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Homological mirror symmetry for toric del Pezzo surfaces. (English) Zbl 1106.14026

The author proves the homological mirror conjecture for toric del Pezzo surfaces. The main theorem states indeed the equivalence between the bounded derived category of coherent sheaves on a toric del Pezzo surface \(X\) and the derived category of the directed Fukaya category of the mirror \(W\) of \(X\). This extends works of P. Seidel [in: Symplectic Geometry and Mirror Symmetry, Proc. 4th KIAS int. conf. Seoul, 2000, 429–465 (2001; Zbl 1079.14259)], and D. Auroux, L. Katzarkov and D. Orlov [preprint, math.AG/0404281]. The present paper was submitted before the homological mirror conjecture for not necessarily toric del Pezzo surfaces was proved in the preprint of D. Auroux, L. Katzarkov and D. Orlov [preprint, math.AG/0506166].
The explicit proof of the theorem is given for the toric del Pezzo surface \(Y\) obtained as the blow-up of three points in general position in the projective plane. This proof can be performed for any toric del Pezzo surface to obtain the main result.
The author first details the structure of the bounded derived category \(D^{b}\text{Coh}(Y)\) of coherent sheaves on \(Y\), by explicitely giving a full exceptional collection and by calculating all non zero Ext-groups between the exceptional objects of the sequence. On the other hand, on the mirror \(W\), the directed Fukaya category \(\text{Fuk}^{\to}W\) is constructed, explicitely giving the distinguished basis of vanishing cycles and all the Hom-groups between them. The equivalence between \(D^{b}\text{Coh}(Y)\) and \(D^{b}\text{Fuk}^{\to}W\) is then obtained by giving isomorphisms between the Hom-groups of the vanishing cycles and the Ext-groups of the exceptional objects.
Six figures throughout the paper help the reader in following proofs and constructions. A final appendix contains explicit calculations for the compositions of morphisms between the exceptional objects of \(D^{b}\text{Coh}(Y)\).

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J26 Rational and ruled surfaces
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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References:

[1] Auroux, D., Katzarkov, L., Orlov, D.: Mirror symmetry for weighted projective planes and their noncommutative deformations. http://arxiv.org/list/math.AG/0404281, 2004. · Zbl 1175.14030
[2] Auroux, D., Katzarkov, L., Orlov, D.: Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves. http://arxiv.org/list/math.AG/0506166, 2005 · Zbl 1110.14033
[3] Batyrev, Astérisque, 218, 9 (1993) · Zbl 0806.14041
[4] Beilinson, A.: Coherent sheaves on \(####\) and problems in linear algebra. Funkts. Anal. i. Prilozhen. 12, no. 3, 68-69 (1978) · Zbl 0402.14006
[5] Bondal, A.: Representation of associative algebras and coherent sheaves. Izv. Ross. Akad. Nauk Ser. Mat. 53, no. 1, 25-44 (1989)
[6] Eguchi, T.; Hori, K.; Xiong, C.-S., Gravitational quantum cohomology, Int. J. Mod. Phys. A, 12, 1743-1782 (1997) · Zbl 1072.32500
[7] Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory. Preprint, available at http://www.math.kyoto-u.ac.jp/ fukaya, 2000 · Zbl 1181.53003
[8] Givental, A.: Homological geometry and mirror symmetry. In: Proceedings of the International Congress of Mathematicians. Proceedings, Zurich 1994. Basel: Birkhäuser, 1994, pp. 472-480 · Zbl 0863.14021
[9] Hori, K., Iqbal, A., Vafa, C.: D-Branes and mirror symmetry. http://arxiv.org/list/hep-th/0005247, 2000
[10] Hori, K., Vafa, C.: Mirror symmetry. http:// arxiv.org/list/hep-th/0002222, 2000 · Zbl 1044.14018
[11] Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians. Proceedings, Zurich 1994. Basel: Birkhäuser, 1994, pp. 120-139 · Zbl 0846.53021
[12] Kouchnirenko, A.: Polyédres de Newton et nombres de Milnor. Invent. Math. 32, no. 1, 1-31 (1976) · Zbl 0328.32007
[13] Orlov, D.: Projective bundles, monoidal transformations and derived categories of coherent sheaves. Izv. Ross. Akad. Nauk Ser. Math. 56, no. 4, 852-862 (1992) · Zbl 0798.14007
[14] Robbin, J., Salamon D.: Maslov index for path. Topology 32, no. 4, 827-844 (1993) · Zbl 0798.58018
[15] Seidel, P.: Graded Lagrangian submanifolds. Bull. Soc. Math. France 128, no. 1, 103-149 (2000) · Zbl 0992.53059
[16] Seidel, P.: Vanishing cycles and mutations. In: European Congress of Mathematics, Vol. II. Proceedings, Barcelona 2000. Basel: Birkhäuser, 2001, pp. 65-85 · Zbl 1042.53060
[17] Seidel, P.: More about vanishing cycles and mutations. In: Symplectic geometry and mirror symmetry. Proceedings, Seoul 2000. Singapore:World Sci. Publishing, 2001, pp. 429-465 · Zbl 1079.14529
[18] Seidel, P.: Homological mirror symmetry for the quartic surface. http://arxiv.org/list/math.AG/0310414 (2003) · Zbl 1334.53091
[19] Witten, Nucl. Phys. B, 403, 159 (1993) · Zbl 0910.14020 · doi:10.1016/0550-3213(93)90033-L
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