On the equality of periods of Kontsevich–Zagier

Effective periods are defined by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of Q-rational functions over Q-semi-algebraic domains in R^d. The Kontsevich-Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables and Stokes formula. In this paper, we discuss about possible geometric interpretations of this conjecture, viewed as a generalization of the Hilbert's third problem for compact semi-algebraic sets as well as for rational polyhedron equipped with piece-wise algebraic forms. Based on partial known results for analogous Hilbert's third problems, we study obstructions of possible geometric schemas to prove this conjecture.