Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

Abouzaid, Mohammed; Auroux, Denis; Katzarkov, Ludmil

doi:10.1007/s10240-016-0081-9

Abouzaid, Mohammed ¹ ; Auroux, Denis ² ; Katzarkov, Ludmil ³

¹ Department of Mathematics, Columbia University 10027 New York NY USA
² Department of Mathematics, UC Berkeley 94720-3840 Berkeley CA USA
³ Fakultät für Mathematik, University of Vienna 1090 Vienna Austria

Publications Mathématiques de l'IHÉS, Tome 123 (2016), pp. 199-282.

Résumé

We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V \times C$ along $H \times 0$ , under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to $H$ . The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.

MR Zbl

DOI : 10.1007/s10240-016-0081-9

Mots clés : Modulus Space, Toric Variety, Exceptional Divisor, Maslov Index, Instanton Correction

Affiliations des auteurs :

Abouzaid, Mohammed ¹ ; Auroux, Denis ² ; Katzarkov, Ludmil ³

@article{PMIHES_2016__123__199_0,
     author = {Abouzaid, Mohammed and Auroux, Denis and Katzarkov, Ludmil},
     title = {Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {199--282},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {123},
     year = {2016},
     doi = {10.1007/s10240-016-0081-9},
     zbl = {1368.14056},
     mrnumber = {3502098},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-016-0081-9/}
}

TY  - JOUR
AU  - Abouzaid, Mohammed
AU  - Auroux, Denis
AU  - Katzarkov, Ludmil
TI  - Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces
JO  - Publications Mathématiques de l'IHÉS
PY  - 2016
SP  - 199
EP  - 282
VL  - 123
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://archive.numdam.org/articles/10.1007/s10240-016-0081-9/
DO  - 10.1007/s10240-016-0081-9
LA  - en
ID  - PMIHES_2016__123__199_0
ER  -

%0 Journal Article
%A Abouzaid, Mohammed
%A Auroux, Denis
%A Katzarkov, Ludmil
%T Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces
%J Publications Mathématiques de l'IHÉS
%D 2016
%P 199-282
%V 123
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U http://archive.numdam.org/articles/10.1007/s10240-016-0081-9/
%R 10.1007/s10240-016-0081-9
%G en
%F PMIHES_2016__123__199_0

Abouzaid, Mohammed; Auroux, Denis; Katzarkov, Ludmil. Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces. Publications Mathématiques de l'IHÉS, Tome 123 (2016), pp. 199-282. doi : 10.1007/s10240-016-0081-9. http://archive.numdam.org/articles/10.1007/s10240-016-0081-9/

Bibliographie
Cité par

[1.] M. Abouzaid, Family Floer cohomology and mirror symmetry, in Proceedings of the 2014 ICM, vol. II, p. 815, | arXiv

[2.] M. Abouzaid and D. Auroux, Homological mirror symmetry for hypersurfaces in ${(C^{*})}^{n}$ , in preparation.

[3.] Abouzaid, M.; Auroux, D.; Efimov, A. I.; Katzarkov, L.; Orlov, D. Homological mirror symmetry for punctured spheres, J. Am. Math. Soc., Volume 26 (2013), pp. 1051-1083 | DOI | MR | Zbl

[4.] M. Abouzaid and S. Ganatra, Generating Fukaya categories of LG models, in preparation.

[5.] M. Abouzaid and P. Seidel, Lefschetz fibration methods in wrapped Floer cohomology, in preparation.

[6.] Auroux, D. Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol., Volume 1 (2007), pp. 51-91 | MR | Zbl

[7.] Auroux, D. Special Lagrangian fibrations, wall-crossing, and mirror symmetry, Surveys in Differential Geometry (2009), pp. 1-47 | Zbl

[8.] Auroux, D. Infinitely many monotone Lagrangian tori in $R^{6}$ , Invent. Math., Volume 201 (2015), pp. 909-924 | DOI | MR | Zbl

[9.] Bondal, A.; Orlov, D. Derived categories of coherent sheaves, Proc. International Congress of Mathematicians, vol. II (2002), pp. 47-56 | Zbl

[10.] Castaño-Bernard, R.; Matessi, D. Some piece-wise smooth Lagrangian fibrations, Rend. Semin. Mat. (Torino), Volume 63 (2005), pp. 223-253 | MR | Zbl

[11.] Castaño-Bernard, R.; Matessi, D. Lagrangian 3-torus fibrations, J. Differ. Geom., Volume 81 (2009), pp. 483-573 | DOI | MR | Zbl

[12.] Chan, K.; Lau, S.-C.; Leung, N. C. SYZ mirror symmetry for toric Calabi-Yau manifolds, J. Differ. Geom., Volume 90 (2012), pp. 177-250 | DOI | MR | Zbl

[13.] Cho, C.-H. Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus, Int. Math. Res. Not., Volume 2004 (2004), pp. 1803-1843 | DOI | MR | Zbl

[14.] Cho, C.-H. Products of Floer cohomology of torus fibers in toric Fano manifolds, Commun. Math. Phys., Volume 260 (2005), pp. 613-640 | DOI | MR | Zbl

[15.] Cho, C.-H.; Oh, Y.-G. Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math., Volume 10 (2006), pp. 773-814 | DOI | MR | Zbl

[16.] P. Clarke, Duality for toric Landau-Ginzburg models, | arXiv

[17.] Efimov, A. I. Homological mirror symmetry for curves of higher genus, Adv. Math., Volume 230 (2012), pp. 493-530 | DOI | MR | Zbl

[18.] Fukaya, K. Floer homology for families—a progress report, Integrable Systems, Topology, and Physics (2002), pp. 33-68 | DOI | Zbl

[19.] Fukaya, K. Cyclic symmetry and adic convergence in Lagrangian Floer theory, Kyoto J. Math., Volume 50 (2010), pp. 521-590 | DOI | MR | Zbl

[20.] Fukaya, K.; Oh, Y.-G.; Ohta, H.; Ono, K. Lagrangian Intersection Floer Theory: Anomaly and Obstruction I and II (2009) | Zbl

[21.] Fukaya, K.; Oh, Y.-G.; Ohta, H.; Ono, K. Lagrangian Floer theory on compact toric manifolds I, Duke Math. J., Volume 151 (2010), pp. 23-174 | DOI | MR | Zbl

[22.] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian surgery and holomorphic discs, Chapter 10 of [20], available at: http://www.math.kyoto-u.ac.jp/~fukaya/fukaya.html.

[23.] Gross, M. Topological mirror symmetry, Invent. Math., Volume 144 (2001), pp. 75-137 | DOI | MR | Zbl

[24.] Gross, M. Examples of special Lagrangian fibrations, Symplectic Geometry and Mirror Symmetry (2001), pp. 81-109 | DOI | Zbl

[25.] M. Gross, L. Katzarkov and H. Ruddat, Towards mirror symmetry for varieties of general type, | arXiv

[26.] Gross, M.; Siebert, B. Mirror symmetry via logarithmic degeneration data I, J. Differ. Geom., Volume 72 (2006), pp. 169-338 | DOI | MR | Zbl

[27.] Gross, M.; Siebert, B. From real affine geometry to complex geometry, Ann. Math., Volume 174 (2011), pp. 1301-1428 | DOI | MR | Zbl

[28.] Hori, K. Mirror symmetry and quantum geometry, Proc. ICM, vol. III (2002), pp. 431-443 | Zbl

[29.] K. Hori and C. Vafa, Mirror symmetry, | arXiv

[30.] D. Joyce, Lectures on Calabi-Yau and special Lagrangian geometry, | arXiv

[31.] Kadeisvili, T. On the theory of homology of fiber spaces, Usp. Mat. Nauk, Volume 35 (1980), pp. 183-188 | MR

[32.] Kapustin, A.; Katzarkov, L.; Orlov, D.; Yotov, M. Homological mirror symmetry for manifolds of general type, Cent. Eur. J. Math., Volume 7 (2009), pp. 571-605 | MR | Zbl

[33.] Katzarkov, L. Birational geometry and homological mirror symmetry, Real and Complex Singularities (2007), pp. 176-206 | DOI | Zbl

[34.] Kontsevich, M. Homological algebra of mirror symmetry, Proc. International Congress of Mathematicians (1995), pp. 120-139 | DOI | Zbl

[35.] M. Kontsevich, Lectures at ENS, Paris, Spring 1998, notes taken by J. Bellaiche, J.-F. Dat, I. Marin, G. Racinet and H. Randriambololona, unpublished.

[36.] Kontsevich, M.; Soibelman, Y. Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (2001), pp. 203-263 | DOI | Zbl

[37.] Kontsevich, M.; Soibelman, Y. Affine structures and non-Archimedean analytic spaces, The Unity of Mathematics (2006), pp. 321-385 | DOI | Zbl

[38.] Lekili, Y.; Maydanskiy, M. The symplectic topology of some rational homology balls, Comment. Math. Helv., Volume 89 (2014), pp. 571-596 | DOI | MR | Zbl

[39.] S. Mau, K. Wehrheim and C. Woodward, $A_{\infty}$ functors for Lagrangian correspondences, preprint, | arXiv

[40.] McLean, R. C. Deformations of calibrated submanifolds, Commun. Anal. Geom., Volume 6 (1998), pp. 705-747 | DOI | MR | Zbl

[41.] Mikhalkin, G. Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology, Volume 43 (2004), pp. 1035-1065 | DOI | MR | Zbl

[42.] Milnor, J. Spin structures on manifolds, Enseign. Math., Volume 9 (1963), pp. 198-203 | MR | Zbl

[43.] Orlov, D. Triangulated categories of singularities and equivalences between Landau-Ginzburg models, Sb. Math., Volume 197 (2006), pp. 1827-1840 | DOI | MR | Zbl

[44.] H. Rullgård, Polynomial amoebas and convexity, preprint (2001).

[45.] Seidel, P. A long exact sequence for symplectic Floer cohomology, Topology, Volume 42 (2003), pp. 1003-1063 | DOI | MR | Zbl

[46.] Seidel, P. Fukaya categories and Picard-Lefschetz theory, Zurich Lect. in Adv. Math. (2008) | Zbl

[47.] Seidel, P. Homological mirror symmetry for the genus two curve, J. Algebraic Geom., Volume 20 (2011), pp. 727-769 | DOI | MR | Zbl

[48.] Seidel, P. Fukaya $A_{\infty}$ -structures associated to Lefschetz fibrations I, J. Symplectic Geom., Volume 10 (2012), pp. 325-388 | DOI | MR | Zbl

[49.] Sheridan, N. On the homological mirror symmetry conjecture for pairs of pants, J. Differ. Geom., Volume 89 (2011), pp. 271-367 | DOI | MR | Zbl

[50.] Smith, I. Floer cohomology and pencils of quadrics, Invent. Math., Volume 189 (2012), pp. 149-250 | DOI | MR | Zbl

[51.] Strominger, A.; Yau, S.-T.; Zaslow, E. Mirror symmetry is T-duality, Nucl. Phys. B, Volume 479 (1996), pp. 243-259 | DOI | MR | Zbl

[52.] Tu, J. On the reconstruction problem in mirror symmetry, Adv. Math., Volume 256 (2014), pp. 449-478 | DOI | MR | Zbl

[53.] K. Wehrheim and C. T. Woodward, Exact triangle for fibered Dehn twists, | arXiv

Cité par Sources :