Enumerative geometry of del Pezzo surfaces (Q6567157)

Since the pioneering work [\textit{P. Candelas} et al., AMS/IP Stud. Adv. Math. 9, 31--95 (1998; Zbl 0904.32019)] computing the number of rational curves of a given degree in a geometric quintic, mirror symmetry has provided a new efficient way of computing enumerative invariants to Fano manifolds.\N\N\textit{D. Auroux} [J. Gökova Geom. Topol. GGT 1, 51--91 (2007; Zbl 1181.53076)] investigated the holomorphic discs with boundary in the non-tropic Lagrangian fibration constructed by \textit{M. Gross} [in: Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14--18, 2000. Singapore: World Scientific. 81--109 (2001; Zbl 1034.53054)] outside of a singular anti-canonical divisor \(D\) of some toric Fano manifold \(Y\), conjecturing the existence of the special Lagrangians in the complement of the smooth anti-canonical divisor \(X=Y/D\) and having predictions on the SYZ mirror (Conjecture 7.3). The existence of the special Lagrangian was settled by Collins, Jacobs and Lin [\textit{T. C. Collins} et al., Duke Math. J. 170, No. 7, 1291--1375 (2021; Zbl 1479.14046)]. The remaining half is settled in this paper (Theorem 5.5).\N\NTheorem. There exists an open neighborhood \(U\) such that the special Lagrangian torus fiber contained in \(U\) bounds a unique holomorphic disc of Maslov index two disc contained in \(U\).\N\NInspired by Strominger, Yau and Zaslow's conjecture [\textit{A. Strominger} et al., Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] and the related heuristic in symplectic geometry, Gross, Hacking and Keel [\textit{M. Gross} et al., Publ. Math., Inst. Hautes Étud. Sci. 122, 65--168 (2015; Zbl 1351.14024)] constructed the mirrors for the pair \((Y,D)\), where \(Y\) is a projective surface with \(D\in |-K_{Y}|\) wheel of rational curves. Gross, Hacking and Keel further introduced the notion of broken lines and theta functions as the weighted count of broken lines. The mirror can then be realized as the spectrum of the algebra generated by the theta functions. The construction was extended to the case when \(D\) is smooth in [\textit{M. Carl} et al., Acta Math. Sin., Engl. Ser. 40, No. 1, 329--382 (2024; Zbl 1536.14028)]. It is shown (Theorem 5.6) that\N\NTheorem. The weighted counts of the broken lines coincide with the countings of the Maslov index two discs are the coefficients of the superpotential in Lagrangian Floer theory.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] reviews the special Lagrangian fibration constructed in [\textit{T. C. Collins} et al., Duke Math. J. 170, No. 7, 1291--1375 (2021; Zbl 1479.14046)] and its properties.\N\N\item[\S 3] reviews the tropical geometry on the base of the special Lagrangian with the complex affine structure.\N\N\item[\S 4] establishes the tropical/holomorphic correspondence for Maslov index zero discs with boundary on special Lagrangian fibers.\N\N\item[\S 5] establishes the equivalence between counting Maslov index two discs with the same boundary conditions and the weighted count of broken lines.\N\N\item[\S 6] considers the relation of certain open Gromov-Witten invariants of Maslov index zero discs with relative Gromov-Witten invariants of maximal tangency.\N\N\item[\S 7] provides some explicit calculations in the case of \(\mathbb{P}^{2} \), giving connections to the preceding work.\N\end{itemize}