We investigate the question: for which functions $f(x_1,…,x_n),~g(x_1,…,x_n)$ the asymptotic expansion of the integral $int g(x_1,…,x_n) e^{frac{f(x_1,…,x_n)+x_1y_1+dots+x_ny_n}{hbar}}dx_1…dx_n$ consists only of the first term. We reveal a hidden projective invariance of the problem which establishes its relation with geometry of projective hypersurfaces of the form ${(1:x_1:…:x_n:f)}$. We also construct various examples, in particular we prove that Kummer surface in $P^3$ gives a solution to our problem. This is a joint work with Maxim Kontsevich.

IHES and Brock Univ.