Conifold transitions via affine geometry and mirror symmetry

Abstract

Mirror symmetry of Calabi–Yau manifolds can be understood via a Legendre duality between a pair of certain affine manifolds with singularities called tropical manifolds. In this article, we study conifold transitions from the point of view of Gross and Siebert; see [J. Differential Geom. 72 (2006) 169–338], [J. Algebraic Geom. 19 (2010) 679–780] and [Ann. of Math. 174 (2011) 1301–1428]. We introduce the notions of tropical nodal singularity, tropical conifolds, tropical resolutions and smoothings. We interpret known global obstructions to the complex smoothing and symplectic small resolution of compact nodal Calabi–Yau manifolds in terms of certain tropical $2$–cycles containing the nodes in their associated tropical conifolds. We prove that the existence of such cycles implies the simultaneous vanishing of the obstruction to smoothing the original Calabi–Yau and to resolving its mirror. We formulate a conjecture suggesting that the existence of these cycles should imply that the tropical conifold can be resolved and its mirror can be smoothed, thus showing that the mirror of the resolution is a smoothing. We partially prove the conjecture for certain configurations of nodes and for some interesting examples.