singularities Yd,n. Consequently Yd,n is (n + 1)-dimensional Calabi-Yau variety.

3. Hodge numbers of a finite quotient

In [Bur20] we determined Hodge numbers of varieties Xd,n based on Chen-Ruan cohomology and systematic study of orbits of the action of Gd,n. The method is very complex and we were not able to generalize it to the case of Calabi-Yau manifold other than elliptic curves. 3.1. Chen-Ruan cohomology. In [CR04] W. Chen and Y. Ruan introduced new cohomology theory for orbifold. Let X be a projective variety and G be a finite group which acts on X. 3.1. Definition. For a variety X/G define the Chen-Ruan cohomology by  C(g) i,j M M i−age(g), j−age(g) (3.1) Horb(X/G) :=  H (U) , [g]∈Conj(G) U∈Λ(g) where Conj(G) is the set of conjugacy classes of G (we choose a representative g of each conjugacy class), C(g) is the centralizer of g, Λ(g) denotes the set of irreducible connected components of the set fixed by g ∈ G and age(g) is the age of the matrix of linearised action of g near a point of U. i,j i,j The dimension of Horb(X/G) will be denoted by horb(X/G). 4 D. BUREK

3.2. Remark. The deﬁnition makes sense since age is locally constant along each component of Fix(g). The components of Fix(g) are not necessarily invariant under the action of C(g), so in 3.1 we need to consider the action of C(g) on the inner direct sum (not on each summand separately). An important feature of Chen-Ruan cohomology is the possibility of computing Hodge numbers of a crepant resolution of singularities of a quotient variety, without referring to an explicit construction of such a resolution. 3.3. Theorem ([Yas04]). Let G be a ﬁnite group acting on an algebraic smooth variety X. If there exists a crepant resolution X/G of variety X/G, then the following equality holds ¬ hi,j(X/G) = hi,j (X/G) . ¬ orb For a systematic exposition of the orbifold Chen-Ruan cohomology see [ALR07].

3.2. Hodge numbers of Xd,n. Let Xi be a variety of Calabi-Yau type of dimension ni with purely non-symplectic automorphism φi,d : Xi → Xi of order d. Suppose that there exists a crepant resolution Xd,n of the quotient variety n−1 (X1 × X2 × ... × Xn)/Zd ,

then Xd,n is of Calabi-Yau type. The aim of this section is to give a formula for the Hodge number of Xd,n using Chen-Ruan cohomology. We shall use the following obvious lemma:

3.4. Lemma. Let V1,V2,...,Vn be vector spaces over ﬁeld of k containing ζd. Assume that αi ∈ End(Vi) for i = 1, 2, . . . n is an automorphism of order d. Then the ﬁxed locus of Gd,n acting on V1 × V2 × ... × Vn by

g1 gn αg : Gd,n 3 g 7→ α1 × ... × αn ∈ End(V1 × V2 × ... × Vn) equals d−1 M (V1) m ⊗ ... ⊗ (Vn) m , ζd ζd m=0

where (Vi) m denotes m-th eigenspace of d action on Vi, for i ∈ {1, 2, . . . , n} and m ∈ {0, 1, . . . , d − 1}. ζd Z

Proof. Since the action of Gd,n is diagonalizable we can consider only the tensor product of eigenvectors mi v1 ⊗ ... ⊗ vn where αi(vi) = ζd vi. If mi 6= mj for some 1 ≤ i < j ≤ n then consider an element gij = 0,..., 1 , . . . d − 1 ,... 1 . |{z} | {z } i−th place j−th place Since we have mi+(d−1)mj αgij (v1 ⊗ ... ⊗ vn) = ζd v1 ⊗ ... ⊗ vn the vector v1 ⊗ ... ⊗ vn is ﬁxed by αgij iﬀ d | mi + (d − 1)mj which is equivalent to mi = mj.

mi Observe ﬁrst that for i ∈ {1, 2, . . . , n}, 0 ≤ mi ≤ d−1, the automorphism φi,d has local diagonalization mi near a point of Fix φi,d of the following form

α1mi αni mi ζd , . . . , ζd ,

α1 αni where ζd · ... · ζd = ζd. Consequently m age φmi = i + λ , i,d d i

where λi is a non-negative integer, moreover λi is constant along every component of Fix(αi). 3.5. Deﬁnition. Deﬁne n m o X = x ∈ Fix φmi : age φmi = i + λ near x i,mi,λi i,d i,d d i and X X p,q p+λi q+λi j FXi,mi,j(X,Y ) = dim H (Xi,mi,λi ) X Y . C ζd λi≥0 0≤p,q≤dim Xi L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 5

3.6. Theorem. Under the above assumptions d−1 n d−1 ! p,q X Y X pd m p q (3.2) h (Xd,n) = (XY ) · FXi,m,j(X,Y ) [X Y ], j=0 i=1 m=0 where P[XpY q] denotes the coeﬃcients of polynomial P in XpY q. Proof. By the Yasuda Theorem 3.3 p,q p,q n−1 H (Xd,n) = Horb (X1 × X2 × ... × Xn)/Zd . Now by the K¨unnethformula for Hodge groups it follows that

!Gn,d p,q X X X p1,q1 pn,qn H (Xd,n) = H (X1,m1,λ1 ) ⊗ ... ⊗ H (Xn,mn,λn ) , n n m∈Gn,d λ∈N p,q∈N 1 |p|=p− d |p|−|λ| 1 |q|=q− d |q|−|λ|

where λ = (λ1, λ2, . . . , λn), p = (p1, p2, . . . , pn), q = (q1, q2, . . . , qn) and |λ| = λ1 + λ2 + ... + λn, |p| = p1 + p2 + ... + pn, |q| = q1 + q2 + ... + qn.

Since φd,n (Xi,mi,λi ) = Xi,mi,λi , the group Gn,d acts separately on each summand of the inner sum, hence by the lemma 3.4 we get d p,q X X X X p1,q1 pn,qn j j H (Xd,n) = H (X1,m1,λ1 ) ⊗ ... ⊗ H (Xn,mn,λn ) . ζd ζd n n m∈Gn,d λ∈N p,q∈N j=0 1 |p|=p− d |p|−|λ| 1 |q|=q− d |q|−|λ| Taking dimensions and forming the generating function we get d X p,q p q X X X X X p1−λ1,q1−λ1 j h (Xd,n) X Y = h (X1,m1,λ1 ) · ... · orb ζd n n 0≤p,q≤dim Xd,n 0≤p,q≤dim Xd,n m∈Gn,d λ∈N j=0 p,q∈N |p|=p |q|=q

m1+...+mn pn−λn,qn−λn p q j d · h (Xn,mn,λn ) X Y · (XY ) = ζd d   m1 X X X X p1−λ1,q1−λ1 p1 q1 j d =  h (X1,m1,λ1 ) X Y · (XY )  × ... × ζd n m∈Gn,d λ∈N j=0 p1,q1≥0   mn X pn−λn,qn−λn pn qn j d ×  h (Xn,mn,λn ) X Y · (XY )  = ζd pn,qn≥0

d−1   m1 X X X X p1,q1 p1 q1 λ1+ j d =  h (X1,m1,λ1 ) X Y  (XY ) · ... · ζd n m∈Gn,d λ∈N j=0 p1,q1≥0   mn X pn,qn pn qn λn+ j d ·  h (Xn,mn,λn ) X Y  (XY ) . ζd pn,qn≥0 X X Enlarging the exterior sum (...) to (...) we introduce only terms with fractional powers of n m∈Gn,d m∈Zd p,q p q X and Y , hence h (Xd,n) is the coeﬃcient in X Y of the following Puiseux polynomial:

d−1   m1 X X X X p1,q1 p1 q1 λ1+ j d  h (X1,m1,λ1 ) X Y  (XY ) · ... · ζd n n m∈Zd λ∈N j=0 p1,q1≥0   mn X pn,qn pn qn λn+ j d ·  h (Xn,mn,λn ) X Y  (XY ) = ζd pn,qn≥0 d−1 n d−1 X Y X pd m = (XY ) · FXi,mi,j(X,Y ). j=0 i=1 m=0 6 D. BUREK

Therefore, in order to compute Poincar´epolynomial of Xd,n for any 1 ≤ i ≤ n it is enough to produce the following tables H j HH 0 1 ··· j ··· d − 1 k HH

0 FXi,0,0 FXi,0,1 FXi,0,j FXi,0,d−1

1 FXi,1,0 FXi,1,1 FXi,1,j FXi,1,d−1

2 FXi,2,0 FXi,2,1 ··· FXi,2,j ··· FXi,2,d−1

......

d − 1 FXi,d−1,0 FXi,d−1,1 FXi,d−1,j FXi,d−1,d−1 = vXi,j

Table 1. FXi,j,k

for any considered variety Xi. Then we compute scalar product of vectors vXi,j and q pd pd 2 d d−1 vd := 1, (XY ), (XY ) ,..., (XY )

for 1 ≤ j ≤ n. Finally we multiply all values of vXi,j ◦ vd for j ∈ {1, 2, . . . , n} and add all products for i ∈ {1, 2, . . . , n}. From the above Theorem 3.6 it is particularly easy to compute the outer Hodge numbers i,0 H (Xd,n) = 0, 0 < i < dim Xd,n. However in this case it follows directly from orbifold formula 3.1 and K¨unneth’s formula for Hodge groups we have

G i,0 i,0 Gd,n M i1,0 in,0 d,n H (Xd,n) = H (X1 × X2 × ... × Xn) = H (X1) ⊗ ... ⊗ H (Xn) =

i1+...+in=i n M i1,0 in,0 Zd M i1,0 d in,0 d = H (X1) ⊗ ... ⊗ H (Xn) = H (X1)Z ⊗ ... ⊗ H (Xn)Z .

i1+...+in=i i1+...+in=i Consequently we get the following Corollary:

3.7. Corollary. If X1,X2,...,Xn are Calabi-Yau manifolds then Xd,n is also a Calabi-Yau manifold.

3.3. Hodge numbers of Xd,n. In the following section we apply theorem 3.6 to the case Xi = Ed, reproving formulas from [Bur20]. 3.3.1. d = 6. The explicit terms of Table 1 are given in Table 2.

H j HH 0 1 2 3 4 5 k HH 0 1 + XY X 0 0 0 Y

1 1 0 0 0 0 0

2 2 0 0 1 0 0

3 2 0 1 0 1 0

4 2 0 0 1 0 0

5 1 0 0 0 0 0

Therefore by Theorem 3.6: Table 2. FE6,k,j(X,Y )

p,q n h (E6 /G6,n) = (¬ √ n = (1 + XY ) · p6 (XY )0 + 1 · 6 XY + 2 · p6 (XY )2 + 2 · p6 (XY )3 + 2 · p6 (XY )4 + 1 · p6 (XY )5 + L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 7

√ n + X · p6 (XY )0 + 0 · 6 XY + 0 · p6 (XY )2 + 0 · p6 (XY )3 + 0 · p6 (XY )4 + 0 · p6 (XY )5 + √ n + 0 · p6 (XY )0 + 0 · 6 XY + 0 · p6 (XY )2 + 1 · p6 (XY )3 + 0 · p6 (XY )4 + 0 · p6 (XY )5 + √ n + 0 · p6 (XY )0 + 0 · 6 XY + 1 · p6 (XY )2 + 0 · p6 (XY )3 + 1 · p6 (XY )4 + 0 · p6 (XY )5 + √ n + 0 · p6 (XY )0 + 0 · 6 XY + 0 · p6 (XY )2 + 1 · p6 (XY )3 + 0 · p6 (XY )4 + 0 · p6 (XY )5 + ) √ n + Y · p6 (XY )0 + 0 · 6 XY + 0 · p6 (XY )2 + 0 · p6 (XY )3 + 0 · p6 (XY )4 + 0 · p6 (XY )5 [XpY q] =

( √ n = Xn + Y n + 1 + XY + 6 XY + 2p6 (XY )2 + 2p6 (XY )3 + 2p6 (XY )4 + p6 (XY )5 +

n ) n p6 p6 p q + 2 · (XY ) 2 + (XY )2 + (XY )4 [X Y ].

3.3.2. d = 4. The explicit terms of Table 1 are given in Table 3.

HH j H 0 1 2 3 k HH 0 1 + XY X 0 Y

1 2 0 0 0

2 3 0 1 0

3 2 0 0 0

Table 3. FE4,k,j(X,Y ) Therefore by Theorem 3.6: ( ) √ n n p,q n n n 4 p4 2 p4 3 p4 2 p q h (E4 /G4,n) = X + Y + 1 + XY + 2 XY + 3 (XY ) + 2 (XY ) + (XY ) [X Y ]. ¬

3.3.3. d = 3. The explicit terms of Table 1 are given in Table 4.

H j HH 0 1 2 k HH 0 1 + XY X Y

1 3 0 0

2 3 0 0

Table 4. FE3,k,j(X,Y ) Therefore by Theorem 3.6:

n √ no p,q n n n 3 p3 2 p q h (E3 /G3,n) = X + Y + 1 + XY + 3 XY + 3 (XY ) [X Y ] = ¬ √ 3n = Xn + Y n + 1 + 3 XY [XpY q].

3.3.4. d = 2. The explicit terms of Table 1 are given in Table 5. 8 D. BUREK

H j HH 0 1 k HH 0 1 + XY X + Y

1 4 0

Table 5. FE2,k,j(X,Y ) Therefore by Theorem 3.6:

√ n p,q n n n o p q h (E2 /G2,n) = (X + Y ) + 1 + XY + 4 XY [X Y ]. ¬

3.4. Hodge numbers of Yd,n. In the following section we shall derive formulas for Hodge numbers of manifolds Yd,n. As in the previous section we shall separately treat cases 2, 3, 4 and 6.

k 3.4.1. Y6,n. In the formulas for Hodge numbers of Y6,n several invariants appear which describe Fix(γd ) and were introduced in [CG16]:

S6 – K3 surfaces with a non-symplectic automorphism γ6 : S6 → S6 of order 6, 2 3 2 E6 – elliptic curve with the Weierstrass equation y = x + 1, and automorphism α6(x, y) = (ζ6 x, −y), 2 where ζ6 denotes a ﬁxed 6-th root of unity satisfying ζ6 = ζ3, 2 γ6 r = dim H (S6, C) , 2 m = dim H (S6, ) i for i ∈ {1, 5}, C ζ6 2 α = dim H (S6, ) i for i ∈ {2, 4}, C ζ6 2 β = dim H (S , ) 3 , 6 C ζ6 3 2 3 Fix(α6) = Fix(α6) = {f1}, Fix(α6) = {f1, f2, f3}, where α6(f2) = f3, α6(f3) = f2 and Fix(α6) = {f1, f4, f5, f6}, where α6(f4) = f5, α6(f5) = f6 and α6(f6) = f4,

Fix (γ6) = {K1,...K`−1} ∪ {D} ∪ {P1,...,Pp(2,5) } ∪ {Q1,...,Qp(3,4) }, where – the set {K1,...K`−1}∪{D} consists of curves which are ﬁxed by γ6 together with the curve D of maximal genus g(D), in fact Ki are rational,

– {P1,...,Pp(2,5) } is the set of points such that linearisation of γ6 near the ﬁxed point is 2 5 represented by the diagonal matrix diag(ζ6 , ζ6 ),

– {Q1,...,Qp(3,4) } is the set of points such that linearisation of γ6 near the ﬁxed point is 3 4 represented by the diagonal matrix diag(ζ6 , ζ6 ), 2 0 0 0 0 Fix γ6 = {L1,...,Lk−2b−1} ∪ {G} ∪ {(A1,A1),..., (Ab,Ab)} ∪ {(R1,R1),..., (Rn0 ,Rn0 )} ∪ {P1,...,

Pp(2,5) }, where 2 – the set {L1,...Lk−2b−1} ∪ {G} consists of curves which are fixed by γ6 together with the curve G of maximal genus g(G), in fact Li are rational, 0 0 0 2 – {(A1,A1),..., (Ab,Ab)} is the set of all pairs (Ai,Ai) of curves which are fixed by γ6 and 0 γ6(Ai) = Ai (curves of the third type), 0 0 2 0 – {(R1,R1),..., (Rn0 ,Rn0 )} is the set of pairs of points fixed by γ6 and such that γ6(Ri) = Ri, 3 0 00 0 00 Fix γ6 = {(M1,M1,M1 ),..., (Ma,Ma,Ma )} ∪ {T1,...,TN−3a−2} ∪ {F1,F2}, where 0 00 0 00 3 – the set {(M1,M1,M1 ),..., (Ma,Ma,Ma )} consists of curves which are fixed by γ6 and such 0 0 00 00 that γ6(Mi) = Mi , γ6(Mi ) = Mi and γ6(Mi ) = Mi, 3 – the set {T1,...,TN−3a−2} ∪ {F1,F2} consists of curves fixed by γ6 (and so γ6 ), together with curves F1 and F2 of maximal genera g(F1) and g(F2), respectively.

The explicit terms of Table 1 are given in Table 6. L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 9 XY · 1) 5 0 0 0 0 0 − m + ( 2 Y − ) 6 /γ XY 1 · F ( a g − ) + ) 2 Y F XY 4 0 0 0 0 ( + · g X α ( ) + ) 1 6 F ( /γ g 2 F 1 2 ( g + − a · ) · 6 )) 6 G/γ ( g G/γ ( − g ) XY − G · ( ) XY 3 0 0 0 b XY g · G · ( β b g ) + + Y + ( ) + 0 + XY Y n · X 0 + ( + ) n · b X ( + · b X,Y ( ,k,j − 6 ) 6 S /γ F XY 1 · F ( a g − ) + ) 2 Y F XY 2 0 0 0 0 ( + · g X α ( Table 6. ) + ) 1 6 F ( /γ g 2 F 1 2 ( g + − a XY · 1) − 1 0 0 0 0 0 m + ( 2 X + )+ · Y )) + XY 6 + + /γ X 2 XY XY )( F · · 6 ( XY + 1 XY XY g ` · 4) , · · ) ) (3 a G/γ 5) b p , ( ) + XY ) + 2 6 g · (2 − Y + − p r 0 0 /γ + k + 1 + N + 5) F , XY 2 ( X ) ( · (2 g ) + ( · p 6 XY ) + ( 5) ) XY , · + ( + XY Y · 0 (2 D ( 0 a ) ( n G/γ p + n b 2 g + + + − − X + ( + b ` ( b k g · N − − + k + ( k j H H 0 1 2 3 4 5 H H k H 10 D. BUREK

From 3.2 and tables 6, 2 we get Poincar´epolynomial of Y6,n :

5 √ ! X 6 p6 2 p6 3 p6 4 p6 5 FS6,0,j + XYFS6,1,j + (XY ) FS6,2,j + (XY ) FS6,3,j + (XY ) FS6,4,j + (XY ) FS6,5,j × j=0 n−1 √ ! 6 p6 2 p6 3 p6 4 p6 5 × FE,0,j + XYFE6,1,j + (XY ) FE6,2,j + (XY ) FE6,3,j + (XY ) FE6,4,j + (XY ) FE6,5,j =

√ 2 6 = (XY ) + r · XY + 1 + XY · ` + p(2,5) · XY + p(3,4) · XY + g(D) · (X + Y ) + ` · XY + p6 2 0 S6 + (XY ) · k − b + n · XY + p(2,5) · XY + g G/φ6 (X + Y ) + (k − b) · XY + + p6 (XY )3 · N − 2a + g F1 + g F2 · (X + Y ) + (N − 2a) · XY + γ6 γ6 p6 4 0 + (XY ) · k − b + n + p(2,5) + g(G/γ6)(X + Y ) + (k − b) · XY + ! p6 5 + (XY ) · ` + p(2,5) + p(3,4) + g(D) · (X + Y ) + ` · XY ·

√ n−1 · 1 + XY + 6 XY + 2p6 (XY )2 + 2p6 (XY )3 + 2p6 (XY )4 + p6 (XY )5 +

1 + X2 + (m − 1) · XY · Xn−1 + α · XY + p6 (XY )3 · a + g(F ) + g(F )− 2 1 2 ! n−1 − g F1 − g F2 · (X + Y ) + a · XY · p6 (XY )3 + γ6 γ6

p6 2 0 + β · XY + (XY ) · b + n · XY + g(G) − g(G/γ6) · (X + Y ) + b · XY +

! n−1 p6 4 0 p6 2 p6 4 + (XY ) · b + n + g(G) − g(G/γ6) · (X + Y ) + b · XY · (XY ) + (XY ) +

1 p6 3 F1 F2 + α · XY + (XY ) · a + g(F1) + g(F2) − g − g · (X + Y )+ 2 γ6 γ6 ! n−1 + a · XY · p6 (XY )3 + Y 2 + (m − 1) · XY · Y n−1.

From this formula we can compute Euler characteristic by evaluating the above formula at 6-th roots of unity. Therefore we can compute this as a sum of six geometric sequences, hence we expect that there exists recurrence of degree at most 6. In fact two of them coincides and ﬁnally we obtain recurrence of order 4.

3.8. Corollary. The Euler characteristic of Y6,n is equal to an−1, where  an = 12an−1 − 19an−2 − 12an−3 + 20an−4,   a0 = e(Y6,1) = 24,   a = e(Y ) = 4 + 2r − 2m + 4l + 6p + 2p − 4g(D) + 8k − 4b + 6w − 4g(G/γ ) − 4g(G)+  1 6,2 (2,5) (3,4) 6   + 4N − 4a − 2g(F1/γ6) − 2g(F2/γ6) − 2g(F1) − 2g(F2),

a2 = e(Y6,3) = 80 + 16r − 2m − 2α + 64l + 66p(2,5) + 32p(3,4) − 64g(D) + 68k − 64b + 36w−    − 64g(G/γ6) − 4g(G) + 32N − 64a − 32g(F1/γ6) − 32g(F2/γ6),   a3 = e(Y6,4) = 380 + 166r − 6m − 4α + 660l + 666p(2,5) + 330p(3,4) − 660g(D) + 672k − 660b + 342w−   − 660g(G/γ ) − 12g(G) + 332N − 660a − 330g(F /γ ) − 330g(F /γ ) − 2g(F ) − 2g(F ).  6 1 6 2 6 1 2 L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 11

Therefore

1 e(Y ) = 46 − r + 2m − α + 2l + p − 2g(D) − 2k − 2b − 3w − 2g(G/γ ) + 4g(G) − 2N− 6,n 3 (3,4) 6 ! 1 r − 2a − g(F /γ ) − g(F /γ ) + 3g(F ) + 3g(F ) · (−1)n−1 − − 23 + + 2m + 2α+ 1 6 2 6 1 2 3 2 ! n−1 + 2l + p(3,4) − 2g(D) − 2k − 2b − 3w − 2g(G/γ6) + 4g(G) + N − 2a − g(F1/γ6) − g(F2/γ6) · 2 +

1 + 2 + r + 3α − 2l − 2p − p + 2g(D) − 2k + 2b − w + 2g(G/γ ) + 2N + 2a + g(F /γ )+ 3 (2,5) (3,4) 6 1 6 ! 1 r + g(F /γ ) − 3g(F ) − 3g(F ) − − − 2l − 2p − p + 2g(D) − 2k + 2b − w + 2g(G/γ )− 2 6 1 2 3 2 (2,5) (3,4) 6 ! − N + 2a + g(F1/γ6) + g(F2/γ6) − 1 .

3.4.2. Y4,n. Let us keep the following notation for invariants of K3 surface and elliptic curve given in [CG16]:

S4 – K3 surfaces with a non-symplectic automorphism γ4 : S4 → S4 of order 4 2 3 E4 – elliptic curve with the Weierstrass equation y = x + x, and automorphism α4 is given by α4(x, y) = (−x, iy),

2 γ4 r = dim H (S4, C) , 2 m = dim H (S4, ) i for i ∈ {1, 2}, C ζ4 2 α = dim H (S , ) 2 , 4 C ζ4 3 2 Fix(α4) = Fix(α4) = {f1, f2}, Fix(α4) = {f1, f2, f3, f4}, where α4(f3) = f4 and α4(f4) = f3, 2 0 0 0 Fix γ4 = L1 ∪ L2 ∪ ... ∪ LN−b−2a−1 ∪ {D} ∪ {(A1,A1), (A2,A2),..., (Aa,Aa)} ∪ {B1,B2,...,Bb}, where 0 0 0 0 – {(A1,A1), (A2,A2),..., (Aa,Aa)} is the set of all pairs (Ai,Ai) of curves which are ﬁxed by 2 0 γ4 and γ4(Ai) = Ai, 2 – {B1,B2,...,Bb} is the set of curves which are ﬁxed by γ4 and are non-invariant by γ4.

Fix (γ4) = {R1,R2,...Rk−1} ∪ {G} ∪ {P1,P2,...,Pn1 } ∪ {Q1,Q2,...,Qn2 }, where – the set {R1,R2,...Rk−1} ∪ {G} contains of curves which are ﬁxed by γ4 together with the curve G of maximal genus g(G), curves Ri are rational,

– {P1,P2,...,Pn1 } is the set of points which are ﬁxed by γ4 not laying on the curve D,

– {Q1,Q2,...,Qn2 } is the set of points which are ﬁxed by γ4 laying on the curve D. The corresponding table 1 has the form Table 7. From Theorem 3.6 and tables 7, 3 we get Poincar´epolynomial of Y4,n :

√ 2 4 (XY ) + r · XY + 1 + k + (n1 + n2) · XY + g(G) · (X + Y ) + k · XY · XY +

D p4 2 + N − a + g · (X + Y ) + (N − a) · XY · (XY ) + k + n1 + n2+ γ4 ! √ n−1 + g(G) · (X + Y ) + k · XY · p4 (XY )3 · 1 + XY + 2 4 XY + 3p4 (XY )3 + 2p4 (XY )3 +

+ X2 + (m − 1) · XY · Xn−1 + (22 − r − 2m) · XY + a + g(D) − g D (X + Y )+ γ4 n−1 + a · XY · p4 (XY )2 · p4 (XY )2 + Y 2 + (m − 1) · XY · Y n−1.

Similarly as before we obtain the following corollary 12 D. BUREK XY · 1) − 3 0 0 0 m + ( 2 Y XY · a ) + Y + XY · X ( ) m 2 2 0 0 4 γ − r D − g (22 − ) D ( g + ) a X,Y ( ,k,j 4 XY S · F 1) − 1 0 0 0 m + ( Table 7. 2 X XY XY · · k ) XY a · − k ) + Y N ) + + Y + 1 X ) + ( ( + · Y ) X XY + ( · G · ( r 0 ) X g ( + G · + ( 2 g ) 4 + γ XY 2 XY · ( n ) D 2 + n g 1 + n + 1 + a n k − + ( N k j H H 0 1 2 3 H H k H L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 13

3.9. Corollary. The Euler characteristic of Y4,n is equal to an−1, where  an = 9an−1 + an−2 − 9an−3,  a0 = e(Y4,1) = 24, a1 = e(Y4,2) = 2r + 8k + 4n1 + 4n2 + 6N − 4a + 4 − 14g(D) − 2m,  a2 = e(Y4,3) = 64 + 20r + 80k + 40n1 + 40n2 − 120g(D) + 40N − 40a. Therefore ! 1 r e(Y ) = − N + g(D) + m + 12 · (−1)n−1 − − N + a + 3g(D) − 2k − n − n − − 1 · 9n−1+ 4,n 2 1 2 2 ! 1 r + N + a + g(D) − 2k − 2m − n − n − + 23 . 2 1 2 2

3.4.3. Y3,n. Let us keep the following notation for invariants of K3 surface and elliptic curve given in [CG16]:

S3 – K3 surfaces with a non-symplectic automorphism γ3 : S3 → S3 of order 3, 2 3 E3 – elliptic curve with the Weierstrass equation y = x + 1, and automorphism α3 is given by α3(x, y) = (ζ3x, y),

2 γ3 r = dim H (S3, C) , 2 m = dim H (S3, C)ζ3 , 2 Fix (γ3) = Fix γ3 = {f1, f2, f3}, Fix (γ3) = L1 ∪ L2 ∪ ... ∪ Lk−1 ∪ C ∪ {P1,P2,...,Ph}, where – the set {L1,L2,...Lk−1} ∪ {C} consists of curves which are ﬁxed by γ3 together with the curve C of maximal genus g(C), in fact Li are rational,

– {P1,P2,...,Pn} is the set of points which are ﬁxed by γ3. The corresponding table 1 has the following form (Table 8):

HH j H 0 1 2 k HH 0 (XY )2 + r · XY + 1 X2 + (m − 1) · XY Y 2 + (m − 1) · XY

1 k + h · XY + g(C) · (X + Y ) + k · XY 0 0

2 k + h + g(C) · (X + Y ) + k · XY 0 0

Table 8. FS3,k,j(X,Y )

From Theorem 3.6 and tables 8, 4 we get Poincar´epolynomial of Y3,n :

√ (XY )2 + r · XY + 1 + k + h · XY + g(C) · (X + Y ) + k · XY · 3 XY + k + h+

! √ n−1 + g(C) · (X + Y ) + k · XY · p3 (XY )2 · 1 + XY + 3 3 XY + 3p3 (XY )2 +

+ X2 + (m − 1) · XY · Xn−1 + Y 2 + (m − 1) · XY · Y n−1.

Similarly as before we obtain the following corollary

3.10. Corollary. The Euler characteristic of Y3,n is equal to an−1, where  a = 7a + 8a ,  n n−1 n−2 a0 = e(Y3,1) = 24,  a1 = e(Y3,2) = 2r + 6h + 12k + 4 − 12g(C) − 2m. 14 D. BUREK

Therefore 1 1 e(Y ) = 188 − 2r − 6h − 12k + 12g(C) + 2m · (−1)n−1 + 28 + 2r + 6h + 12k − 12g(C) − 2m · 8n−1. 3,n 9 9

3.4.4. Y2,n. For a K3 surface S2 with involution we have the following invariants:

S2 – K3 surfaces with a non-symplectic automorphism γ2 : S2 → S2 involution,

E2 – arbitrary elliptic curve with involution α2(x, y) = (x, −y),

2 γ2 r = dim H (S2, C) , m = dim H2(S , ) , 2 C ζ2 Fix (α2) = {a, b, c, d},

Fix (γ2) = C1 ∪ C2 ∪ ... ∪ CN where the set {C1,C2,...,CN } contains of curves which are ﬁxed by γ2 with sum of genera equals N 0. The corresponding table 1 has the following form (Table 9):

H j HH 0 1 k HH 0 (XY )2 + r · XY + 1 X2 + Y 2 + (m − 2) · XY

1 N + N 0 · (X + Y ) + N · XY 0

Table 9. FS2,k,j(X,Y )

From Theorem 3.6 and tables 9, 5 we get Poincar´epolynomial of Y2,n : ! √ √ n−1 (XY )2 + r · XY + 1 + N + N 0 · (X + Y ) + N · XY · XY · 1 + XY + 4 XY +

+ X2 + Y 2 + (m − 2) · XY · (X + Y )n−1.

Using the following relations (see [Voi93]): r = 10 + N − N 0 and m = 12 − N + N 0 we can rewrite the above formula in terms of N and N 0, i.e. ! √ (XY )2 + (10 + N − N 0) · XY + 1 + N + N 0 · (X + Y ) + N · XY · XY ×

√ n−1 × 1 + XY + 4 XY + X2 + Y 2 + (10 − N + N 0) · XY · (X + Y )n−1.

4. Relations among invariants of S6 We shall use orbifold Euler characteristic, topological and the holomorphic Lefchetz numbers in order to ﬁnd relations between parameters attached to K3 surfaces.

4.1. Stringy Euler Characteristic. One of the most important cohomological invariant of the finite quotient of compact manifolds conjectured by “physicists” is physicists (stringy) Euler characteristic. Let G be a finite group acting on a compact, smooth differentiable manifold X. For any g ∈ G let Xg := {x ∈ X : g(x) = x} . In [Dix+85; Dix+86] L. Dixon, J. Harvey, C. Vafa and E. Witten proposed the following orbifold Euler number: 1 X (4.1) e (X/G) := e(Xg ∩ Xh). orb #G (g,h)∈G×G gh=hg

It is expected that orbifold Euler characteristic eorb (X/G) coincides with the topological Euler characteristic e(X/G) of any crepant resolution of X/G. ¬ L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 15

4.1. Theorem ([Roa89] (abelian case), [Bat99]). Let G ⊂ SLn(C) be a ﬁnite group acting on smooth algebraic variety X. If there exists a crepant resolution X/G of variety X/G, then the following equality holds ¬

e(X/G) = eorb (X/G) . ¬ Some explicit examples of a group G and possible calculations were studied by F. Hirzebruch and T. H¨oferin [HH90].

4.2. Topological and holomorphic Lefchetz’s numbers. In the following section we review basic information about topological and holomorphic Lefschetz numbers. We refer to [GH94] and [Pet86]. Let X be a compact oriented manifold and f : X → X any continuous map. The intersection number of the graph Γf of f and diagonal ∆X in X × X is equal to the so called topological Lefschetz number given by the following formula:

X q ∗ q q Ltop(f) := (−1) tr (f : H (X) → H (X)) . q≥0

Assuming X is a complex manifold and f a holomorphic map then intersection Γf · ∆X may be computed in the case when connected components C of the Fix(f) are non-degenerate i.e. C is a manifold and for any P ∈ C the linear map (id − dfP ) is invertible. 4.2. Theorem ([Uen76]). Let X be a compact complex manifold and f : X → X a holomorphic map with non-degenerate ﬁxed locus then X (4.2) Ltop(f) = e(C), C∈Fix(f) where the sum is taken over connected components of Fix(f).

Formula 4.2 holds in particular when f n = id for some n ≥ 1 (see [Car57]). The map f acts not only on the de Rham cohomology of X but on the Dolbeault cohomology too. Therefore there is a hope that the action of f on H∗∗(X) will be reﬂected in local properties of f around the ﬁxed point locus.

4.3. Deﬁnition. Let X be a compact complex manifold and f : X → X a holomorphic map. The number

X q ∗ 0,q Lhol(f) := (−1) tr f |H (X) q≥0 is called holomorphic Lefschetz number.

According to [AS68b] and [AS68a] the holomorphic Lefschetz number can be computed in diﬀerent way. For our purpose assume that X has dimension 2 and let G be a ﬁnite group of automorphisms of X.

4.4. Theorem ([AS68b], [AS68a]). For any g ∈ G the following formula holds X X Lhol(g) = a(Pj) + b(Ck), j∈J k∈K where sets {Pj}j∈J and {Ck}k∈K denote ﬁxed points and ﬁxed curves in Fix(g) and 1 1 − g(C) ζC2 a(P ) := and b(C) := − 2 det(1 − g|Tg) 1 − ζ (1 − ζ) where TP is a tangent space at P and ζ is an eigenvalue of g on the normal bundle of Fix(g).

4.3. Invariants of S6. There are many relations among numerical invariants of S6. Some of them were pointed out in [CG16] and [Bur18]. Most of them follow from adopted notation 3.4.1 and Riemann- Hurwitz formula (see [CG16]). In [Bur18] we got new relation by comparing Stringy Euler Characteristic of Y6,2 with the Euler characteristic computed from orbifold Hodge numbers. In the present section we shall use the same idea as in [Bur18], moreover we shall examine another formulas i.e. Hurwitz formula and holomorphic and topological Lefschetz numbers. 16 D. BUREK

2 4.3.1. Riemann-Hurwitz formula. Some isolated ﬁxed points of γ6 lie on curves in Fix γ6 and all of G them are of type (3, 4). The canonical map πG : G → can be considered as a covering of G of degree γ6 two ramiﬁed at

p(3,4) − 2 · (k − 2b − ` − 1) | {z } number of rational curves invariant by γ6 points lying on G, which have ramiﬁcation index 2, therefore by the Reimann-Hurwitz formula we get: G 2 − 2g(G) = 2 2 − 2g − p(3,4) − 2(k − 2b − ` − 1) , φ6 thus G 1 (4.3) g = 2g(G) − p(3,4) + 2k − 4b − 2` . φ6 4 F Assuming g(D) = 0. Let F := F1 ∪ F2. The canonical map πF : F → is triple covering of F , γ6 ramiﬁed at

p(3,4) + p(2,5) − 2 · (N − 3a − 1 − `) | {z } number of rational curves invariant by γ6 points, with ramiﬁcation index equal to 3, so F1 F2 2 − 2g(F1) + 2 − 2g(F2) = 3 2 − 2g + 2 − 2g − γ6 γ6

− 2(p(3,4) + p(2,5) − 2(N − 3a − 1 − `)). Consequently 1 F1 F2 (4.4) g + g = 2g(F1) + 2g(F2) − 2p(2,5) − 2p(3,4) + 4N − 12a − 4l . γ6 γ6 6

4.3.2. Topological Lefschetz number. Using 4.2 we deduce:

2 3 4 5 Ltop (γ6) = 1 + 1 + r + ζ6 · (1 + m − 1) + ζ6 · α + ζ6 · β + ζ6 · α + ζ6 · (m − 1 + 1) = 2 + r + m − α − β, therefore

2 + r + m − α − β = e (Fix (γ6)) = 2` − 2g(D) + p(3,4) + p(2,5), which gives relation

(4.5) 2 + r + m − α − β − 2` + 2g(D) − p(3,4) − p(2,5) = 0. And similarly 2 2 Ltop γ6 = (1 + r + β + 1) + ζ3(1 + m − 1 + α) + ζ3 (α + m − 1 + 1) = = (1 + r + β + 1) − 1 − m − α + 1 = −α + β + r + 2 − m, so 2 −α + β + r + 2 − m = e Fix γ6 = 2k − 2g(G), giving relation (4.6) − α + β + r + 2 − m − 2k + 2g(G) = 0. In the same way:

3 Ltop γ6 = (1 + r + 2α + 1) − (1 + β + 2m − 2 + 1) = 2 + r + 2α − β − 2m, so 3 2 + r + 2α − β − 2m = e Fix γ6 = 2N − 2g(F1) − 2g(F2), giving relation

(4.7) 2 + r + 2α − β − 2m − 2N + 2g(F1) + 2g(F2) = 0. L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 17

4.3.3. Holomorphic Lefschetz number. According to the section 4.2 we shall deduce another relations using holomorphic Lefschetz formulas. On the one hand 2 X i ∗ i 5 Lhol (γ6) = (−1) tr (γ6) |H (S, OS) = 1 + ζ6 . i=0 Let as compute numbers a(P ) and b(C) from theorem 4.4:

1 1 1 a(P ) = = = , j 1 0 ζ3 0 (1 − ζ3) (1 − ζ4) det 1 − γ∗ det − 6 6 6 6 |T 4 Pj 0 1 0 ζ6 1 1 1 a(Q ) = = = j ∗ 1 0 ζ2 0 (1 − ζ2) (1 − ζ5) det 1 − φS6 det − 6 6 6 6 |T 5 Qj 0 1 0 ζ6 and similarly

2 1 − g(Ki) ζ6 · Ki 1 ζ6 · (2g(Ki) − 2) 1 −2 · ζ6 1 + ζ6 b(Ki) = − 2 = − 2 = − 2 = 2 , 1 − ζ6 (1 − ζ6) 1 − ζ6 (1 − ζ6) 1 − ζ6 (1 − ζ6) (1 − ζ6) 2 1 − g(D) ζ6 · D 1 − g(D) ζ6 · (2 − 2g(D)) (1 + ζ6) · (1 − g(D)) b(D) = − 2 = − 2 = 2 . 1 − ζ6 (1 − ζ6) 1 − ζ6 (1 − ζ6) (1 − ζ6) Therefore `−1 p(2,5) p(3,4) X X X Lh (γ6) = b(Ki) + b(D) + a(Pj) + a(Qj) = i=1 j=1 j=1

1 + ζ6 (1 + ζ6) · (1 − g(D)) p(3,4) p(2,5) = (` − 1) · 2 + 2 + 3 4 + 2 5 (1 − ζ6) (1 − ζ6) (1 − ζ6 ) (1 − ζ6 ) (1 − ζ6 ) (1 − ζ6 ) which is equivalent to

5 1 + ζ6 (1 + ζ6) · (1 − g(D)) p(3,4) p(2,5) 1 + ζ6 = (` − 1) · 2 + 2 + 3 4 + 2 5 . (1 − ζ6) (1 − ζ6) (1 − ζ6 ) (1 − ζ6 ) (1 − ζ6 ) (1 − ζ6 ) Multiplying this equality by denominators and after some manipulations we get the following relation: p (4.8) 3 + 3` − 3g(D) − (3,4) − p = 0, 2 (2,5) which agrees with [Dil12].

4.3.4. Stringy Euler Characteristic. Using stringy Euler characteristic (see section 4.1) for n = 1,..., 6 we have

24 for n = 1,  8l − 8g(D) + 8p + 4p + 8k − 8g(G) + 8n0 + 4N − 4g(F ) − 4g(F ) for n = 2,  (2,5) (3,4) 1 2   0 96 + 128l − 128g(D) + 88p(2,5) + 64p(3,4) + 48k − 48g(G) + 48n + 16N−  for n = 3,  − 16g(F ) − 16g(F )  1 2   0 672 + 1320l − 1320g(D) + 888p(2,5) + 660p(3,4) + 456k − 456g(G) + 456n +  for n = 4, + 168N − 168g(F1) − 168g(F2)   0 6720 + 13312l − 13312g(D) + 8888p(2,5) + 6656p(3,4) + 4464k + 4464n −  for n = 5,   − 4464g(G) + 1664N − 1664g(F1) − 1664g(F2)   0 66720 + 133288l − 133288g(D) + 88888p(2,5) + 66644p(3,4) + 44488k + 44488n −  for n = 6.  − 44488g(G) + 16664N − 16664g(F ) − 16664g(F )  1 1

Comparing es(Y6,n) and 3.8 for n = 1 ... 6 we obtain new relations:

(4.9) − 2α + 10 + N − r − g(F1) − g(F2) = 0 18 D. BUREK

G (4.10) − m + 2 + r − 2l − p(2,5) − p(3,4) + 2g(D) − 2b − w − 2g + 2g(G) γ6 F1 F2 − 2a − g − g + g(F1) + g(F2) = 0. γ6 γ6

0 3 G (4.11) − n − 3 + · r − 6l − 2p(2,5) − 3p(3,4) + 6g(D) + 2k − 6b − 6g + 2 γ6 3 3 3 F1 F2 + 4g(G) + · N − 6a − 3g − 3g + · g(F1) + · g(F2) = 0. 2 γ6 γ6 2 2 To summarize we have the following relations among invariants of S6:

4.5. Proposition. The following relations hold between invariants of S6 : 1) 0 = 2m + r + α + β − 20, 0 2) 0 = n − p(2,5) − 2n ,

3) 0 = 2 + r + m − α − β − 2` + 2g(D) − p(2,5) − p(3,4), 4) 0 = −α + β + r + 2 − m − 2k + 2g(G),

5) 0 = 2 + r + 2α − β − 2m − 2N + 2g(F1) + 2g(F2),

6) 0 = −2α + 10 + N − r − g(F1) − g(F2), p 7) 0 = 3 + 3` − 3g(D) − (3,4) − p , 2 (2,5) 1 8) 0 = −g G/φS6 + 2g(G) − p + 2k − 4b − 2` , 6 4 (3,4) 1 9) 0 = −g F /φS6 − g F /φS6 + 2g(F ) + 2g(F ) − 2p − 2p + 1 6 2 6 6 1 2 (2,5) (3,4) + 4N − 12a − 4l, (assuming g(D) = 0), G 10) 0 = −m + 2 + r − 2l − p(2,5) − p(3,4) + 2g(D) − 2b − w − 2g + 2g(G) γ6 F1 F2 − 2a − g − g + g(F1) + g(F2), γ6 γ6 0 3 G 11) 0 = −n − 3 + · r − 6l − 2p(2,5) − 3p(3,4) + 6g(D) + 2k − 6b − 6g + 2 γ6 3 3 3 F1 F2 + 4g(G) + · N − 6a − 3g − 3g + · g(F1) + · g(F2). 2 γ6 γ6 2 2 4.1. Question. Some relations from 4.5 were known. Is there a geometrical way to prove last four relations from 4.5?

5. Zeta functions of a finite quotient In this section we prove an analogous formula to 3.6 for Zeta function of a finite quotients. Since Zeta function may be different for any member in the family and as it is an arithmetic function depending on prime power q, the problem is much more subtle. Let V,W be finite dimensional vector spaces over a field K and take L ∈ End(V ),M ∈ End(W ). We define their characteristic polynomials: f(t) := det(1 − t · L) and g(t) := det(1 − t · M). Over an algebraic closure K of K we have factorisations

f(t) = (1 − λ1t)(1 − λ2t) ... (1 − λdim V t) and g(t) = (1 − µ1t)(1 − µ2t) ... (1 − µdim W t),

for some λ1, λ2, . . . , λdim V , µ1, µ2, . . . , µdim W ∈ which are in fact eigenvalues of endomorphisms L and K K M respectively. K Denoting by f ⊗ g the characteristic polynomial of L ⊗ M: f ⊗ g(t) = det(1 − t · L ⊗ M), can be written as: dim V dim W Y Y f ⊗ g(t) := (1 − λiµjt). i=1 j=1 L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 19

One can see that dim V dim W Y Y f ⊗ g(t) = g(λit) = f(µjt). i=1 j=1 Moreover one can check that the this polynomial can be computed using resultant i.e.

t f ⊗ g(t) = res f(s), sdeg(g) · g . s s a c The tensor product of polynomials extends uniquely to the case of rational functions f = , g = , b d where a, b, c, d ∈ K[t] by taking:

a c (a ⊗ c) · (b ⊗ d) ⊗ = . b d (a ⊗ d) · (b ⊗ c) Rose in [Ros07] introduced the orbifold Frobenius morphisms on the Chen-Ruan orbifold cohomology and use it deﬁne orbifold zeta function

∗ (5.1) ZCR(X , t) := det 1 − Froborb t | HCR X ×Fq Fq, Ql ,

∗ where HCR(X , Ql) is `-adic Chen Ruan cohomology and Froborb is the orbifold Frobenius morphism deﬁned in Section 3 and 5, respectively of [Ros07]. It was also proven (Corollary 6.4) that for a crepant resolution X → X of the course moduli scheme X of X the orbifold cohomological zeta function of X coincide with¬ classical zeta function of X i.e. ¬ ZH∗ (X , t) = Zq(X, t). CR ¬ Let Xi be a variety of Calabi-Yau type with automorphism φi,d : Xi → Xi of order d. Suppose that there exists a crepant resolution Xd,n of the quotient variety

n−1 (X1 × X2 × ... × Xn)/Zd .

5.1. Theorem. The Zeta function Zq (Xd,n)(T ) equals the product of factors of the rational function in T

(−1)n+1 d−1 n d−1 ! Y O Y m d (5.2)  ZXi,m,j q T  j=0 i=1 m=0 which contain only integral powers of q, where

(−1)ki+1 Y Y ∗ ki λi j ZXi,m,j(T ) = det 1 − Frobq T | H (Xi,m,λi ) (q T ). ζd λi≥0 0≤ki≤2 dim Xi Proof. Similarly as in the proof of 3.6 merging formulas 5.1 and 3.1 we have the following formula for zeta function of Xd,n:

d−1 n (−1)ki+1 Y Y Y Y O ki Zq (Xd,n)(T ) = det 1 − Frobq T | H (Xi,mi,λi ) =

m∈Gd,n λ≥0 |m| j=0 i=1 |k|=k−2 d −λ d−1 n k +1 Y Y Y Y O (−1) i mi ki d +λi = det 1 − Frobq T | H (Xi,mi,λi ) q T =

m∈Gd,n λ≥0 |k|=k j=0 i=1 |k|+n d−1 n !(−1) Y Y Y Y O mi ki d +λi = det 1 − Frobq T | H (Xi,mi,λi ) q T ,

m∈Gd,n λ≥0 |k|=k j=0 i=1

where λ = (λ1, λ2, . . . , λn), k = (k1, k2, . . . , kn), m = (m1, m2, . . . , mn) and |λ| = λ1 + λ2 + ... + λn, |k| = k1 + k2 + ... + kn, |m| = m1 + m2 + ... + mn. 20 D. BUREK Y Y Extending the exterior product (...) to (...) we introduce only factors containing fractional n m∈Gn,d m∈Zd powers of q, denote the resulting rational function by

α1 α2 αs Let W = W1 · W2 · ... · Ws be the decomposition of W into product of irreducible polynomials 1 Wi ∈ Z[q d ,T ], for αi ∈ Z. Then

Y α1 Zq (Xd,n)(T ) = {Wi : Wi ∈ Z[q, T ], i = 1, . . . , s} .

Therefore, in order to compute zeta function of Xd,n for any i ∈ {1, 2, . . . , n} it is enough to produce the following tables

H j HH 0 1 ··· j ··· d − 1 k HH

0 ZXi,0,0 ZXi,0,1 ZXi,0,j ZXi,0,d−1

1 ZXi,1,0 ZXi,1,1 ZXi,1,j ZXi,1,d−1

2 ZXi,2,0 ZXi,2,1 ··· ZXi,2,j ··· ZXi,2,d−1

......

d − 1 ZXi,d−1,0 ZXi,d−1,1 ZXi,d−1,j ZXi,d−1,d−1 = vXi,j

Table 10. ZXi,j,k

for any considered variety Xi. Then we evaluate vector vXi,j on

√ d pd 2 pd d−1 vd := T, qT, q T,..., q T

and multiply all its terms. Then we take tensor product for all i ∈ {1, 2, . . . , n} and take product over j ∈ {0, 1, . . . , d − 1}. Finally we take (−1)n+1 power of the result and form product of factors containing only integral powers of q.

5.1. Explicit computation of the zeta function.

5.1.1. Zeta function of Borcea-Voisin Calabi-Yau threefold. In the presenting section we compute zeta function of classical Borcea-Voisin Calabi-Yau threefold. Let (S, αS) be a K3 surface admitting a non-symplectic involution αS. Consider an elliptic curve E with non-symplectic involution αE. Let us keep notation from the section 3.4.4. The polynomial ZE,1,0 depends on the number of points in Fix(αE) which are deﬁned over Fq. The corresponding table 10 for elliptic curve E has the following form: L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 21

H j HH 0 1 k HH 1 0 1 − a T + qT 2 (1 − T )(1 − qT ) q

1 All points/ : , Fq (1 − T )4 1 1 Two points/ : , 1 Fq (1 − T )3(1 + T ) 1 One point/ : Fq (1 − T )2(1 + T + T 2)

Table 11. ZE,k,j(T )

Now let us make analysis of corresponding table 10 for K3 surface S. The polynomial ZS,0,0: In that case:

∗∗ αS ZS,0,0 = Zq (H (S) ) .

The Frobenius map acting on r curves induces permutation π ∈ Sr with decomposition into disjoint cycles of lengths a1, a2, . . . , as for some natural s. Since the Hodge diamond of αS invariant part is equal to 1 0 0 0 r 0 0 0 1

we have 1 ZS,0,0 = s . Y (1 − T )(1 − qT ) 1 − (qT )ai 1 − q2T i=1

The polynomial ZS,0,1: One can see that

2 H´et (S, Ql) = T (S) ⊕ NS(S)−1.

In general we cannot say much more but in special cases we can ﬁnd explicit description of the polynomial ZS,0,1. In particular if S is one of K3 surfaces appearing in Borcea-Voisin construction then the polynomial can be read out from Theorem 5.6 of [GLY13]. The polynomial ZS,1,0: Let Cg be the curve of maximal genus g in Fix(αS). Then we see that

1 ∗ det 1 − t · Frobq |H´et(Cg, Ql) ZS,1,0 = Zq (H´et (Fix(αS))) = s . Y (1 − T )(1 − qT ) (1 − (qT )as ) i=1

Otherwise the local zeta function ZS,1,0 is equal to

1 s . Y (1 − T )(1 − qT ) (1 − (qT )as ) i=1

The polynomial ZS,1,1: This polynomial is obviously equal to 1. Therefore the corresponding table 10 has the following form (Table 12). 22 D. BUREK

HH j H 0 1 k HH 1 s 0 Y Z (1 − T )(1 − qT ) (1 − (qT )ai ) 1 − q2T S,0,1 i=1

1 det 1 − t · Frobq |H´et(Cg, Ql) s 1 Y 1 (1 − T )(1 − qT ) 1 − T ai 1 − (qT )ai i=1

Table 12. ZS,k,j(T )

n We shall compute the Zeta function of (S × E )/Gd,n in the case when S has particularly nice arithmetic properties. Let S be the K3 surface studied in [AOP02]. It can be deﬁned as double cover of P2 branched along the union of six lines given by XYZ(X + λY )(Y + Z)(Z + X) = 0. One can see that it is a K3 surface with an obvious non-symplectic involution.

The resolution of the singularities of that surface is obtained in the following way: ﬁrstly we blow up 3 triple points that deﬁnes 24 points on the double cover, then we blow up resulting variety at 15 double points. The corresponding table 10 has the following form (Table 13).

HH j H 0 1 k HH 1 1 0 19 2 2 2 (1 − T )(1 − qT ) (1 − q T ) (1 − γqqT ) (1 − γqπ T ) (1 − γqπ T )

1 1 1 (1 − T )9(1 − qT )9

Table 13. ZS,k,j(T )

From Theorem 5.1 we have the following formula for zeta function of the classical Borcea-Voisin threefold:

! S × E √ √ Zq = ZE,0,0(T ) · ZE,1,0 ( q · T ) ⊗ ZS,0,0(T ) · ZS,1,0 ( q · T ) × Z2 ¬ ! √ √ × ZE,0,1(T ) · ZE,1,1 ( q · T ) ⊗ ZS,0,1(T ) · ZS,1,1 ( q · T ) . L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 23

Now according to table 11 we have three cases depending on number of ﬁxed points of αE deﬁned over Fq. Resulting zeta functions are summarized in the following table:

ZE,1,0 Y2,2 = Zq (S × E)/Z2 ¬

2 3 2 2 4 2 2 2 4 2 2 1 1 − aq yq qT + yq q T 1 − aq π yq T + π yq qT 1 − aq π yq T + π yq qT (1 − T )4 (1 − T ) (1 − qT )56 (1 − q2T )56 (1 − q3T )

2 3 2 2 4 2 2 2 4 2 2 1 1 − aq yq qT + yq q T 1 − aq π yq T + π yq qT 1 − aq π yq T + π yq qT (1 − T )3(1 + T ) (1 − T ) (1 − qT )47 (1 + qT )9 (1 + q2T )9 (1 − q2T )47 (1 − q3T )

2 3 2 2 4 2 2 2 4 2 2 1 1 − aq yq qT + yq q T 1 − aq π yq T + π yq qT 1 − aq π yq T + π yq qT (1 − T )2(1 + T + T 2) (1 − T ) (1 − qT )38 (1 − q2T )38 (1 + q2T + q4T 2)9 (1 + qT + q2T 2)9 (1 − q3T )

Table 14. Zeta function of Y2,2

5.1.2. Zeta function of Y6,n. We shall give details of computation of the Zeta function of a Borcea-Voisin Calabi-Yau n-fold with a very particular shape of the Hodge diamond. 2 3 Consider an elliptic curve E6 with the Weierstrass equation y = x +1 together with a non-symplectic automorphism of order 6. Let S6 be the K3 surface no. 18 from Table 1 of [Dil12]. Then S6 is isomorphic to an elliptic K3 surface X → P1 whose Weierstrass equation is y2 = x3 + λ(z − 1)2z5

and on which we have the following ζ6-action: 2 α:(x, y, t) → (ζ3 x, y, z). 1 ∗ The elliptic fibrations S6 → P has 1 type IV fiber and 2 type II fibers. The surface S6 has the following invariants (see [Dil12] and relations 4.5):

0 r m n n k a p3,4 p2,5 ` N b α β 19 1 9 0 6 0 6 9 3 10 0 0 1

After careful study of the resolution of singularities of the surface S6 the corresponding table 10 has the following form (Table 15).

HH j H 0 1 2 3 4 5 k HH 1 1 1 1 0 19 2 1 1 (1 − T )(1 − qT ) (1 − q T ) 1 − βqT 1 − cqqT 1 − βqT

1 1 1 1 1 1 1 (1 − T )3(1 − qT )18

1 2 1 1 1 1 1 (1 − T )6(1 − qT )15

1 3 1 1 − δ T 1 1 − δ T 1 (1 − T )10(1 − qT )10 q q

1 4 1 1 1 1 1 (1 − T )15(1 − qT )6

1 5 1 1 1 1 1 (1 − T )18(1 − qT )3

Table 15. ZS6,k,j 24 D. BUREK

Local zeta functions of E6 are written in the following table:

HH j H 0 1 2 3 4 5 k HH 1 0 1 − α T 1 1 1 1 − α T (1 − T )(1 − qT ) q q

1 1 1 1 1 1 1 1 − T

1 1 2 1 1 1 1 (1 − T )2 1 − T

1 1 1 3 1 1 1 (1 − T )2 1 − T 1 − T

1 1 4 1 1 1 1 (1 − T )2 1 − T

1 5 1 1 1 1 1 1 − T

Table 16. ZE6,k,j

Applying Theorem 5.1 we get " 1 1 1 1 Zq (S6 × E6)/ 6 = · √ · · · Z 6 p6 p6 ¬ (1 − T )(1 − q · T ) (1 − q · T ) (1 − q2 · T )2 (1 − q3 · T )2 ! 1 1 1 1 · · ⊗ × √ √ × (1 − p6 q4 · T )2 (1 − p6 q5 · T ) (1 − T )(1 − q · T )19(1 − q2 · T ) (1 − 6 q · T )3(1 − 6 q · q · T )18 1 1 1 × × × × (1 − p6 q2 · T )6(1 − p6 q2 · qT )15 (1 − p6 q3 · T )10(1 − p6 q3 · q · T )10 (1 − p6 q4 · T )15(1 − p6 q4 · q · T )6 !# " !# 1 1 × × (1 − αqT ) ⊗ × p6 p6 (1 − q5 · T )18(1 − q5 · q · T )3 1 − βqT " ! # " ! !# 1 p6 3 1 1 1 × ⊗ (1 − q · δq · T ) × · ⊗ × p6 p6 p6 1 − q3 · T 1 − q2 · T 1 − q4 · T 1 − cq · q · T " ! # " !# 1 p6 3 1 × ⊗ (1 − q · δq · T ) × (1 − αq · T ) ⊗ = p6 3 1 − q · T 1 − βq · T (1 − α β T )(1 − δ T )(1 − δ T )(1 − α β T ) = q q q q q q . (1 − T )(1 − qT )103(1 − q2T )103(1 − q3T )

Using the same method we are able to compute zeta functions of higher dimensional quotients n−1 n−1 (S6 × E6 )/Z6 . Unfortunately for higher value of n computations are much more involved but can be¬ easily carried out with a help of Maple. In the table below we collect results for n = 2, 3, 4, for n ≥ 5 formulas are too long to display.

n n−1 n−1 Zq(Y6,n) = Zq (S6 × E6 )/Z6 ¬ (1 − α β T )(1 − δ T )(1 − δ T )(1 − α β T ) 2 q q q q q q (1 − T )(1 − qT )103(1 − q2T )103(1 − q3T )

1 3 340 2 2 2 1402 3 340 2 2 4 (1 − T ) (1 − qT ) 1 − αq βq T (1 − αq βq T ) (1 − q T ) (1 − q T ) (1 − q cqT ) (1 − q T )

3 2 2 3 1 − αq βq T 1 − q δq T 1 − q δq T 1 − αq βq T 4 868 2 9548 2 3 3 9548 4 868 5 (1 − T ) (1 − qT ) (1 − q T ) (1 − q cq T ) (1 − q cq T ) (1 − q T ) (1 − q T ) (1 − q T )

Table 17. Zeta function of Y6,2, Y6,3, Y6,4 L-FUNCTIONS OF HIGHER DIMENSIONAL CALABI-YAU MANIFOLDS OF BORCEA-VOISIN TYPE 25

Below we present Hodge diamonds of varieties Y6,2, Y6,3, Y6,4.

0 0 1 0 340 0 0 0 00 00 0 0 103 0 11404 00 1 11 1 1 00 00

103 00 340 00

00 0 0

1 1

∗∗ ∗∗ H (Y6,2) H (Y6,3)

0 0

0 868 0

00 000 0

09549 00 0

111 00 1

9549 00 00

0 0 00

0868 0

∗∗ H (Y6,4) 26 D. BUREK

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