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

Abstract

The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski [29], predicts a correspondence between weight and perverse filtrations in the context of mirror symmetry. In this paper, we offer to reexamine 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. A mirror object is a multi-potential version of the Landau-Ginzburg (LG) model, called hybrid LG model, which encodes the mirrors of all irreducible components of D. We explain how the mirror P=W conjecture is induced by the relative version of homological mirror symmetry for such pairs. The key idea is to find a combinatorial description of the perverse filtration. The requisite combinatorics naturally appears in the course of studying Hodge numbers of hybrid LG models. We discuss this topic in the second half of the paper and use it to extend the work of to the hybrid LG setting. In application, we discover an interesting upper bound of the multiplicativity of the perverse filtration, which is an independent interest. Contents 1. Introduction 1.1. Motivation: the mirror P=W conjecture 1.2. Extended Fano/LG correspondence 1.3. Landau-Ginzburg Hodge numbers 1.4. Acknowledgement 2. The mirror P=W conjecture 2.1. The weight filtration 2.2. The perverse filtration 2.3. The mirror P=W conjecture 3. Fano/LG correspondence 3.1. B-side categories 3.2. A-side categories 3.3. From HMS to mirror P=W 4. Extended Fano/LG Correspondence 4.1. The case of two components 4.2. The general case 5. Landau-Ginzburg Hodge numbers 5.1. The smooth case 5.2. The case of two components 5.3. The general case 6. Appendix 6.1. Hodge theory of simplicial varieties 6.2. The combinatorial perverse filtration References

Complexes of stable $\infty$-categories

“…As indicated to us by the authors, this theory will provide a means to introduce cubical categorical diagrams in terms of partially wrapped Fukaya categories which one would expect to mirror the cubical diagrams in §7.1. Variants of this homological mirror symmetry conjecture for categorical cubes were already recorded in [Lee22]. The expected relation to our formulation will be explained below.…”

“…This "mirror symmetry conjecture for cubes" has for example been described in the recent article [Lee22]. Once this type of cubical mirror symmetry is established, our Conjecture 7.3.4 then essentially reduces to a statement that our Fukaya-Seidel complexes from Definition 7.2.5 arise as totalization of the cubical wrapped Fukaya-category diagrams that one expects to associate to a multi-potential Landau-Ginzburg model (cf.…”