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Journal of the London Mathematical Society 1995 51(1):27-40; doi:10.1112/jlms/51.1.27
© 1995 by London Mathematical Society
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© The London Mathematical Society

Lech's Conjecture on Deformations of Singularities and Second Harrison Cohomology

Jörg Jahnel

Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26 W-5300 Bonn 3, Germany
Universität zu Köln, Mathematisches Institut Weyertal 86-90, W-5000 Köln 41, Germany

Received 6 January 1993. Revision received 2 April 1993.

Let B0 be a local singularity of dimension d. Then we consider the problem of Lech, whether for every deformation (A, m) -> (B, n) of B0 the inequality Formula ≤ Formula between the Hilbert functions is true, and give a positive answer in the case that the formal versal deformation of B0 is a base change of an algebraic family (R, M) -> (S, N), where R is regular and dim S = dim R + d.

So one would hope to lift versal deformations in that way. There are obstructions against this in certain second Harrison cohomology groups.


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