Mirror symmetry for quasi-smooth Calabi–Yau hypersurfaces in weighted projective spaces

Abstract

We consider a d-dimensional well-formed weighted projective space P(w¯) as a toric variety associated with a fan Σ(w¯) in Nw¯⊗R whose 1-dimensional cones are spanned by primitive vectors v0,v1,…,vd∈Nw¯ generating a lattice Nw¯ and satisfying the linear relation ∑iwivi=0. For any fixed dimension d, there exist only finitely many weight vectors w¯=(w0,…,wd) such that P(w¯) contains a quasi-smooth Calabi–Yau hypersurface Xw defined by a transverse weighted homogeneous polynomial W of degree w=∑i=0dwi. Using a formula of Vafa for the orbifold Euler number χorb(Xw), we show that for any quasi-smooth Calabi–Yau hypersurface Xw the number (−1)d−1χorb(Xw) equals the stringy Euler number χstr(Xw¯∗) of Calabi–Yau compactifications Xw¯∗ of affine toric hypersurfaces Zw¯ defined by non-degenerate Laurent polynomials fw¯∈ℂ[Nw¯] with Newton polytope conv({v0,…,vd}). In the moduli space of Laurent polynomials fw¯ there always exists a special point fw¯0 defining a mirror Xw¯∗ with a Z∕wZ-symmetry group such that Xw¯∗ is birational to a quotient of a Fermat hypersurface via a Shioda map.