Flat meromorphic connections of Frobenius manifolds with tt*-structure
Jiezhu Lin, Claude Sabbah

TL;DR
This paper studies the geometric structures of Frobenius manifolds with CDV-structures, focusing on flat meromorphic connections and conditions for their formal isomorphisms to converge, especially in semi-simple cases.
Contribution
It establishes a formal isomorphism between two natural holomorphic bundles with connections for semi-simple CDV-structures and provides explicit convergence criteria when the super-symmetric index vanishes.
Findings
Existence of a formal isomorphism between bundles for semi-simple CDV-structures.
Convergence of the formal isomorphism when the super-symmetric index Q is zero.
Explicit conditions for convergence in the case of dimension two.
Abstract
The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDV-structure by C. Hertling. For any CDV-structure on a Frobenius manifold M, the pull-back of the (1,0)-tangent bundle of M to the product of M by the complex line carries two natural holomorphic structures equipped with flat meromorphic connections. We show that, for any semi-simple CDV-structure, there is a formal isomorphism between these two bundles compatible with connections. Moreover, if we assume that the super-symmetric index Q vanishes, we give a necessary and sufficient condition for such a formal isomorphism to be convergent, and we make it explicit for dim M = 2.
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