Finiteness and infiniteness results for Torelli groups of (hyper-)Kähler manifolds

The Torelli group $$\mathcal T(X)$$ of a closed smooth manifold X is the subgroup of the mapping class group $$\pi _0(\mathrm {Diff}^+(X))$$ consisting of elements which act trivially on the integral cohomology of X. In this note we give counterexamples to Theorem 3.4 by Verbitsky (Duke Math J 162(15):2929–2986, 2013) which states that the Torelli group of simply connected Kahler manifolds of complex dimension $$\ge 3$$ is finite. This is done by constructing under some mild conditions homomorphisms $$J: \mathcal T(X) \rightarrow H^3(X;\mathbb Q)$$ and showing that for certain Kahler manifolds this map is non-trivial. We also give a counterexample to Theorem 3.5 (iv) in (Verbitsky in Duke Math J 162(15):2929–2986, 2013) where Verbitsky claims that the Torelli group of hyperkahler manifolds are finite. These examples are detected by the action of diffeomorphsims on $$\pi _4(X)$$ . Finally we confirm the finiteness result for the special case of the hyperkahler manifold $$K^{[2]}$$ .