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

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$ 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 j 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 j 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 (!X ) = ζd!X ; n;0 where !X 2 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 lineari- sation 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 lineari- sation 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 lineari- sation 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) n 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 ) n Fix(η), (3) Fix(η3) n 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) n 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 f2; 3; 4; 6g: Assume moreover, k that automorphism η1 satisfies condition Ad and Fix(η2 ) for k j 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 f2; 3; 4; 6g: Consider the following group n n−1 Gd;n := f(m1; m2; : : : ; mn) 2 Zd : m1 + m2 + ::: + mn = 0g ' 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 f2; 3; 4; 6g: Assume moreover, that auto- k morphism φ1;d satisfies condition Ad and Fix(φi;d) for k j 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.
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