Finiteness and infiniteness results for Torelli groups of (hyper-)K\xe4hler 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 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]}.