Fukaya \(A_{\infty}\)-categories associated to Lefschetz fibrations. II \(\tfrac{1}{2}\)
Let \(E\to \mathbb{C}\) be a symplectic Lefschetz fibration allowing a fiberwise compactification \(\bar{E}\to\mathbb{C}\). Then, taking a bulk term \(b\in H^2(E;\mathbb{C}^\ast)\) and its deformed version \(\bar{b}\in H^2(E;\mathbb{C}q^\times)\), the twisted Fukaya \(A_\infty\)-category \(\mathcal{F}_b(\pi)\) and its twisted formal deformation \(\mathcal{F}_{q,\bar{b}}(\bar{\pi})\) are associated [\textit{K. Fukaya} et al., Sel. Math., New Ser. 17, No. 3, 609--711 (2011; Zbl 1234.53023); the author, in: Proceedings of the international congress of mathematicians, ICM'02. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor. 351--360 (2002; Zbl 1014.53052)]. Then the following question arises: Question. Can we choose the bulk term so that the formal deformation induced by fiberwise compactification is trivial? The author remarks that, since compactification adds a new holomorphic disc changing the \(A_\infty\)-structure, the answer to this question is negative in known relative Fukaya categories in the ordinary sense. In this paper, instead of working on the entire Fukaya category \(\mathcal{F}\), its subcategory \(\mathcal{A}\), obtained to fix a basis of Lefschetz thimbles \(\{L_1,\ldots,L_m\}\), is considered. In \S3, via considerations of Gromov-Witten invariants, a differential equation involving \(\bar{b}\) (Assumption 3.1. In Lemma 3.2, this equation is shown to always have a solution) is presented. Then the main theorem announced in this paper is the following partial answer to this question. Theorem 4.1. Suppose that \(\bar{b}\) satisfies Assumption 3.1. Then \(\mathcal{A}_{q,\bar{b}}\) is a trivial deformation. Corollary 4.2. There is a distinguished class of bulk \(\bar{b}\), all related to each other by suitable reparametrizations (an exact description is given in (2.13)), such that \(\mathcal{A}_{q,\bar{b}}\) is a trivial deformation The proof of Theorem 4.1 is only sketched. It uses Kaledin's triviality condition of \(\mathcal{A}_{q,\bar{b}}\). See Lemma 4.3. and [\textit{D. Kaledin}, Mosc. Math. J. 7, No. 4, 643--652 (2007; Zbl 1163.18006); \textit{V. A. Lunts}, J. Algebra 323, No. 4, 878--898 (2010; Zbl 1227.18013)], and applies methods of \textit{N. Sheridan} [Invent. Math. 199, No. 1, 1--186 (2015; Zbl 1344.53073)]. As a consequence, for a Lefschetz fibration arising from a Lefschetz pencil on a del Pezzo surface which is not \(\mathbb{C}P^2\) blown up at 1 or 2 points, then the deformation with \(\bar{b}=1\) is shown to be trivial (\S6.1). Basic ingredients such as Lefschetz fibrations, Fukaya categories, Gromov-Witten invariants and some homological algebra are explained in \S2. Further studies of Gromov-Witten invariants are given in \S3. As stated above, they are essential to state Theorem 4.1. After stating the main theorem, \S4 discusses how additional structures present in Fukaya categories of Lefschetz fibrations behave under fiberwise compactification. The discussions are rather conjectural. For instance, it is conjectured that if the deformation \(\mathcal{A}_{q,\bar{b}}\) is trivial, then its subcategory \(\mathcal{B}_{q,\bar{b}|M}\), analogous to \(\mathcal{B}\), the subcategory of \(\mathcal{F}\) the full \(A_\infty\)-subcategory of the Fukaya category of the fiber formed by the vanishing cycles in the fiber \(M\) (details are described in \S2.3), has the structure of a deformation of noncommutative divisors on \(\mathcal{A}\) (Conjecture 4.9, with a detailed conjecture on its leading order). Before stating the applications of the main theorem (and corollary) in \S6, some elementary examples, which do not come from Lefschetz pencils, but illustrate the general idea of fiberwise compactification, and the associated deformation of Fukaya category, are explained in \S5. In \S7, the conjectures in \S4 are tested for Hirzebruch surfaces with the aid of the discussions in \S6, and using the mirror map, the expected results from Conjecture 4.9 are derived. As an additional detail for the algebraic part of review of homological algebra (\S2.5), an abstract Maurer-Cartan theory is exposed in the appendix of [\textit{W. M. Goldman} and \textit{J. J. Millson}, Publ. Math., Inst. Hautes Étud. Sci. 67, 43--96 (1988; Zbl 0678.53059)].
scientific article; zbMATH DE number 6666960