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Calabi–Yau threefolds with a curve of singularities and counterexamples to the Torelli problem II

Published online by Cambridge University Press:  16 October 2000

BALÁZS SZENDRŐI
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB; e-mail: balazs@maths.warwick.ac.uk

Abstract

This paper is a continuation of [15]. In that paper, I introduced a general framework which allows one to produce ‘weak’ counterexamples to Torelli for Calabi–Yau threefolds: deformation families containing non-isomorphic varieties Yt, Y+t with isomorphic Hodge theory on the third cohomology. The varieties arise as deformations of threefolds Y that are resolutions of singular varieties X with rather special properties (cf. Section 1). In [15], I discussed two families containing suitable X that do provide a counterexample and a third family with remarkably similar properties where however the existence of a nontrivial generic automorphism destroys the counterexample. This shows that explicit examples are necessary; there is a precise set of conditions one needs to check carefully in order to obtain counterexamples to Torelli (Theorem 1·1). In fact, there exist several families with renitent automorphisms (Remark 4·5).

Type
Research Article
Copyright
2000 Cambridge Philosophical Society

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