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Generalized Homological Mirror Symmetry and cubics

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Abstract

We discuss an approach to studying Fano manifolds based on Homological Mirror Symmetry. We consider some classical examples from a new point of view.

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References

  1. M. Abouzaid, D. Auroux, and L. Katzarkov, “Homological Mirror Symmetry for Blowups,” Preprint (Massachusetts Inst. Technol., 2008).

  2. V. V. Golyshev, “Classification Problems and Mirror Duality,” in Surveys in Geometry and Number Theory, Ed. by N. Young (Cambridge Univ. Press, Cambridge, 2007), LMS Lect. Note Ser. 338, pp. 88–121; arXiv:math/0510287.

    Google Scholar 

  3. V. A. Iskovskikh, “Fano 3-folds. I,” Izv. Akad. Nauk SSSR, Ser. Mat. 41(3), 516–562 (1977) [Math. USSR, Izv. 11, 485–527 (1977)]; “Fano 3-folds. II,” Izv. Akad. Nauk SSSR, Ser. Mat. 42 (3), 506–549 (1978) [Math. USSR, Izv. 12, 469–506 (1978)].

    MATH  MathSciNet  Google Scholar 

  4. V. A. Iskovskikh and Yu. I. Manin, “Three-Dimensional Quartics and Counterexamples to the Lüroth Problem,” Mat. Sb. 86(1), 140–166 (1971) [Math. USSR, Sb. 15, 141–166 (1971)].

    MathSciNet  Google Scholar 

  5. K. Kato and Ch. Nakayama, “Log Betti Cohomology, Log Étale Cohomology, and Log de Rham Cohomology of Log Schemes over ℂ,” Kodai Math. J. 22(2), 161–186 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  6. L. Katzarkov, M. Kontsevich, and T. Pantev, “Hodge Theoretic Aspects of Mirror Symmetry,” arXiv: 0806.0107.

  7. L. Katzarkov and V. Przyjalkowski, “Generalized Homological Mirror Symmetry, Perverse Sheaves and Questions of Rationality,” in Topology of Stratified Spaces, Berkeley, MSRI, Sept. 2008, Ed. by F. Bogomolov, A. Libgober, and S. Cappell (Cambridge Univ. Press, Cambridge, 2009), Math. Sci. Res. Inst. Publ. (in press).

    Google Scholar 

  8. C. Leung, “Geometric Aspects of Mirror Symmetry (with SYZ for Rigid CY Manifolds),” Proc. Int. Congr. Chin. Math. 2001; arXiv:math/0204168.

  9. D. O. Orlov, “Projective Bundles, Monoidal Transformations, and Derived Categories of Coherent Sheaves,” Izv. Ross. Akad. Nauk, Ser. Mat. 56(4), 852–862 (1992) [Russ. Acad. Sci., Izv. Math. 41 (1), 133–141 (1993)].

    Google Scholar 

  10. V. Przyjalkowski, “On Landau-Ginzburg Models for Fano Varieties,” Commun. Number Theory Phys. 1(4), 713–728 (2008); arXiv: 0707.3758.

    MATH  MathSciNet  Google Scholar 

  11. S. Sethi, “Supermanifolds, Rigid Manifolds and Mirror Symmetry,” Nucl. Phys. B 430, 31–50 (1994); arXiv: hep-th/9404186.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Miami, Coral Gables, FL, 33124, USA

    L. Katzarkov

  2. Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    V. Przyjalkowski

Authors
  1. L. Katzarkov
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  2. V. Przyjalkowski
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To the cherished memory of our unforgettable teacher V.A. Iskovskikh

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Katzarkov, L., Przyjalkowski, V. Generalized Homological Mirror Symmetry and cubics. Proc. Steklov Inst. Math. 264, 87–95 (2009). https://doi.org/10.1134/S0081543809010118

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  • DOI: https://doi.org/10.1134/S0081543809010118

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