P=W Phenomena

TL;DR

This paper discusses the mirror P=W conjecture, relating the weight and perverse Leray filtrations on cohomology of mirror log Calabi--Yau manifolds, and explores its connections to classical mirror symmetry and nonabelian Hodge theory.

Contribution

It provides an overview of recent progress towards the mirror P=W conjecture and proposes a potential link to the classical P=W conjecture via the SYZ framework.

Findings

Progress towards the mirror P=W conjecture is summarized.

A possible connection between mirror P=W and classical P=W conjectures is proposed.

The conjecture's relation to the SYZ formulation of mirror symmetry is discussed.

Abstract

In this paper, we describe recent work towards the mirror P=W conjecture, which relates the weight filtration on a 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, and de Cataldo and Migliorini. 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 P=W conjecture provides a possible bridge between the mirror P=W conjecture and the well-known P=W conjecture in nonabelian Hodge theory.