Finiteness and infiniteness results for Torelli groups of (hyper-)K\"ahler manifolds

Abstract

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 of Verbitsky's paper "Mapping class group and a global Torelli theorem for hyperk\"ahler manifolds" (Duke Math.~J.~162 (2013), no.~15, 2929-2986) which states that the Torelli group of simply connected K\"ahler manifolds of complex dimension $\ge 3$ is finite. This is done by constructing under some mild conditions homomorphisms $J: \mathcal T(X) \to H^3(X;\mathbb Q)$ and showing that for certain K\"ahler manifolds this map is non-trivial. We also give a counterexample to Theorem 3.5 (iv) in this paper where Verbitsky claims that the Torelli group of hyperk\"ahler 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 hyperk\"ahler manifold $K^{[2]}$.