When the Fourier transform is one loop exact?

We investigate the question: for which functions $f(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n),g(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)$ the asymptotic expansion of the integral $\int g(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)e^{\frac{f(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)+x_1{y}_1+\mathrm{.}\mathrm{.}\mathrm{.}+x_n{y}_n}{\mathrm{\hslash }}}d{x}_1\mathrm{.}\mathrm{.}\mathrm{.}d{x}_n$ 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 $\{(1:x_1:\mathrm{.}\mathrm{.}\mathrm{.}:x_n:f)\}$. We also construct various examples, in particular we prove that Kummer surface in ${\mathbb{P}}^3$ gives a solution to our problem.