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There is a theorem due to Gaifullin--Ignashchenko stating that the Dehn invariant of any flexible polyhedron in the $n$-dimensional Euclidean space ($n\geq 3$) is constant during the flexion.

Is there an analogue of this theorem featuring the Grothendieck class of varieties instead of the Dehn invariant? Probably no, but the infinitesimal chance that there is makes me eager to ask this question.

Families of varieties would have to be severely restricted for this to make sense (consider abelian varieties to get some idea).

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  • $\begingroup$ I have seen somewhere a conjecture about the invariance of the full spectrum of the Laplacian in this context. This would imply the known invariance of the volume. $\endgroup$ – F. C. May 25 at 7:09
  • $\begingroup$ What would flexibility mean? (And which motivic measure would you want to take as replacement for volume?) $\endgroup$ – Matthias Wendt Jul 9 at 19:05

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