Conjecture on irreducibility of Fermi varieties for periodic Schrödinger operators

Let $d\geq 2$, let $V$ be a periodic potential, and let $F_{\lambda}(V)$ denote the Fermi variety at energy $\lambda$. An analytic set is irreducible if it cannot be represented as the union of two nonempty proper analytic subsets. Fermi-variety irreducibility conjecture. Then $F_{\lambda}(V)/\mathbb{Z}^d$ is irreducible, possibly except for finitely many $\lambda\in\mathbb{C}$. This is the Fermi-variety counterpart of the expected irreducibility of Bloch varieties and addresses the global analytic geometry underlying periodic spectral theory. The supplied text gives no resolution status.