The current work explores the crystalline perovskite oxide, strontium hafnate, as a potential high-k gate dielectric for Ge-based transistors. SrHfO3 (SHO) is grown directly on Ge by atomic layer deposition and becomes crystalline with epitaxial registry after post-deposition vacuum annealing at ∼700 °C for 5 min. The 2 × 1 reconstructed, clean Ge (001) surface is a necessary template to achieve crystalline films upon annealing. The SHO films exhibit excellent crystallinity, as shown by x-ray diffraction and transmission electron microscopy. The SHO films have favorable electronic properties for consideration as a high-k gate dielectric on Ge, with satisfactory band offsets (>2 eV), low leakage current (<10^-5 A/cm² at an applied field of 1 MV/cm) at an equivalent oxide thickness of 1 nm, and a reasonable dielectric constant (k ∼ 18). The interface trap density (Dit) is estimated to be as low as ∼2 × 10¹² cm^-2 eV^-1 under the current growth and anneal conditions. Some interfacial reaction is observed between SHO and Ge at temperatures above ∼650 °C, which may contribute to increased Dit value. This study confirms the potential for crystalline oxides grown directly on Ge by atomic layer deposition for advanced electronic applications.