On the relative opers in dimension one

TL;DR

This paper studies relative opers and differential operators over families of complex one-dimensional manifolds, establishing a bijective correspondence between certain classes of these structures.

Contribution

It introduces the notion of relative opers from the second fundamental form and proves a bijection with relative differential operators with identity symbol.

Findings

Establishes a correspondence between relative opers and differential operators.

Defines relative opers via the second fundamental form.

Analyzes the structure of relative differential operators with identity symbol.

Abstract

We investigate the relative opers over the complex analytic family of compact complex manifolds of relative dimension one. We introduce the notion of relative opers arising from the second fundamental form associated with a relative holomorphic connection. We also investigate the relative differential operators over the complex analytic family of compact complex manifolds whose symbol is the identity automorphism. We show that the set of equivalent relative opers arising from the second fundamental form is in bijective correspondence with the set of equivalent relative differential operators whose symbol is the identity automorphism.