We investigate a potential relationship between mirror symmetry for
Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and
Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a
so-called Tyurin degeneration to a union of two Fano varieties, then one should
be able to construct a mirror to that Calabi-Yau by gluing together the
Landau-Ginzburg models of those two Fano varieties. We provide evidence for
this correspondence in a number of different settings, including
Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds,
Dolgachev-Nikulin mirror symmetry for K3 surfaces, and an explicit family of
threefolds that are not realized as complete intersections in toric varieties.Comment: v2: Section 5 has been completely rewritten to accommodate results
removed from Section 5 of arxiv:1501.04019. v3: Final version, to appear in
String-Math 2015, forthcoming volume in the Proceedings of Symposia in Pure
Mathematics serie
