P=W Phenomena

L. Katzarkov, V. V. Przyjalkowski, A. Harder

Research output: Contribution to journalArticlepeer-review

Abstract

Abstract: In this paper, we describe recent work towards the mirror P=W 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 P=W conjecture provides a possible bridge between the mirror P=W conjecture and the well-known P=W conjecture in non-Abelian Hodge theory.

Original languageEnglish (US)
Pages (from-to)39-49
Number of pages11
JournalMathematical Notes
Volume108
Issue number1-2
DOIs
StatePublished - Jul 1 2020

Keywords

  • $\mathrm P=\mathrm W$ conjecture
  • mirror symmetry
  • mixed Hodge structure
  • perverse Leray filtration

ASJC Scopus subject areas

  • Mathematics(all)

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