The quantum D-module of product varieties

Abstract

We study the quantum connection of product varieties in the framework of quantum cohomology. Our first main result shows that, near the origin of the Novikov variables, the quantum spectrum of \(X \times Y\) converges to the set of pairwise sums of the spectra of \(X\) and \(Y\). This arises from the leading contribution of the connection matrices \(K_X \otimes \mathrm{id}\) and \(\mathrm{id} \otimes K_Y\), while mixed curve classes contribute only at higher order. Our second main result establishes a formal isomorphism of quantum \(D\)-modules $ \mathrm{QDM}(X \times Y)^{\mathrm{la}} \cong \mathrm{QDM}(X)^{\mathrm{la}} \otimes \mathrm{QDM}(Y)^{\mathrm{la}}$, compatible with the quantum connection. As applications, we show that atoms, birational invariants arising from quantum cohomology, factor multiplicatively for product varieties, and we deduce the existence of a motivic measure associated with atoms.