Exponential Gauss-Manin Connections

Abstract: Given a family of algebraic varieties $X \rightarrow S$, a Gauss-Manin connection is a natural connection over $S$ on the fibrewise de Rham cohomology sheaf $\mathcal{H}^\bullet_{X/S}$. An exponential Gauss-Manin connection arises similarly when the fibrewise cohomology is twisted by a function $f : X \rightarrow \mathbb{P}^1$. The configuration of singularities of $f$ determines the correct homology theory $\mathcal{H}^f_\bullet$ that pairs with the twisted cohomology $\mathcal{H}^\bullet_f$; this pairing is the integral representation of solutions to the given exponential Gauss-Manin system. We will say that to give a “motivic description” of a given differential system is to describe it as an exponential Gauss-Manin connection as above.

In this talk, I will explain the setup for exponential Gauss-Manin connections. Instead of giving a detailed construction, I will focus on a few concrete examples. Specifically, I will give the motivic description of the Airy differential equation, as well as the example of the simple harmonic oscillator.