Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence

TL;DR

This paper explores the mirror P=W conjecture within mirror symmetry for Fano pairs, introducing hybrid LG models to study filtrations and proposing a relative homological mirror symmetry framework.

Contribution

It introduces hybrid LG models as mirrors of Fano pairs and connects the mirror P=W conjecture to a relative homological mirror symmetry conjecture.

Findings

Discovered an upper bound on the multiplicativity of the perverse filtration.

Linked the mirror P=W conjecture to a relative homological mirror symmetry framework.

Analyzed the topological and Hodge-theoretic properties of hybrid LG models.

Abstract

The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski, predicts a correspondence between weight and perverse filtrations in the context of mirror symmetry. In this paper, we revisit this conjecture through the lens of mirror symmetry for a Fano pair $(X,D)$, where $X$ is a smooth Fano variety and $D$ is a simple normal crossing divisor. We introduce its mirror object as a multi-potential analogue of a Landau-Ginzburg (LG) model, which we call the hybrid LG model. This model is expected to capture the mirrors of all irreducible components of $D$. We study the topological aspects, particularly the perverse filtration, and the Hodge theory of hybrid LG models, building upon the work of Katzarkov-Kontsevich-Pantev. As an application, we discover an interesting upper bound on the multiplicativity of the perverse filtration for a hybrid LG model. Additionally, we propose a relative version of the homological mirror symmetry conjecture and explain how the mirror P=W conjecture naturally emerges from it.