The mirror of Calabi-Yau orbifold.

*(English)*Zbl 0817.14018It has recently been recognized that the relation between exactly solvable conformal field theory and Calabi-Yau (CY) spaces necessarily involves hypersurfaces in weighted projective 4-space. A surprising symmetry which pairs different CY spaces with Euler numbers differed by \(\chi \leftrightarrow - \chi\) was found by examining a large such class of manifolds [cf. B. R. Greene and M. R. Plesser, Nuclear Phys., Particle Phys., B 338, No. 1, 15-37 (1990)]. This duality shows that topologically distinct CY pairs yield isomorphic conformal theories. Such symmetry indicates that this class of CY spaces is potentially of much higher phenomenological interest for the string theorists. As the simplest example, consider the Fermat quintic in \(\mathbb{P}^ 4\), \(z^ 5_ 1 + z^ 5_ 2 + z^ 5_ 3 + z^ 5_ 4 + z^ 5_ 5 = 0\), which is invariant under the action of the subgroup \(G\) of \(SL_ 5 (\mathbb{C})\) consisting of all the order 5 diagonal matrices. The mirror partner of this Fermat quintic is the CY resolution of its quotient variety by \(G\). Its Euler number equals to 200 \((= - \chi\)(quintic)), by the “orbifold Euler number” formula. In this paper, we shall study the above construction on a more general setting for the hypersurfaces in weighted projective space.

Consider the quasihomogeneous polynomial functions \(f(z)\), \(h(z)\) of degree \(d\) w.r.t. \(n_ i\)’s, which are invariant under the action of \(G\), \(G'\), respectively. Assume \(f(z)\), \(h(z)\) have the only isolated critical point at origin. Denote \(\widehat X = \) the CY resolution of \(X\) \((= X(f,G))\), \(\widehat X' = \) the CY resolution of \(X'\) \((= X(h,G'))\).

Theorem. We have the following duality between the Hodge numbers of \(\widehat X\) and \(\widehat X'\): \(h^{1,1} (\widehat X) = h^{2,1} (\widehat X')\), \(h^{2,1} (\widehat X) = h^{1,1} (\widehat X')\). – As a consequence, \(\chi (\widehat X) = - \chi (\widehat X')\).

The Hodge numbers of \(\widehat X\), \(\widehat X'\) depend only on the groups \(G\), \(G'\). We may assume both polynomials \(f(z)\) and \(g(z)\) equal to the Fermat polynomial. Then the preceding theorem is a consequence of the following:

Theorem. When \(f(z) = g(z) =\) the Fermat polynomial for \(n_ i\)’s, \(f(z) = z^{d_ 1}_ 1 + z_ 2^{d_ 2} + z^{d_ 3}_ 3 + z^{d_ 4}_ 4 + z^{d_ 5}_ 5\), \(d_ i = d/n_ i \geq 3\), there exist \(\mathbb{C}\)-isomorphisms between the Hodge groups of \(\widehat X\) and \(\widehat X'\): \(H^{1,1} (\widehat X) \simeq H^{2,1} (\widehat X')\), \(H^{2,1} (\widehat X) \simeq H^{1,1} (\widehat X')\).

Consider the quasihomogeneous polynomial functions \(f(z)\), \(h(z)\) of degree \(d\) w.r.t. \(n_ i\)’s, which are invariant under the action of \(G\), \(G'\), respectively. Assume \(f(z)\), \(h(z)\) have the only isolated critical point at origin. Denote \(\widehat X = \) the CY resolution of \(X\) \((= X(f,G))\), \(\widehat X' = \) the CY resolution of \(X'\) \((= X(h,G'))\).

Theorem. We have the following duality between the Hodge numbers of \(\widehat X\) and \(\widehat X'\): \(h^{1,1} (\widehat X) = h^{2,1} (\widehat X')\), \(h^{2,1} (\widehat X) = h^{1,1} (\widehat X')\). – As a consequence, \(\chi (\widehat X) = - \chi (\widehat X')\).

The Hodge numbers of \(\widehat X\), \(\widehat X'\) depend only on the groups \(G\), \(G'\). We may assume both polynomials \(f(z)\) and \(g(z)\) equal to the Fermat polynomial. Then the preceding theorem is a consequence of the following:

Theorem. When \(f(z) = g(z) =\) the Fermat polynomial for \(n_ i\)’s, \(f(z) = z^{d_ 1}_ 1 + z_ 2^{d_ 2} + z^{d_ 3}_ 3 + z^{d_ 4}_ 4 + z^{d_ 5}_ 5\), \(d_ i = d/n_ i \geq 3\), there exist \(\mathbb{C}\)-isomorphisms between the Hodge groups of \(\widehat X\) and \(\widehat X'\): \(H^{1,1} (\widehat X) \simeq H^{2,1} (\widehat X')\), \(H^{2,1} (\widehat X) \simeq H^{1,1} (\widehat X')\).

##### MSC:

14J30 | \(3\)-folds |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

32G20 | Period matrices, variation of Hodge structure; degenerations |

14D07 | Variation of Hodge structures (algebro-geometric aspects) |