Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-14T05:20:31.389Z Has data issue: false hasContentIssue false
  • English
  • Français

Toric systems and mirror symmetry

Part of: Special varieties Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  28 August 2013

Raf Bocklandt
Raf Bocklandt*
Affiliation:
School of Mathematics and Statistics, Herschel Building, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In their paper [Exceptional sequences of invertible sheaves on rational surfaces, Compositio Math. 147 (2011), 1230–1280], Hille and Perling associate to every cyclic full strongly exceptional sequence of line bundles on a toric weak del Pezzo surface a toric system, which defines a new toric surface. We interpret this construction as an instance of mirror symmetry and extend it to a duality on the set of toric weak del Pezzo surfaces equipped with a cyclic full strongly exceptional sequence.

Keywords

dimershelicesdel Pezzo surfacestoric systemshomological mirror symmetry

MSC classification

Secondary: 14M25: Toric varieties, Newton polyhedra 14J33: Mirror symmetry 14J45: Fano varieties 14J32: Calabi-Yau manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 11 , November 2013 , pp. 1839 - 1855
Copyright
© The Author(s) 2013 

References

Abouzaid, M., Morse homology, tropical geometry, and homological mirror symmetry for toric varieties, Selecta Math. 15 (2009), 189–270.CrossRefGoogle Scholar
Abouzaid, M., Auroux, D., Efimov, A., Katzarkov, L. and Orlov, D., Homological mirror symmetry for punctured spheres, Preprint (2011), arXiv:1103.4322.Google Scholar
Abouzaid, M. and Seidel, P., An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010), 627–718.Google Scholar
Auroux, D., Katzarkov, L. and Orlov, D., Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves, Invent. Math. 166 (2006), 537–582.CrossRefGoogle Scholar
Auroux, D., Katzarkov, L. and Orlov, D., Mirror symmetry for weighted projective planes and their noncommutative deformations, Ann. of Math. (2) 167 (2008), 867–943.CrossRefGoogle Scholar
Bender, M. and Mozgovoy, S., Crepant resolutions and brane tilings II: tilting bundles, Preprint (2009), arXiv:0909.2013.Google Scholar
Bocklandt, R., Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra 212 (2008), 14–32.Google Scholar
Bocklandt, R., Calabi–Yau algebras and weighted quiver polyhedra, Math. Z., to appear, Preprint (2009), arXiv:0905.0232.Google Scholar
Bocklandt, R., Noncommutative mirror symmetry for punctured surfaces, Preprint (2011), arXiv:1111.3392.Google Scholar
Bocklandt, R., Generating toric noncommutative crepant resolutions, J. Algebra 364 (2012), 119–147.Google Scholar
Bocklandt, R., Consistency conditions for dimer models, Glasg. Math. J. 54 (2012), 429–447.Google Scholar
Bridgeland, T., t-structures on some local Calabi-Yau varieties, J. Algebra 289 (2005), 453–483.CrossRefGoogle Scholar
Bridgeland, T. and Stern, D., Helices on del Pezzo surfaces and tilting Calabi–Yau algebras, Preprint (2009), arXiv:0909.1732.Google Scholar
Broomhead, N., Dimer models and Calabi-Yau algebras, Mem. Amer. Math. Soc. 215 (2011), 1011.Google Scholar
Craw, A. and Ishii, A., Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124 (2004), 259–307.Google Scholar
Davison, B., Consistency conditions for brane tilings, J. Algebra 338 (2011), 1–23.Google Scholar
Feng, B., He, Y., Kennaway, K. and Vafa, C., Dimer models from mirror symmetry and quivering amoebae, Adv. Theor. Math. Phys. 12 (2008), 489–545.Google Scholar
Ginzburg, V., Calabi–Yau algebras, Preprint (2006), arXiv:math/0612139.Google Scholar
Hanany, A. and Vegh, D., Quivers, tiling, branes and rhombi, J. High Energy Phys. 10 (2007), 029; 35 pages.Google Scholar
Hanany, A., Herzog, C. P. and Vegh, D., Brane tilings and exceptional collections, J. High Energy Phys. 07 (2006), 001; 44 pages, doi:10.1088/1126-6708/2006/07/001.Google Scholar
Hille, L. and Perling, M., Exceptional sequences of invertible sheaves on rational surfaces, Compositio Math. 147 (2011), 1230–1280.Google Scholar
Hille, L. and Perling, M., Tilting bundles on rational surfaces and quasi-hereditary algebras, Preprint (2011), arXiv:1110.5843.Google Scholar
Ishii, A. and Ueda, K., On moduli spaces of quiver representations associated with brane tilings, in Higher dimensional algebraic varieties and vector bundles, RIMS Kôkyûroku Bessatsu, vol. B9 (Research Institute for Mathematical Sciences, Kyoto, 2008), 127–141.Google Scholar
Ishii, A. and Ueda, K., Dimer models and exceptional collections, Preprint (2009), arXiv:0911.4529.Google Scholar
Ishii, A. and Ueda, K., A note on consistency conditions on dimer models, in Higher dimensional algebraic geometry, RIMS Kôkyûroku Bessatsu, vol. B24 (Research Institute for Mathematical Sciences, Kyoto, 2011), 143–164.Google Scholar
Katzarkov, L., Birational geometry and homological mirror symmetry, in Real and complex singularities (World Scientific, Hackensack, NJ, 2007), 176–206.CrossRefGoogle Scholar
Keller, B., Introduction to A-infinity algebras and modules, Homology, Homotopy Appl. 3 (2001), 1–35.CrossRefGoogle Scholar
Kenyon, R., An introduction to the dimer model, in School and conference on probability theory, ICTP Lecture Notes, vol. 17 (Abdus Salam International Centre for Theoretical Physics, Trieste, 2004), 267–304.Google Scholar
King, A., Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515–530.Google Scholar
Kontsevich, M., Homological algebra of mirror symmetry, in Proceedings of the International Congress of Mathematicians, Zürich 1994 (Birkhäuser, Basel, 1995), 120–139.Google Scholar
Mozgovoy, S., Crepant resolutions and brane tilings I: toric realization, Preprint (2009), arXiv:0908.3475.Google Scholar
Mozgovoy, S. and Reineke, M., On the noncommutative Donaldson–Thomas invariants arising from brane tilings, Adv. Math. 223 (2010), 1521–1544.CrossRefGoogle Scholar
Orlov, D., Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Proc. Steklov Inst. Math. 246 (2004), 227–248.Google Scholar
Orlov, D., Triangulated categories of singularities and equivalences between Landau–Ginzburg models, Sb. Math. 197 (2006), 1827–1840.Google Scholar
Perling, M., Examples for exceptional sequences of invertible sheaves on rational surfaces, Sémin. Congr. 25 (2010), 369–389.Google Scholar
Van den Bergh, M., Non-commutative crepant resolutions, in The legacy of Niels Hendrik Abel: the Abel Bicentennial, Oslo 2002 (Springer, Berlin, 2002), 749–770.Google Scholar