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Mirror symmetry for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces | | Victor Batyrev
; Karin Schaller
; | | Date: |
8 Jun 2020 | | Abstract: | We consider a $d$-dimensional well-formed weighted projective space
$mathbb{P}(overline{w})$ as a toric variety associated with a fan
$Sigma(overline{w})$ in $N_{overline{w}} otimes mathbb{N}$ whose
$1$-dimensional cones are spanned by primitive vectors $v_0, v_1, ldots, v_d
in N_{overline{w}}$ generating a lattice $N_{overline{w}}$ and satisfying
the linear relation $sum_i w_i v_i =0$. For any fixed dimension $d$, there
exist only finitely many weight vectors $overline{w} = (w_0, ldots, w_d)$
such that $mathbb{P}(overline{w})$ contains a quasi-smooth Calabi-Yau
hypersurface $X_w$ defined by a transverse weighted homogeneous polynomial $W$
of degree $w = sum_{i=0}^d w_i$. Using a formula of Vafa for the orbifold
Euler number $chi_{
m orb}(X_w)$, we show that for any quasi-smooth
Calabi-Yau hypersurface $X_w$ the number $(-1)^{d-1}chi_{
m orb}(X_w)$ equals
the stringy Euler number $chi_{
m str}(X_{overline{w}}^*)$ of Calabi-Yau
compactifications $X_{overline{w}}^*$ of affine toric hypersurfaces
$Z_{overline{w}}$ defined by non-degenerate Laurent polynomials
$f_{overline{w}} in mathbb{C}[N_{overline{w}}]$ with Newton polytope
$ ext{conv}({v_0, ldots, v_d})$. In the moduli space of Laurent polynomials
$f_{overline{w}}$ there always exists a special point $f_{overline{w}}^0$
defining a mirror $X_{overline{w}}^*$ with a $mathbb{Z}/wmathbb{Z}$-symmetry
group such that $X_{overline{w}}^*$ is birational to a quotient of a Fermat
hypersurface via a Shioda map. | | Source: | arXiv, 2006.4465 | | Services: | Forum | Review | PDF | Favorites | |
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