On an example of Aspinwall and Morrison

Balázs Szendrői
2003 Proceedings of the American Mathematical Society  
In this paper, a family of smooth multiply-connected Calabi-Yau threefolds is investigated. The family presents a counterexample to global Torelli as conjectured by Aspinwall and Morrison. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 622 BALÁZS SZENDRŐI 4.12], birational equivalence implies isomorphism between (rational) Hodge structures. However, in the present case the situation should be entirely different. Conjecture 0.2. For
more » ... eral t ∈ B, the threefolds Y ξ i t for i = 0, . . . , 4 are not birationally equivalent to one another. One obvious direct approach to this conjecture is to aim to understand the various birational models of a fixed fibre Y t . Birational models of minimal threefolds can be studied via their cones of nef divisors in the Picard group; so this method requires an explicit understanding of the nef cone of Y t . Anétale cover Z t of Y t is a toric hypersurface. A recent conjecture [3, Conjecture 6.2.8] of Cox and Katz aims at giving a complete understanding of the nef cone of toric Calabi-Yau hypersurfaces. However, it is proved in [14] that in fact the conjecture of Cox and Katz fails for Z t . At this point the computation of the nef cone of Y t seems rather hopeless. A different approach to Conjecture 0.2 is required. To conclude the Introduction, let me point out that the varieties Y t are multiply connected with fundamental group Z/5Z (Proposition 1.5 and Proposition 1.7). This is a curious fact. The construction of Aspinwall and Morrison requires in an essential way that members of the mirror Calabi-Yau family should have a nontrivial (and in fact non-cyclic) fundamental group. Computations of Gross [7, Section 3] connect torsion in the integral cohomologies of mirror Calabi-Yau threefolds, and these computations imply that the cohomology (and hence homology) of Y t should have torsion of some kind. However, the direct relationship between failure of Torelli and the fundamental group seems rather mysterious; compare also Remark 2.6. Notation and conventions. All schemes and varieties are defined over C. A Calabi-Yau threefold is a normal projective threefold X with canonical Gorenstein singularities satisfying K X ∼ 0 and H 1 (X, O X ) = 0. Some statements use the language of toric geometry; my notation follows Fulton [5] and Cox-Katz [3, Chapter 3]. If A is a Z-module, then A free denotes the torsion-free part.
doi:10.1090/s0002-9939-03-07084-9 fatcat:5siltq6nmzdrjhp6vb6xb57bae