Generalized Homological Mirror Symmetry and cubics

We discuss an approach to studying Fano manifolds based on Homological Mirror Symmetry. We consider some classical examples from a new point of view

Phenomena

In this paper, we describe recent work towards the mirror conjecture, which relates the weight filtration on the cohomology of a log Calabi–Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi–Yau manifold taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers of mirror dual compact Calabi–Yau manifolds, incorporating tools and ideas which appear in the fascinating and groundbreaking works of de Cataldo, Hausel, and Migliorini [1] and de Cataldo and Migliorini [2]. We give a broad overview of the motivation for this conjecture, recent results towards it, and describe how this result might arise from the SYZ formulation of mirror symmetry. This interpretation of the mirror conjecture provides a possible bridge between the mirror conjecture and the well-known conjecture in non-Abelian Hodge theory

Compactifications of spaces of Landau-Ginzburg models

This paper reviews results and techniques from the authors' previous work "Symplectomorphism group relations and degenerations of Landau-Ginzburg models" and applies them in basic examples. The main example is the $A_n$ category where we observe a relationship to stability conditions and directed quiver representations. We conclude with a brief survey of applications to the birational geometry of del Pezzo surfaces