Convergence of the normalized solution of the Maurer-Cartan equation in the Barannikov-Kontsevich construction. (Q1396359)

A Frobenius manifold is a quadruple \((M,{\mathfrak I}^f_M, g,A)\), where \(M\) is a supermanifold in one of the standard categories \((C^\infty)\), analytic, algebraic, formal, etc.), \({\mathfrak I}^f_M\) is the sheaf of flat vector fields tangent to an affine structure, \(g\) is a flat Riemannian metric such that \({\mathfrak I}^f_M\) is the sheaf of \(g\)-flat tangent fields, and \(A\) is an even symmetric tensor \(A: S^3({\mathfrak I}_M)\to{\mathfrak O}_M\), where \({\mathfrak O}_M\) is the sheaf of germs of functions on \(M\). For any flat vector fields \(X\), \(Y\), and \(Z\), a locally defined function \(\Phi\) satisfying \(A(X,Y,Z)= XYZ\Phi\) is called potential, and in a formal Frobenius manifold the potential \(\Phi\) is a formal power series. The Barannikov-Kontsevich construction [\textit{S. Barannikov} and \textit{M. Kontsevich}, Int. Math. Res. Not. 1998, 201--215 (1998; Zbl 0914.58004)] belong to the class of formal Frobenius manifolds, for which the author gives a detailed proof of the convergence of the normalized solution of the Maurer-Cartan equation and of the potential, which implies that the Barannikov-Kontsevich construction gives a large class of holomorphic Frobenius manifolds.