Higher residues and canonical pairing on the twisted de Rham cohomology (Q6602149)

Let \(f \in Q = \mathbb C [x_1, \dots, x_n]\) be a polynomial for which the origin is the only critical point of the map \(f: \mathbb C^n \to \mathbb C\). Letting \(H_f^{(0)} = H_n (\Omega^\bullet_{Q/\mathbb C} [[u]], -df+ud)\) and \(H_{-f}^{(0)} = H_n (\Omega^\bullet_{Q/\mathbb C} [[u]], df+ud)\), \textit{K. Saito} introduced the higher residue pairings \(H_f^{(0)} \times H_{-f}^{(0)} \stackrel{\mathrm{id} \times (-1)^n}{\longrightarrow} H_f^{(0)} \times H_{-f}^{(0)} \cong \mathbb C [[u]]\), where the last isomorphism runs through hypercohomology of the de Rham complex, the Künneth formula, and the residue map [\textit{A. Buryak}, Mosc. Math. J. 20, No. 3, 475--493 (2020; Zbl 1462.14042)]. These pairings play a role in the theory of Frobenius manifolds [\textit{C. Hertling}, Frobenius manifolds and moduli spaces for singularities. Cambridge: Cambridge University Press (2002; Zbl 1023.14018)].\N\NThe main result is an explicit formula for these pairings. The authors give an application to work of \textit{D, Shklyarov} on the relation between the canonical pairing in the dg category of matrix factorizations of an isolated singularity and Saito's higher residue pairings on the de Rham cohomology associated with the singularity [Adv. Math. 292, 181--209 (2016; Zbl 1397.14007)]. There Shklyarov shows that the two pairings are the same up to sign and conjectures the appropriate sign to be \((-1)^{\frac{n(n+1)}{2}}\). This conjecture was first proved by \textit{Brown and Walker} [J. Noncommut. Geom. 16, 1479--1523 (2022; Zbl 1520.14031)] and later by \textit{B. Kim} [``Hirzebruch-Riemann-Roch for global matrix factorizations'', Preprint, \url{arXiv:2106.00435}]. Here the authors give another proof using their formula.