Here is a rather naïve question: to apply the classical holomorphic Lefschetz theorem, you often work with biholomorphisms of a complex compact manifold whose fixed points are non degenerate (i.e. $1$ is not an eigenvalue of the differential) even if work of O'Brian http://www.jstor.org/stable/1998512 allows to get rid of the degeneracy assumption and compute explicitly the multiplicities at the fixed points, provided they remain isolated.

However, I have encountered no interesting examples where some fixed point are isolated and degenerate. For instance, this condition implies (using Bochner's linearization) that the automorphism is not of finite order.

Ideally I would like to construct such an exemple with the manifold $X$ satisfying the additional assumptions $h^0(X, \, T_X)=h^1(X, \, T_X)=0$. A finite product of $\mathbb{P}^2$ blown up at 4 generic points seems a reasonable candidate, but I'm not sure this is the simplest path to start.