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Abstract: The affine cancellation problem, which asks whether complex affine varietieswith isomorphic cylinders are themselves isomorphic, has a positive solutionfor two dimensional varieties whose coordinate rings are unique factorizationdomains, in particular for the affine plane, but counterexamples are foundwithin normal surfaces Danielewski surfaces and factorial threefolds oflogarithmic Kodaira dimension equal to 1. The latter are therefore remote fromthe affine three-space, the first unknown case where the base of one cylinderis an affine space. Locally trivial Ga-actions play a significant role in theseexamples. Threefolds admitting free Ga-actions are discussed, especially aclass of varieties with negative logarithmic Kodaira dimension which are totalspaces of nonisomorphic Ga-bundles. Some members of the class are shown to beisomorphic as abstract varieties, but it is unknown whether any members of theclass constitute counterexamples to cancellation.

Autor: Adrien Dubouloz IMB, David R. Finston, Parag Deepak Mehta

Fuente: https://arxiv.org/

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