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

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 $C^{\mathrm{\infty }}$-isotopically trivial automorphisms, resp. cohomologically trivial automorphisms, have a number of connected components which can be arbitrarily large.