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Through the looking glass. (English) Zbl 0935.32020
Phong, Duong H. (ed.) et al., Mirror symmetry III. Proceedings of the conference on Complex geometry and mirror symmetry. Montréal, Canada, 1995. New York, NY, Cambridge, MA: AMS, International Press. AMS/IP Stud. Adv. Math. 10, 263-277 (1999).
Summary: It is frequently possible to produce new Calabi-Yau threefolds from old ones by a process of allowing the complex structure to degenerate to a singular one, and then performing a resolution of singularities. (Some care is needed to ensure that the Calabi-Yau condition be preserved.) There has been speculation that all Calabi-Yau threefolds could be linked in this way, and considerable evidence has been amassed in this direction. We propose here a natural way to relate this construction to the string-theoretic phenomenon known as “mirror symmetry”. We formulate a conjecture which in principle could predict mirror partners for all Calabi-Yau threefolds, provided that all were indeed linked by the degeneration/resolution process. The conjecture produces new mirrors from old, and so requires some initial mirror manifold construction – such as Greene-Plesser orbifolding – as a starting point.
For the entire collection see [Zbl 0908.00032].

32Q25 Calabi-Yau theory (complex-analytic aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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