A note on the automorphism group of a compact complex manifold

TL;DR

This paper provides explicit examples of compact complex 3-folds with automorphisms isotopic to the identity via smooth diffeomorphisms but not through biholomorphisms, impacting the study of Teichmüller stacks.

Contribution

It introduces specific examples illustrating a nuanced difference in automorphism isotopies relevant to higher-dimensional Teichmüller theory.

Findings

Existence of automorphisms isotopic to identity via smooth but not holomorphic deformations

Examples relevant for the construction of higher-dimensional Teichmüller stacks

Highlights subtle distinctions in automorphism groups of complex manifolds

Abstract

In this note, we give explicit examples of compact complex 3-folds which admit automorphisms that are isotopic to the identity through C $\infty$-diffeomorphisms but not through biholomorphisms. These automorphisms play an important role in the construction of the Te-ichm{\"u}ller stack of higher dimensional manifolds.