Perverse curves and mirror symmetry (Q2832761)

A perverse curve is a pair \((Z,\mathcal F_Z)\) of a possibly reducible curve \(Z\) with a perverse sheaf \(\mathcal F_Z\) supporting a cohomological mixed Hodge complex. \textit{M. Gross}, \textit{L. Katzarkov} and the author [``Towards mirror symmetry for varieties of general type'', Preprint, \url{arXiv:1202.4042}] suggest that the mirror of a curve \(Z\) of genus \(g\geq 2\) is a perverse curve \((\check{Z}, \mathcal F_{\check{Z}})\), where \(\check{Z}\) is a union of \(3g-3\) projective lines meeting in \(2g-2\) points with exactly three components meeting at each point, and \(\mathcal F_{\check{Z}}\) is the sheaf of vanishing cycles. The critical locus of a Strominger-Yau-Zaslow fibration of a Calabi-Yau threefold shows a striking similarity to a perverse curve. This paper relates mirror symmetry for Calabi-Yau threefolds to that of curves of \(g\geq 2\) via a perverse curve. The main theorem states that for perverse curves \((Z, \mathcal F_Z)\) and \((\check{Z}, \mathcal F_{\check{Z}})\) arising from mirror partners of the Batyrev construction of Calabi-Yau threefolds after a maximal projective partial crepant resolution of the ambient toric Fano varieties, their Hodge diamonds are related by mirror duality.