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Higher dimensional analogon of Borcea-Voisin Calabi-Yau manifolds, their Hodge numbers and $L$-functions

Dominik Burek

Abstract

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

Higher dimensional analogon of Borcea-Voisin Calabi-Yau manifolds, their Hodge numbers and $L$-functions

Abstract

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

Paper Structure

This paper contains 8 sections, 10 theorems, 30 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Kummer type Calabi-Yau manifolds
  3. Borcea-Voisin construction
  4. Generalisation
  5. Hodge numbers of a finite quotient
  6. Chen-Ruan cohomology
  7. Hodge numbers of $\mathcal{X}_{d,n}$
  8. Hodge numbers of $X_{d,n}$

Key Result

Theorem 2.1

The quotient $(S\times E)/(\gamma_{S}\times \alpha_{E})$ has a crepant resolution of singularities $X$ which is a Calabi-Yau threefold. Moreover where $N$ is the number of curves in $\operatorname{Fix}(\gamma_{S})$ and $N'$ is the sum of their genera.

Theorems & Definitions (19)

  • Theorem 2.1: BorceaCV
  • Theorem 2.2: C
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5: Burii3, CH
  • Theorem 2.6
  • Theorem 2.7: CH, Burii3
  • Theorem 2.8
  • Definition 3.1
  • Remark 3.2
  • ...and 9 more