Homological Mirror Symmetry for the genus two curve
Author:
Paul Seidel
Journal:
J. Algebraic Geom. 20 (2011), 727-769
DOI:
https://doi.org/10.1090/S1056-3911-10-00550-3
Published electronically:
January 26, 2010
MathSciNet review:
2819674
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We prove a form of Homological Mirror Symmetry for the genus two curve (on the symplectic or $A$-model side).
References
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- P. Seidel. Homological mirror symmetry for the quartic surface. Preprint math.SG/0310414, 2003.
- Paul Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 (2003), no. 5, 1003–1063. MR 1978046, DOI https://doi.org/10.1016/S0040-9383%2802%2900028-9
- Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780
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References
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- M. Akaho. Intersection theory for Lagrangian immersions. Math. Res. Lett., 12:543–550, 2005. MR 2155230 (2006h:53093)
- M. Akaho and D. Joyce. Immersed Lagrangian Floer theory. Preprint arXiv: 0803.0717.
- V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko. Singularities of differentiable maps. Birkhäuser, 1988. MR 966191 (89g:58024)
- P. Aspinwall and S. Katz. Computation of superpotentials for D-branes. Comm. Math. Phys., 264:227–253, 2006. MR 2212222 (2007b:81178)
- M. Atiyah. Vector bundles over an elliptic curve. Proc. London Math. Soc. 7:414–452, 1957. MR 0131423 (24:A1274)
- D. Auroux, L. Katzarkov and D. Orlov. Mirror symmetry for weighted projective planes and their noncommutative deformations. Ann. of Math. (2), 167:867–943, 2008. MR 2415388 (2009f:53142)
- D. Auroux. Mirror symmetry and T-duality in the complement of an anticanonical divisor. Preprint arXiv:0706.3207. MR 2386535 (2009f:53141)
- P. Balmer and M. Schlichting. Idempotent completion of triangulated categories. J. Algebra, 236:819–834, 2001. MR 1813503 (2002a:18013)
- V. Baranovsky and J. Pecharich. On equivalences of derived and singular categories. Preprint arXiv:0907.3717, 2009.
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- A. Efimov. Homological mirror symmetry for curves of higher genus. Preprint arXiv:0907.3903, 2009.
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- K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono. Lagrangian Floer theory on compact toric manifolds. II: Bulk deformations. Preprint arXiv:0810.5654.
- K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono. Lagrangian intersection Floer theory - anomaly and obstruction. In press.
- Kenji Fukaya. Mirror symmetry of abelian varieties and multi-theta functions. J. Algebraic Geom., 11:393–512, 2002. MR 1894935 (2003m:14059)
- B. Fang, C.-C. Liu, D. Treumann and E. Zaslow. The coherent-constructible correspondence and Homological Mirror Symmetry for toric varieties. Preprint arXiv:0901.4276, 2009.
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- G. Hochschild, B. Kostant, and A. Rosenberg. Differential forms on regular affine algebras. Trans. Amer. Math. Soc., 102:383–408, 1962. MR 0142598 (26:167)
- D. Huybrechts and R. Thomas. $\mathbb {P}$-objects and autoequivalences of derived categories. Math. Res. Lett. 13:87–98, 2006. MR 2200048 (2006j:14015)
- L. Katzarkov. Birational geometry and homological mirror symmetry. In Real and complex singularities, pages 176–206. World Sci. Publ., 2007. MR 2336686 (2008g:14062)
- A. Kapustin, L. Katzarkov, D. Olov and M. Yotov. Homological Mirror Symmetry for manifolds of general type. Preprint, 2009.
- M. Kontsevich. Deformation quantization of Poisson manifolds I. Lett. Math. Phys., 66:157–216, 2003. MR 2062626 (2005i:53122)
- M. Kontsevich. Homological algebra of mirror symmetry. In Proceedings of the International Congress of Mathematicians (Zürich, 1994), pages 120–139. Birkhäuser, 1995. MR 1403918 (97f:32040)
- M. Kontsevich and Y. Soibelman. Homological mirror symmetry and torus fibrations. In Symplectic geometry and mirror symmetry, pages 203–263. World Scientific, 2001. MR 1882331 (2003c:32025)
- T. Pantev, L. Katzarkov, M. Kontsevich. Hodge theoretic aspects of mirror symmetry. Preprint arXiv:0806.0107. MR 2483750 (2009j:14052)
- B. Keller. Introduction to $A$-infinity algebras and modules. Homology Homotopy Appl. (electronic), 3:1–35, 2001. MR 1854636 (2004a:18008a)
- T. Lada and M. Markl. Strongly homotopy Lie algebras. Comm. Algebra, 23:2147–2161, 1995. MR 1327129 (96d:16039)
- M. Markl. Transferring $A_\infty$ (strongly homotopy associative) structures. Rend. Circ. Mat. Palermo (2) Suppl., 79:139–151, 2006. MR 2287133 (2007j:18016)
- D. McDuff. The structure of rational and ruled symplectic $4$-manifolds. Journ. Amer. Math. Soc., 3:679–712, 1990. Erratum: same journal, 5:987–988, 1992. MR 1049697 (91k:58042)
- S. Mehrotra. Triangulated categories of singularities, matrix factorizations and LG models. PhD thesis, Univ. of Pennsylvania, 2005.
- S. Mukai. Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves. Nagoya Math. J. 81:153–175, 1981. MR 607081 (82f:14036)
- I. Nakamura. Hilbert schemes of abelian group orbits. J. Algebraic Geom., 10:757–779, 2001. MR 1838978 (2002d:14006)
- Y.-G. Oh. Floer cohomology of Lagrangian intersections and pseudo-holomorphic discs I. Comm. Pure Appl. Math., 46:949–994, 1993. MR 1223659 (95d:58029a)
- D. Orlov. D-branes of type B in Landau-Ginzburg models and mirror symmetry for the genus two curve. Conference talk, IAS Princeton, 3/12/2008.
- D. Orlov. Formal completions and idempotent completions of triangulated categories of singularities. Preprint arXiv:0901.1859.
- D. Orlov. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc. Steklov Inst. Math., 246:227–248, 2004. MR 2101296 (2006i:81173)
- A. Polishchuk and E. Zaslow. Categorical mirror symmetry: the elliptic curve. Adv. Theor. Math. Phys., 2:443–470, 1998. MR 1633036 (99j:14034)
- A. Polishchuk. Massey and Fukaya products on elliptic curves. Adv. Theor. Math. Phys., 4:1187–1207, 2000. MR 1894854 (2003h:14058)
- P. Rossi. Gromov-Witten theory of orbicurves, the space of tri-polynomials and Symplectic Field Theory of Seifert fibrations. Preprint arXiv:0808.2626.
- M. Schwarz. A quantum cup-length estimate for symplectic fixed points. Invent. Math., 133:353–397, 1998. MR 1632778 (99e:58041)
- P. Seidel. Python code for computing $A_\infty$-deformations, 2008. Can be downloaded from the author’s homepage, http://math.mit.edu/˜seidel.
- P. Seidel. Homological mirror symmetry for the quartic surface. Preprint math.SG/0310414, 2003.
- P. Seidel. A long exact sequence for symplectic Floer cohomology. Topology, 42:1003–1063, 2003. MR 1978046 (2004d:53105)
- P. Seidel. Fukaya categories and Picard-Lefschetz theory. European Math. Soc., 2008. MR 2441780 (2009f:53143)
- P. Seidel and R. Thomas. Braid group actions on derived categories of coherent sheaves. Duke Math. Jour., 108:37–108, 2001. MR 1831820 (2002e:14030)
- A. Subotic. Ph.D. thesis, MIT, in preparation.
- A. Takahashi. Weighted projective lines associated to regular systems of weights of dual type. Preprint arXiv:0711.3907.
- J.-C. Tougeron. Idéaux des fonctions différentiables. Ann. Inst. Fourier, 18:177–240, 1970. MR 0240826 (39:2171)
- A. Quintero Velez. McKay correspondence for Landau-Ginzburg models. Preprint arXiv:0711.4774.
- K. Wehrheim and C. Woodward. Functoriality for Lagrangian correspondences in Floer theory. Preprint arXiv:0708.2851.
- C. A. Weibel. An introduction to homological algebra. Cambridge Univ. Press, 1994. MR 1269324 (95f:18001)
Additional Information
Paul Seidel
Affiliation:
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 2-270, Cambridge, Massachusetts 02139
MR Author ID:
613874
Email:
seidel@math.mit.edu
Received by editor(s):
December 17, 2008
Received by editor(s) in revised form:
August 11, 2009
Published electronically:
January 26, 2010
Additional Notes:
The author was partially supported through NSF grant DMS-0652620