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**Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves.**
*(English)*
Zbl 1110.14033

In the present paper the authors prove the homological mirror symmetry conjecture for Del Pezzo surfaces. Recall that a Landau-Ginzburg model is a pair \((M,W)\), where \(M\) is a non-compact (symplectic and/or complex) manifold and \(W\) is a complex-valued function on \(M\). The idea is that when such a model \((M,W)\) is mirror to a Fano variety \(X\), the complex geometry on \(X\) corresponds to the symplectic geometry of \(M\) and vice versa. The homological mirror symmetry conjecture states that the bounded derived category of coherent sheaves on \(X\) is equivalent to the derived categories of Lagrangian vanishing cycles of \(W\).

Let \(X_K\) be a Del Pezzo surface obtained blowing up \(\mathbb{P}^2\) at \(k\) points. The conjecture is proved by taking as mirror an elliptic fibration \(W_k : M_k \to \mathbb{C}\) with \(k+3\) singular fibers and suitable symplectic form \([B + i \omega]\). Moreover, given a general noncommutative deformation of \(X_K\), there exists a complexified symplectic form \([B + i \omega]\), for which the deformed derived category of \(X_K\) is equivalent to the derived category of Lagrangian vanishing cycles. Conversely, for a generic choice of \([B + i \omega]\), the derived category of Lagrangian vanishing cycles is equivalent to the derived category of coherent sheaves of a noncommutative deformation of a Del Pezzo surface.

In order to prove that, the authors describe in a first time the derived category of \(X_K\) by detailing a strong exceptional collection of objects and morphisms between them. Derived categories of simple degenerations and noncommutative deformations can be described by working on such sets.

In a second time, the construction of the mirror Landau-Ginzburg model is given. The mirror of a given \(X_K\) is an elliptic fibration \(W_k : M_k \to \mathbb{C}\) with \(k+3\) nodal fibers. Moreover, it compactifies to an elliptic fibration \(\bar{W}_k\) over \(\mathbb{P}^1\), in which the fiber above infinity consists of \(9-k\) rational components, and which can be obtained as a deformation of the elliptic fibration \(\bar{W}_0 : \bar{M} \to \mathbb{P}^1\) compactifying the mirror of \(\mathbb{P}^2\). The manifold \(M_k\) is equipped with a symplectic form \(\omega\) and a B-field \(B\) whose cohomology classes are explicitely given by the set of points \(K\).

Once detailed such a construction, the authors recall the definition given in [P. Seidel, Proc. 3rd European Congr. Math., Barcelona 2000, II. Progr. Math., 202, 65–85 (2001; Zbl 1042.53060)] of the category of Lagrangian vanishing cycles of a Landau-Ginzburg model, describe explicitely the derived category for \(\bar{W}_k\) and the cohomology class \([B+i\omega]\). This easily leads to prove the required equivalence of categories.

Let \(X_K\) be a Del Pezzo surface obtained blowing up \(\mathbb{P}^2\) at \(k\) points. The conjecture is proved by taking as mirror an elliptic fibration \(W_k : M_k \to \mathbb{C}\) with \(k+3\) singular fibers and suitable symplectic form \([B + i \omega]\). Moreover, given a general noncommutative deformation of \(X_K\), there exists a complexified symplectic form \([B + i \omega]\), for which the deformed derived category of \(X_K\) is equivalent to the derived category of Lagrangian vanishing cycles. Conversely, for a generic choice of \([B + i \omega]\), the derived category of Lagrangian vanishing cycles is equivalent to the derived category of coherent sheaves of a noncommutative deformation of a Del Pezzo surface.

In order to prove that, the authors describe in a first time the derived category of \(X_K\) by detailing a strong exceptional collection of objects and morphisms between them. Derived categories of simple degenerations and noncommutative deformations can be described by working on such sets.

In a second time, the construction of the mirror Landau-Ginzburg model is given. The mirror of a given \(X_K\) is an elliptic fibration \(W_k : M_k \to \mathbb{C}\) with \(k+3\) nodal fibers. Moreover, it compactifies to an elliptic fibration \(\bar{W}_k\) over \(\mathbb{P}^1\), in which the fiber above infinity consists of \(9-k\) rational components, and which can be obtained as a deformation of the elliptic fibration \(\bar{W}_0 : \bar{M} \to \mathbb{P}^1\) compactifying the mirror of \(\mathbb{P}^2\). The manifold \(M_k\) is equipped with a symplectic form \(\omega\) and a B-field \(B\) whose cohomology classes are explicitely given by the set of points \(K\).

Once detailed such a construction, the authors recall the definition given in [P. Seidel, Proc. 3rd European Congr. Math., Barcelona 2000, II. Progr. Math., 202, 65–85 (2001; Zbl 1042.53060)] of the category of Lagrangian vanishing cycles of a Landau-Ginzburg model, describe explicitely the derived category for \(\bar{W}_k\) and the cohomology class \([B+i\omega]\). This easily leads to prove the required equivalence of categories.

Reviewer: Marcello Bernardara (Nice)

### MSC:

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

14J26 | Rational and ruled surfaces |

14J45 | Fano varieties |

14Dxx | Families, fibrations in algebraic geometry |

### Keywords:

homological mirror symmetry; derived category; Del Pezzo surfaces; Landau-Ginzburg model; Lagrangian vanishing cycles; noncommutative deformations### Citations:

Zbl 1042.53060
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\textit{D. Auroux} et al., Invent. Math. 166, No. 3, 537--582 (2006; Zbl 1110.14033)

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