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Matematicheskie Zametki, 2020, Volume 108, Issue 1, Pages 33–46
DOI: https://doi.org/10.4213/mzm12795
(Mi mzm12795)
 

This article is cited in 5 scientific papers (total in 5 papers)

$\mathrm P=\mathrm W$ Phenomena

L. Katzarkovab, V. V. Przyjalkowskicb, A. Harderd

a University of Miami
b National Research University "Higher School of Economics", Moscow
c Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
d Lehigh University
Full-text PDF (569 kB) Citations (5)
References:
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.
Keywords: $\mathrm P=\mathrm W$ conjecture, mixed Hodge structure, perverse Leray filtration, mirror symmetry.
Funding agency Grant number
Simons Foundation
European Research Council
National Science Foundation DMS-150908
DMS-1265230
DMS-1201475
OISE-1242272 PASI
National Research University Higher School of Economics
Ministry of Science and Higher Education of the Russian Federation 14.641.31.0001
Bulgarian National Science Fund KP-06-DV-7
Contest «Young Russian Mathematics»
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.
Received: 06.06.2019
Revised: 26.09.2019
English version:
Mathematical Notes, 2020, Volume 108, Issue 1, Pages 39–49
DOI: https://doi.org/10.1134/S0001434620070044
Bibliographic databases:
Document Type: Article
UDC: 512.73
Language: Russian
Citation: L. Katzarkov, V. V. Przyjalkowski, A. Harder, “$\mathrm P=\mathrm W$ Phenomena”, Mat. Zametki, 108:1 (2020), 33–46; Math. Notes, 108:1 (2020), 39–49
Citation in format AMSBIB
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Математические заметки Mathematical Notes
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