×

Homological mirror symmetry for punctured spheres. (English) Zbl 1276.53089

Kontsevich’s homological mirror symmetry conjecture for pairs of Calabi-Yau varieties “predicts an equivalence between the derived category of coherent sheaves of one variety and the derived Fukaya category of the other”. This conjecture has been proved so far for elliptic curves, the quartic \(K_3\) surface and their products. Later this conjecture was extended for Fano varieties, where in many cases the mirrors are Landau-Ginzburg models: pairs \((X, W)\) consisting of a variety \(X\) and a holomorphic function \(W:X\to\mathbb{C}\) called superpotential.
More recently L. Katzarkov suggested that mirror symmetry also extends to surfaces of general type, many of which also admit mirror Landau-Ginzburg models. The first positive result in this setting was established by P. Seidel for the genus-\(2\) curve [J. Algebr. Geom. 20, No. 4, 727–769 (2011; Zbl 1226.14028)]. The argument used by Seidel was then extended to higher-genus curves, to pairs of pants and their higher-dimensional analogs and to Calabi-Yau hypersurfaces in the projective space.
In the work under review the authors study homological mirror symmetry for an open genus-\(0\) curve \(C\), which is the complement of a finite set of \(n\geq 3\) points in \(\mathbb{P}^1\). They describe a Landau-Ginzburg model mirror \((X(n),W)\) to \(C\), where \(X(n)\) is a non-compact toric 3-fold and \(W:X(n)\to \mathbb{C}\) is a superpotential. Then they consider the wrapped Fukaya category of \(C\) and the associated triangulated category \(D\mathcal{W}(C)\). The main result states that “this triangulated category is equivalent to the triangulated category of singularities of the singular fiber \(W^{-1}(0)\) of \((X(n), W)\).” A similar result in the case of unramified cyclic covers of punctured spheres is also proved.
The idea of the proof is inspired by that used by Seidel for the genus-2 curve. Only the symplectic side of the homological mirror symmetry is considered in this work. The other side is generally considered to be out of reach of current technologies for the studied examples, due to the singular nature of the critical locus of \(W\).

MSC:

53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14J33 Mirror symmetry (algebro-geometric aspects)
53D40 Symplectic aspects of Floer homology and cohomology
53D12 Lagrangian submanifolds; Maslov index
18E30 Derived categories, triangulated categories (MSC2010)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

Citations:

Zbl 1226.14028
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Mohammed Abouzaid, Morse homology, tropical geometry, and homological mirror symmetry for toric varieties, Selecta Math. (N.S.) 15 (2009), no. 2, 189 – 270. · Zbl 1204.14019
[2] Mohammed Abouzaid, A geometric criterion for generating the Fukaya category, Publ. Math. Inst. Hautes Études Sci. 112 (2010), 191 – 240. · Zbl 1215.53078
[3] Mohammed Abouzaid, A cotangent fibre generates the Fukaya category, Adv. Math. 228 (2011), no. 2, 894 – 939. · Zbl 1241.53071
[4] M. Abouzaid, D. Auroux, L. Katzarkov, Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces, arXiv:1205.0053. · Zbl 1368.14056
[5] M. Abouzaid, K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, in preparation.
[6] Mohammed Abouzaid and Paul Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010), no. 2, 627 – 718. · Zbl 1195.53106
[7] Mohammed Abouzaid and Ivan Smith, Homological mirror symmetry for the 4-torus, Duke Math. J. 152 (2010), no. 3, 373 – 440. · Zbl 1195.14056
[8] Denis Auroux, Ludmil Katzarkov, and Dmitri Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, Ann. of Math. (2) 167 (2008), no. 3, 867 – 943. · Zbl 1175.14030
[9] Denis Auroux, Ludmil Katzarkov, and Dmitri Orlov, Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves, Invent. Math. 166 (2006), no. 3, 537 – 582. · Zbl 1110.14033
[10] R. Bocklandt, Noncommutative mirror symmetry for punctured surfaces, arXiv:1111.3392. · Zbl 1383.16016
[11] A. Bondal, W. D. Ruan, Mirror symmetry for weighted projective spaces, in preparation.
[12] Alexander I. Efimov, Homological mirror symmetry for curves of higher genus, Adv. Math. 230 (2012), no. 2, 493 – 530. · Zbl 1242.14039
[13] Kwokwai Chan and Naichung Conan Leung, Mirror symmetry for toric Fano manifolds via SYZ transformations, Adv. Math. 223 (2010), no. 3, 797 – 839. · Zbl 1201.14029
[14] Cheol-Hyun Cho, Products of Floer cohomology of torus fibers in toric Fano manifolds, Comm. Math. Phys. 260 (2005), no. 3, 613 – 640. · Zbl 1109.53079
[15] Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, and Eric Zaslow, T-duality and homological mirror symmetry for toric varieties, Adv. Math. 229 (2012), no. 3, 1875 – 1911. · Zbl 1260.14049
[16] Kenji Fukaya, Mirror symmetry of abelian varieties and multi-theta functions, J. Algebraic Geom. 11 (2002), no. 3, 393 – 512. · Zbl 1002.14014
[17] K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian Floer theory on compact toric manifolds: survey, Surveys in Differential Geometry, vol. 17, “Algebra and Geometry: In Memory of C. C. Hsiung”, International Press, 2012, pp. 229-298. · Zbl 1382.53001
[18] K. Fukaya, P. Seidel, and I. Smith, The symplectic geometry of cotangent bundles from a categorical viewpoint, Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 1 – 26. · Zbl 1163.53344
[19] S. Ganatra, Symplectic cohomology and duality for the wrapped Fukaya category, Ph.D. thesis, MIT, 2012.
[20] M. Gross, L. Katzarkov, H. Ruddat, Towards mirror symmetry for varieties of general type, arXiv: 1202.4042. · Zbl 1371.14046
[21] K. Hori, C. Vafa, Mirror symmetry, arXiv:hep-th/0002222. · Zbl 1044.14018
[22] Anton Kapustin, Ludmil Katzarkov, Dmitri Orlov, and Mirroslav Yotov, Homological mirror symmetry for manifolds of general type, Cent. Eur. J. Math. 7 (2009), no. 4, 571 – 605. · Zbl 1200.53079
[23] Anton Kapustin and Dmitri Orlov, Vertex algebras, mirror symmetry, and D-branes: the case of complex tori, Comm. Math. Phys. 233 (2003), no. 1, 79 – 136. · Zbl 1051.17017
[24] Bernhard Keller, Introduction to \?-infinity algebras and modules, Homology Homotopy Appl. 3 (2001), no. 1, 1 – 35. Bernhard Keller, Addendum to: ”Introduction to \?-infinity algebras and modules” [Homology Homotopy Appl. 3 (2001), no. 1, 1 – 35; MR1854636 (2004a:18008a)], Homology Homotopy Appl. 4 (2002), no. 1, 25 – 28. · Zbl 0989.18009
[25] Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151 – 190. · Zbl 1140.18008
[26] Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120 – 139. · Zbl 0846.53021
[27] M. Kontsevich, Lectures at ENS, Paris, Spring 1998, notes taken by J. Bellaiche, J.-F. Dat, I. Marin, G.Racinet, and H. Randriambololona, unpublished.
[28] Maxim Kontsevich and Yan Soibelman, Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 203 – 263. · Zbl 1072.14046
[29] M. Kontsevich and Y. Soibelman, Notes on \?_{\infty }-algebras, \?_{\infty }-categories and non-commutative geometry, Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 153 – 219. · Zbl 1202.81120
[30] K. H. Lin, D. Pomerleano, Global matrix factorizations, arXiv:1101.5847. · Zbl 1285.14019
[31] Valery A. Lunts, Categorical resolution of singularities, J. Algebra 323 (2010), no. 10, 2977 – 3003. · Zbl 1202.18006
[32] D. O. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 240 – 262 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3(246) (2004), 227 – 248. · Zbl 1101.81093
[33] Dmitri Orlov, Matrix factorizations for nonaffine LG-models, Math. Ann. 353 (2012), no. 1, 95 – 108. · Zbl 1243.81178
[34] Alexander Polishchuk and Eric Zaslow, Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys. 2 (1998), no. 2, 443 – 470. · Zbl 0947.14017
[35] A. Preygel, Thom-Sebastiani Duality for Matrix Factorizations, arXiv:1101.5834.
[36] Marco Schlichting, Negative \?-theory of derived categories, Math. Z. 253 (2006), no. 1, 97 – 134. · Zbl 1090.19002
[37] P. Seidel, Homological mirror symmetry for the quartic surface, arXiv:math/0310414. · Zbl 1334.53091
[38] Paul Seidel, Fukaya categories and deformations, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 351 – 360. · Zbl 1014.53052
[39] Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. · Zbl 1159.53001
[40] Paul Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geom. 20 (2011), no. 4, 727 – 769. · Zbl 1226.14028
[41] P. Seidel, Some speculations on Fukaya categories and pair-of-pants decompositions, Surveys in Differential Geometry, vol. 17, “Algebra and Geometry: In Memory of C. C. Hsiung”, International Press, 2012, pp. 411-426.
[42] Nick Sheridan, On the homological mirror symmetry conjecture for pairs of pants, J. Differential Geom. 89 (2011), no. 2, 271 – 367. · Zbl 1255.53065
[43] N. Sheridan, Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space, arXiv: 1111.0632. · Zbl 1344.53073
[44] Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, Mirror symmetry is \?-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243 – 259. · Zbl 0896.14024
[45] R. W. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), no. 1, 1 – 27. · Zbl 0873.18003
[46] Kazushi Ueda, Homological mirror symmetry for toric del Pezzo surfaces, Comm. Math. Phys. 264 (2006), no. 1, 71 – 85. · Zbl 1106.14026
[47] Charles Weibel, The negative \?-theory of normal surfaces, Duke Math. J. 108 (2001), no. 1, 1 – 35. · Zbl 1092.14014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.