On topologically trivial automorphisms of compact K\"ahler manifolds and algebraic surfaces

TL;DR

This paper explores automorphisms of compact Kähler manifolds and algebraic surfaces, providing examples where groups of automorphisms with trivial topological or cohomological properties have arbitrarily many connected components.

Contribution

It offers new examples of complex surfaces with automorphism groups exhibiting arbitrarily large connected components, advancing understanding of topologically trivial automorphisms.

Findings

Automorphism groups can have arbitrarily many connected components.

Examples include smooth complex projective surfaces with trivial automorphism groups.

The study distinguishes between topologically and cohomologically trivial automorphisms.

Abstract

In this paper, we investigate automorphisms of compact K\"ahler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of $C^\infty$-isotopically trivial automorphisms, resp. cohomologically trivial automorphisms, have a number of connected components which can be arbitrarily large.