Finiteness and infiniteness results for Torelli groups of (hyper-)Kähler 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 hyperkähler manifolds" (Duke Math.~J.~162 (2013), no.~15, 2929-2986) 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 $J:\mathcal{T}(X)\to H^3(X;\mathbb{Q})$ and showing that for certain Kähler manifolds this map is non-trivial. We also give a counterexample to Theorem 3.5 (iv) in this paper 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 $K^{[2]}$.

10 pages

References in corpus (1)

• Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum

Cited by in corpus (2)

• Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum
• Mapping class group of manifolds which look like $3$-dimensional complete intersections