Irreducibility of the Bloch variety for periodic Schrödinger operators in arbitrary dimension

Let $d\ge 1$ and $V\in C({\mathbb{T}}^d;\mathbb{C})$. We prove that the Bloch variety of $-\mathrm{\Delta }+V$ is irreducible modulo periodicity. The proof combines holomorphic continuation of spectral band functions with integer translations of quasimomentum. Using analytic geometry and perturbation estimates along imaginary rays, we show that, after suitable integer translations, all irreducible components contain the same holomorphic spectral branch.