Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1

Abstract

Holomorphic vector bundles on ℂ × 𝑀, 𝑀 a complex manifold, with meromorphic connections with poles of Poincaré rank 1 along {0} × 𝑀, arise naturally in algebraic geometry. They are called (𝛵𝐸)-structures here. This paper takes an abstract point of view. It gives a complete classification of all (𝛵𝐸)-structures of rank 2 over germs (𝑀, 𝘵⁰) of manifolds. In the case of 𝑀, they separate into four types. Those of the three types have universal unfoldings; those of the fourth type (the logarithmic type) do not. The classification of unfoldings of (𝛵𝐸)-structures of the fourth type is rich and interesting. The paper finds and lists all (𝛵𝐸)-structures which are basic in the following sense: Together they induce all rank 2 (𝛵𝐸)-structures, and each of them is not induced by any other (𝛵𝐸)-structure in the list. Their base spaces 𝑀 turn out to be 2-dimensional F-manifolds with Euler fields. The paper also provides a classification of all rank 2 (𝛵𝐸)-structures over it. Also, this classification is surprisingly rich. The backbone of the paper is normal forms. Though also the monodromy and the geometry of the induced Higgs fields and of the base spaces are important and are considered.