Mirror symmetry for log Calabi-Yau surfaces. I.

*(English)*Zbl 1351.14024Let \(Y\) be a smooth rational projective surface over an algebraically closed field \(\mathbb{K}\) of characteristic zero, and \(D\in|-K_Y|\) be a divisor of a singular nodal curve. The main result is a canonical synthetic construction of a mirror family to such a pair \((Y,D)\), called a Looijenga pair. For \(n\geq3\) the construction gives an embedded smoothing of the \(n\)-cycle \(V_n\) of coordinate planes in \(\mathbb{A}^n\), the “\(n\)-vertex”. The philosophy behind the construction is the tropicalization of the SYZ fibration picture in the spirit of the Gross-Siebert program. The base is replaced by a combinatorial object \(B\), the dual intersection complex of \((Y,D)\), on \(\mathbb{R}^2\) subdivided into cones, and the fibration is replaced by a flat deformation family of \(V_n\).

Relative Gromov-Witten invariants of \((Y,D)\) counting rational curves meeting \(D\) at a single point define an algebra structure on a certain vector space with canonical elements called theta functions, which generalize classical theta functions on Abelian varieties. They are constructed using a tropical analog of a disk with Maslov index 2, the so-called broken line. Deformation families of \(V_n\) are difficult to construct, but the deformations of \(V_n\backslash\{0\}\) are straightforward. The theta functions (log analogs of those from the Tyurin conjecture for polarized \(K3\) surfaces) are used to embed deformations of \(V_n\backslash\{0\}\) produced by “canonical scattering diagrams” into affine space, where the closure can be taken, giving a deformation of \(V_n\). The bulk of the argument is dedicated to proving that such deformations are extendable, and produce global theta functions indexed by elements of \(B\).

The construction so far only produces a family over the completion of \(\mathrm{Spec }\mathbb K[P]\) at the zero dimensional torus orbit, where \(P\) is a finitely generated monoid containing classes of all effective curves on \(Y\) obtained by choosing a strictly convex rational polyhedral cone containing the Mori cone. The second main result shows that the family extends across completions over larger strata. This involves studying products of theta functions and their tropical interpretation.

The third main result is a proof of the Looijenga’s conjecture that a 2-dimensional cusp singularity is smoothable if and only if the exceptional cycle of the dual cusp occurs as an anti-canonical cycle on a smooth projective rational surface. A key step is to represent the conjecture as a mirror symmetry claim in the case of and the intersection matrix \(D_i\cdot D_j\) being negative definite, where \(D=D_1+\cdots+D_n\). Then \(D\) can be analytically contracted into a cusp singularity, and the construction of the paper naturally produces the dual cusp. However, the construction of the theta functions is much more delicate in this context. The sums of monomials (associated with the broken lines), which define them, are always infinite here, and rather technical combinatorial analysis is required to prove their convergence. Moreover, the smoothing of the cusp singularity has to be proved, over and above the smoothing of the \(n\)-vertex.

The authors do not specify in what sense the constructed family is a mirror of \((Y,D)\), but, aside from the structural parallels with SYZ, they expect it to be one in the sense of the homological mirror symmetry when \(\mathbb{K}=\mathbb{C}\). The paper is part of a broader programme extending to the cases where \(D_i\cdot D_j\) is not negative definite. In particular, elsewhere in collaboration with Kontsevich the authors prove a number of conjectures concerning cluster varieties, such as positivity of the Laurent phenomenon and existence of the Fock-Goncharov dual basis. They also expect that further development of the technology of logarithmic Gromov-Witten invariants will lead to generalizing their mirror construction to analogs of Looijenga pairs in higher dimensions.

Relative Gromov-Witten invariants of \((Y,D)\) counting rational curves meeting \(D\) at a single point define an algebra structure on a certain vector space with canonical elements called theta functions, which generalize classical theta functions on Abelian varieties. They are constructed using a tropical analog of a disk with Maslov index 2, the so-called broken line. Deformation families of \(V_n\) are difficult to construct, but the deformations of \(V_n\backslash\{0\}\) are straightforward. The theta functions (log analogs of those from the Tyurin conjecture for polarized \(K3\) surfaces) are used to embed deformations of \(V_n\backslash\{0\}\) produced by “canonical scattering diagrams” into affine space, where the closure can be taken, giving a deformation of \(V_n\). The bulk of the argument is dedicated to proving that such deformations are extendable, and produce global theta functions indexed by elements of \(B\).

The construction so far only produces a family over the completion of \(\mathrm{Spec }\mathbb K[P]\) at the zero dimensional torus orbit, where \(P\) is a finitely generated monoid containing classes of all effective curves on \(Y\) obtained by choosing a strictly convex rational polyhedral cone containing the Mori cone. The second main result shows that the family extends across completions over larger strata. This involves studying products of theta functions and their tropical interpretation.

The third main result is a proof of the Looijenga’s conjecture that a 2-dimensional cusp singularity is smoothable if and only if the exceptional cycle of the dual cusp occurs as an anti-canonical cycle on a smooth projective rational surface. A key step is to represent the conjecture as a mirror symmetry claim in the case of and the intersection matrix \(D_i\cdot D_j\) being negative definite, where \(D=D_1+\cdots+D_n\). Then \(D\) can be analytically contracted into a cusp singularity, and the construction of the paper naturally produces the dual cusp. However, the construction of the theta functions is much more delicate in this context. The sums of monomials (associated with the broken lines), which define them, are always infinite here, and rather technical combinatorial analysis is required to prove their convergence. Moreover, the smoothing of the cusp singularity has to be proved, over and above the smoothing of the \(n\)-vertex.

The authors do not specify in what sense the constructed family is a mirror of \((Y,D)\), but, aside from the structural parallels with SYZ, they expect it to be one in the sense of the homological mirror symmetry when \(\mathbb{K}=\mathbb{C}\). The paper is part of a broader programme extending to the cases where \(D_i\cdot D_j\) is not negative definite. In particular, elsewhere in collaboration with Kontsevich the authors prove a number of conjectures concerning cluster varieties, such as positivity of the Laurent phenomenon and existence of the Fock-Goncharov dual basis. They also expect that further development of the technology of logarithmic Gromov-Witten invariants will lead to generalizing their mirror construction to analogs of Looijenga pairs in higher dimensions.

Reviewer: Sergiy Koshkin (Houston)

##### MSC:

14J33 | Mirror symmetry (algebro-geometric aspects) |

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

32Q25 | Calabi-Yau theory (complex-analytic aspects) |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |

##### Keywords:

Looijenga pair; \(n\)-vertex; tropicalization; relative Gromov-Witten invariants; logarithmic Gromov-Witten invariants; theta functions; Maslov index; broken line; canonical scattering diagram; Tyurin conjecture; dual cusp##### References:

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