On homological mirror symmetry of toric Calabi–Yau threefolds

We use Lagrangian torus fibrations on the mirror $X$ of a toric Calabi–Yau threefold $\check{X}$ to construct Lagrangian sections and various Lagrangian spheres on $X$. We then propose an explicit correspondence between the sections and line bundles on $\check{X}$ and between spheres and sheaves supported on the toric divisors of $\check{X}$. We conjecture that these correspondences induce an embedding of the relevant derived Fukaya category of $X$ inside the derived category of coherent sheaves on $\check{X}$.

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Received 7 April 2015

Accepted 7 February 2018

Published 26 February 2019