Abstract
In this paper, we describe recent work towards the mirror \(\mathrm P=\mathrm W\) 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 \(\mathrm P=\mathrm W\) conjecture provides a possible bridge between the mirror \(\mathrm P=\mathrm W\) conjecture and the well-known \(\mathrm P=\mathrm W\) conjecture in non-Abelian Hodge theory.
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Notes
Note that this agrees with the definition of [2] up to a shift by \(j\).
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Acknowledgments
The authors would like to thank Valery Lunts and Tony Pantev for enlightening conversations.
Funding
A. Harder was supported during part of this work by the Simons Collaboration in Homological Mirror Symmetry. L. Katzarkov was supported by Simons research grant, NSF DMS 150908, ERC Gemis, DMS-1265230, DMS-1201475, OISE-1242272 PASI, Simons collaborative Grant—HMS, Simons investigator grant—HMS; he was supported in part by Laboratory of Mirror Symmetry at National Research University Higher School of Economics, by the Russian Federation Government under grant 14.641.31.0001, and by National Science Fund of Bulgaria, National Scientific Program “Excellent Research and People for the Development of European Science” (VIHREN), project KP-06-DV-7. V. Przyjalkowski was supported in part by Laboratory of Mirror Symmetry at National Research University Higher School of Economics, and by the the Russian Federation Government under grant 14.641.31.0001. He is “Young Russian Mathematics” award winner and would like to thank its sponsors and jury.
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Katzarkov, L., Przyjalkowski, V.V. & Harder, A. \(\mathrm P=\mathrm W\) Phenomena. Math Notes 108, 39–49 (2020). https://doi.org/10.1134/S0001434620070044
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DOI: https://doi.org/10.1134/S0001434620070044