Journal of Symplectic Geometry

Volume 16 (2018)

Number 5

On homological mirror symmetry of toric Calabi–Yau threefolds

Pages: 1249 – 1349



Mark Gross (DPMMS, Centre for Mathematical Sciences, University of Cambridge, United Kingdom)

Diego Matessi (Dipartimento di Matematica, Università degli Studi di Milano, Italy)


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 $\check{X}$.

Received 7 April 2015

Accepted 7 February 2018

Published 26 February 2019