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- W2960444838 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 hyperkahler manifolds (Duke Math.~J.~162 (2013), no.~15, 2929-2986) which states that the Torelli group of simply connected Kahler 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 Kahler 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 hyperkahler 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 hyperkahler manifold $K^{[2]}$." @default.
- W2960444838 created "2019-07-23" @default.
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- W2960444838 date "2019-07-12" @default.
- W2960444838 modified "2023-10-16" @default.
- W2960444838 title "Finiteness and infiniteness results for Torelli groups of (hyper-)Kahler manifolds" @default.
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- W2960444838 doi "https://doi.org/10.48550/arxiv.1907.05693" @default.
- W2960444838 hasPublicationYear "2019" @default.
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