On the Gauss-Manin Connection and Real Singularities
Lars Andersen
TL;DR
This work develops a real-analytic analogue of the Gauss–Manin connection by constructing two real Gauss–Manin systems $\mathfrak{g}^{k\pm}_f$ as pushforwards of the positive and negative Milnor fibrations and endowing them with a $\mathcal{D}$-module structure. It shows that for isolated real singularities and $k=0$, these systems recover the $n$-th homology of the Milnor fibres $H_n(\mathcal{F}_{\eta}^{\pm})$, and it computes the case of ordinary quadratic singularities, where the system is governed by $\mathcal{D}/(D_t\mathcal{D})$ with explicit solutions $u_n(\eta)$. The paper then leverages morsification to express the top Milnor fibre homology in terms of summands associated to quadratic-type critical points, providing a quasi-isomorphism between relative de Rham complexes along a one-parameter morsification and yielding a decomposition of $H_{n-1}(\mathcal{F}^{+})$. As an application, it proposes public-key encryption schemes based on morsification data, with $CCA$-security under the stated hypotheses, illustrating a bridge between real singularity theory, $\mathcal{D}$-modules, and cryptography.
Abstract
We prove that to each real singularity $f: (\mathbb{R}^{n+1}, 0) \to (\mathbb{R}, 0)$ one can associate two systems of differential equations $\mathfrak{g}^{k\pm}_f$ which are pushforwards in the category of $\mathcal{D}$-modules over $\mathbb{R}^{\pm}$, of the sheaf of real analytic functions on the total space of the positive, respectively negative, Milnor fibration. We prove that for $k=0$ if $f$ is an isolated singularity then $\mathfrak{g}^{\pm}$ determines the the $n$-th homology groups of the positive, respectively negative, Milnor fibre. We then calculate $\mathfrak{g}^{+}$ for ordinary quadratic singularities and prove that under certain conditions on the choice of morsification, one recovers the top homology groups of the Milnor fibers of any isolated singularity $f$. As an application we construct a public-key encryption scheme based on morsification of singularities.