Geodesic
$\mathrm P=\mathrm W$ Phenomena
Matematičeskie zametki, Tome 108 (2020) no. 1, pp. 33-46.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Mots-clés : $\mathrm P=\mathrm W$ conjecture, perverse Leray filtration
Keywords: mixed Hodge structure, mirror symmetry. 
@article{MZM_2020_108_1_a2,
     author = {L. Katzarkov and V. V. Przyjalkowski and A. Harder},
     title = {$\mathrm P=\mathrm W$ {Phenomena}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {33--46},
     publisher = {mathdoc},
     volume = {108},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_108_1_a2/}
}
TY  - JOUR
AU  - L. Katzarkov
AU  - V. V. Przyjalkowski
AU  - A. Harder
TI  - $\mathrm P=\mathrm W$ Phenomena
JO  - Matematičeskie zametki
PY  - 2020
SP  - 33
EP  - 46
VL  - 108
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2020_108_1_a2/
LA  - ru
ID  - MZM_2020_108_1_a2
ER  - 
%0 Journal Article
%A L. Katzarkov
%A V. V. Przyjalkowski
%A A. Harder
%T $\mathrm P=\mathrm W$ Phenomena
%J Matematičeskie zametki
%D 2020
%P 33-46
%V 108
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2020_108_1_a2/
%G ru
%F MZM_2020_108_1_a2
L. Katzarkov; V. V. Przyjalkowski; A. Harder. $\mathrm P=\mathrm W$ Phenomena. Matematičeskie zametki, Tome 108 (2020) no. 1, pp. 33-46. http://geodesic.mathdoc.fr/item/MZM_2020_108_1_a2/

[1] M. A. A. de Cataldo, T. Hausel, L. Migliorini, “Topology of Hitchin systems and Hodge theory of character varieties: the case $A_1$”, Ann. of Math. (2), 175:3 (2012), 1329–1407 | DOI | MR | Zbl

[2] M. A. A. de Cataldo, L. Migliorini, “The perverse filtration and the Lefschetz hyperplane theorem”, Ann. of Math. (2), 171:3 (2010), 2089–2113 | DOI | MR | Zbl

[3] M. Kontsevich, Y. Soibelman, “Affine structures and non-Archimedean analytic spaces”, The Unity of Mathematics, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006, 321–385 | MR | Zbl

[4] M. Gross, P. Hacking, S. Keel, “Mirror symmetry for log Calabi–Yau surfaces. I”, Publ. Math. Inst. Hautes Études Sci., 122 (2015), 65–168 | DOI | MR | Zbl

[5] D. Auroux, “Special Lagrangian fibrations, mirror symmetry and Calabi–Yau double covers”, Géométrie différentielle, physique mathématique, mathématiques et société. I, Astérisque, 321, Soc. Math. France, Paris, 2008, 99–128 | MR | Zbl

[6] L. Katzarkov, M. Kontsevich, T. Pantev, “Bogomolov–Tian–Todorov theorems for Landau–Ginzburg models”, J. Differential Geom., 105:1 (2017), 155–117 | MR

[7] A. Harder, L. Katzarkov, V. Przyjalkowski, Landau–Ginzburg Models and $\mathrm P=\mathrm W$ Conjecture, Preprint, 2019

[8] D. Auroux, “Mirror symmetry and $T$-duality in the complement of an anticanonical divisor”, J. Gökova Geom. Topol. GGT, 1 (2007), 51–91 | MR | Zbl

[9] D. Auroux, “Special Lagrangian fibrations, wall-crossing, and mirror symmetry”, Geometry, Analysis, and Algebraic Geometry, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2008, 1–47 | DOI | MR

[10] M. Abouzaid, D. Auroux, L. Katzarkov, “Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces”, Publ. Math. Inst. Hautes Études Sci., 123 (2016), 199–282 | DOI | MR | Zbl

[11] C. Simpson, The Dual Boundary Complex of the $SL_2$ Character Variety of a Punctured Sphere, 2015, arXiv: 1504.05395

[12] A. Harder, Hodge Numbers of Landau–Ginzburg Models, 2017, arXiv: 1708.01174

[13] V. Lunts, V. Przyjalkowski, “Landau–Ginzburg Hodge numbers for mirrors of del Pezzo surfaces”, Adv. Math., 329 (2018), 189–216 | DOI | MR | Zbl

[14] D. Auroux, L. Katzarkov, D. Orlov, “Mirror symmetry for Del Pezzo surfaces: vanishing cycles and coherent sheaves”, Inv. Math., 166:3 (2012), 537–582 | DOI | MR

[15] V. V. Przhiyalkovskii, “Toricheskie modeli Landau–Ginzburga”, UMN, 73:6 (444) (2018), 95–190 | DOI | MR

[16] V. V. Przhiyalkovskii, “Kompaktifikatsii Kalabi–Yau toricheskikh modelei Landau–Ginzburga gladkikh trekhmernykh mnogoobrazii Fano”, Matem. sb., 208:7 (2017), 84–108 | DOI | MR | Zbl

[17] V. V. Przhiyalkovskii, “Slabye modeli Landau–Ginzburga gladkikh trekhmernykh mnogoobrazii Fano”, Izv. RAN. Ser. matem., 77:4 (2013), 135–160 | DOI | MR | Zbl

[18] V. Przyjalkowski, C. Shramov, “On Hodge numbers of complete intersections and Landau–Ginzburg models”, Int. Math. Res. Not. IMRN, 2015:21 (2015), 11302–11332 | DOI | MR | Zbl

[19] I. Cheltsov, V. Przyjalkowski, Kontsevich–Pantev Conjecture for Fano Threefolds, 2018, arXiv: 1809.09218

[20] Y. Shamoto, “Hodge–Tate conditions for Landau–Ginzburg models”, Publ. Res. Inst. Math. Sci., 54:3 (2018), 469–518 | DOI | MR

[21] C. Voisin, Hodge Theory and Complex Algebraic Geometry. I, Cambridge Stud. Adv. Math., 76, Cambridge Univ. Press, Cambridge, 2002 | MR | Zbl

[22] J. Carlson, S. Müller-Stach, C. Peters, Period Mappings and Period Domains, Cambridge Stud. Adv. Math., 85, Cambridge Univ. Press, Cambridge, 2003 | MR | Zbl