TITLE: Finite order automorphisms of complex projective manifolds isotopic to identity
QUESTION [10 upvotes]: Question. Let $V$ be a complex projective manifold of general type (we can even assume that the canonical bundle of $V$ is ample). Suppose $\varphi: V\to V$ is a non-identical automorphism. Can $\varphi$ be isotopic to the identity map (i.e. $\varphi\in Diff_0(V)$)?  
I hope the answer is no, and this can be easily proven when $K_V$ is very ample. 
More generally what restrictions are known on smooth manifolds that admit self-diffeos of finite order that are isotopic to identity?

REPLY [13 votes]: The answer to your question is unknown already for surfaces $S$ of general type. 
Note that, if $S$ is simply connected, by a result of Quinn (see "Isotopy of 4-manifolds", Journal of Differential Geometry 1986) every automorphism acting trivially on rational cohomology must be topologically isotopic to the identity. 
At any rate, it seems that people conjecture that the answer to your question is no for simply connected surfaces of general type. See Catanese's paper "A Superficial Working Guide to Deformations and Moduli" (arXiv:1106.1368), Section 1.4  for further details. In this paper, complex manifolds which do not admit non-trivial automorphisms isotopic to the identity are called rigidified.