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

We consider a d-dimensional well-formed weighted projective space P((w) over bar) as a toric variety associated with a fan Sigma(w) in N-(w) over bar circle times R whose 1- dimensional cones are spanned by primitive vectors v(0), v(1), ... , v(d) is an element of N-(w) over bar generating a lattice N-(w) over bar and satisfying the linear relation Sigma(i) w(i)v(i) = 0. For any fixed dimension d, there exist only finitely many weight vectors (w) over bar = (w(0), ... , w(d)) such that P((w) over bar) contains a quasi-smooth Calabi-Yau hyper-surface X-w defined by a transverse weighted homogeneous polynomial W of degree w = Sigma(d)(i=0) w(i). Using a formula of Vafa for the orbifold Euler number chi(orb)(X-w), we show that for any quasi-smooth Calabi-Yau hypersurface X-w the number (-1)(d-1) chi(orb)(X-w) equals the stringy Euler number chi(str)(X*((w) over bar)) of Calabi-Yau compactifications X*((w) over bar) of affine toric hypersurfaces Z((w) over bar) defined by non-degenerate Laurent polynomials f((w) over bar) is an element of C[N-(w) over bar] with Newton polytope conv({v(0), ... , v(d)}). In the moduli space of Laurent polynomials f((w) over bar) there always exists a special point f((w) over bar)(0) defining a mirror X*((w) over bar) with a Z/wZ-symmetry group such that X*((w) over tilde) is birational to a quotient of a Fermat hypersurface via a Shioda map. (C) 2021 Elsevier B.V. All rights reserved.