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

Abstract

A (TE)-structure \(\nabla \) over a complex manifold M is a meromorphic connection defined on a holomorphic vector bundle over \({\mathbb {C}}\times M\), with poles of Poincaré rank one along \(\{ 0 \} \times M\). Under a mild additional condition (the so-called unfolding condition), \(\nabla \) 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 \(\nabla \) under the inclusions \(\{ z\} \times M \rightarrow {\mathbb {C}}\times M\)\((z\in \mathbb {C}^*)\), we obtain a family of flat connections on vector bundles over M, parameterized by \(z\in {\mathbb {C}}^{*}\). 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.