(T)-structures over 2-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 pull-backs of $\nabla$ under the inclusions $\{z\}\times M\to \mathbb{C}\times M$ we obtain a family of flat connections on vector bundles over $M$, parameterized by $z\in {\mathbb{C}}^{\ast }$. 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 $2$-dimensional $F$-manifolds.

28 pages; with respect to the previous version, some arguments (based on the new Lemma 14) are simplified; the main results are the same; Remarks 17, 20 and 22 are added

Cited by in corpus (2)

• (TE)-structures over the irreducible 2-dimensional globally nilpotent F-manifold germ
• Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1