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

2019/07/12 by Kreck, Matthias, Su, Yang
#14J50 #57R19 #57R50 #57R52 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1907.05693

Abstract

The Torelli group \mathcal T(X) of a closed smooth manifold X is the subgroup of the mapping class group π₀(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 ≥ 3 is finite. This is done by constructing under some mild conditions homomorphisms J: \mathcal T(X) → H³(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 π₄(X). Finally we confirm the finiteness result for the special case of the hyperkähler manifold K^[2].

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