When the Fourier transform is one loop exact?

We investigate the question: for which functions \(f(x_1,\ldots ,x_n),~g(x_1,\ldots ,x_n)\) $f(x_1,\dots ,x_n),g(x_1,\dots ,x_n)$ the asymptotic expansion of the integral \(\int g(x_1,\ldots ,x_n) e^{\frac{f(x_1,\ldots ,x_n)+x_1y_1+\dots +x_ny_n}{\hbar }}dx_1\ldots dx_n\) $\int g(x_1,\dots ,x_n)e^{\frac{f(x_1,\dots ,x_n)+x_1{y}_1+\cdots +x_n{y}_n}{ħ}}d{x}_1\dots d{x}_n$ 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 \(\{(1:x_1:\ldots :x_n:f)\}\) $\{(1:x_1:\dots :x_n:f)\}$ . We also construct various examples, in particular we prove that Kummer surface in \({\mathbb {P}}^3\) ${\mathbb{P}}^3$ gives a solution to our problem.