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

Abstract

The Torelli group T(X) of a closed smooth manifold X is the subgroup of the mapping class group π(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 ≥ 3 is finite. This is done by constructing under some mild conditions homomorphisms J: T(X) → H3(X; 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 π4(X). Finally we confirm the finiteness result for the special case of the hyperkähler manifold K[2].