Calabi-Yau manifolds with a non-Hodge-degenerate mirror dual are Kobayashi non-hyperbolic, assuming the BCOV relation between genus-one Gromov-Witten invariants and Ray-Singer torsion.
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Kobayashi non-hyperbolicity of Calabi-Yau manifolds via mirror symmetry
Calabi-Yau manifolds with a non-Hodge-degenerate mirror dual are Kobayashi non-hyperbolic, assuming the BCOV relation between genus-one Gromov-Witten invariants and Ray-Singer torsion.