(T)-structures over two-dimensional F-manifolds: formal classification

A (TE)-structure ∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla $$\end{document} over a complex manifold M is a meromorphic connection defined on a holomorphic vector bundle over C×M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}\times M$$\end{document}, with poles of Poincaré rank one along {0}×M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ 0 \} \times M$$\end{document}. Under a mild additional condition (the so-called unfolding condition), ∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla $$\end{document} induces a multiplication on TM and a vector field on M (the Euler field), which make M into an F-manifold with Euler field. By taking the pullbacks of ∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla $$\end{document} under the inclusions {z}×M→C×M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ z\} \times M \rightarrow {\mathbb {C}}\times M$$\end{document}(z∈C∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(z\in \mathbb {C}^*)$$\end{document}, we obtain a family of flat connections on vector bundles over M, parameterized by z∈C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\in {\mathbb {C}}^{*}$$\end{document}. The properties of such a family of connections give rise to the notion of (T)-structure. Therefore, any (TE)-structure underlies a (T)-structure, but the converse is not true. The unfolding condition can be defined also for (T)-structures. A (T)-structure with the unfolding condition induces on its parameter space the structure of an F-manifold (without Euler field). After a brief review on the theory of (T)- and (TE)-structures, we determine normal forms for the equivalence classes, under formal isomorphisms, of (T)-structures which induce a given irreducible germ of two-dimensional F-manifolds.