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 Kähler 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 Kähler 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 hyperkähler 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 hyperkähler manifold \(K^{[2]}\).

闭合光滑流形X的Torelli组\（\ mathcal T（X）\）是由元素组成的映射类组\（\ pi _0（\ mathrm {Diff} ^ +（X））\）的子组对X的积分同调微不足道。在本说明中，我们以Verbitsky的定理3.4作反例（Duke Math J 162（15）：2929-2986，2013），该定理指出，简单连接复维\（\ ge 3 \）的Kähler流形的Torelli组是有限的。这是通过在某些温和条件下构造同态\（J：\ mathcal T（X）\ rightarrow H ^ 3（X; \ mathbb Q）\）来完成的并表明对于某些Kähler流形来说，这幅图是不平凡的。我们还对（Verbitsky in Duke Math J 162（15）：2929-2986，2013）中的定理3.5（iv）给出了反例，其中Verbitsky声称Hyperkähler流形的Torelli组是有限的。这些示例通过diffeomorphsims对\（\ pi _4（X）\）的作用来检测。最后，我们确定了超kähler流形\（K ^ {[2]} \）的特殊情况的有限性结果。