We explore coherent ergotropy extraction in quantum batteries from a resource-geometric point of view. For an initial state $\rho$, we quantify the coherent extraction process by the coherent ergotropy ${\mathcal{E}}_c\left(\rho \right)$ and the coherent extraction distance $D_c^{\mathrm{ext}}\left(\rho \right)$ between the active state ${\sigma }_{\rho }$ and the passive state $P_{\rho }$. This defines the geometric power capacity ${\mathrm{\Pi }}_c\left(\rho \right)={\mathcal{E}}_c\left(\rho \right)/D_c^{\mathrm{ext}}\left(\rho \right)$, which measures the coherent ergotropy released unit minimal unitary distance. We prove that, for any driving Hamiltonian satisfying $\parallel V_t\parallel \le \nu$, the actual coherent discharging power is bounded by $P_c^{\mathrm{ext}}(\rho ;V_t)\le \nu {\mathrm{\Pi }}_c\left(\rho \right)$, showing that ${\mathrm{\Pi }}_c\left(\rho \right)$ is a capacity under unit driving norm rather than the power of a particular protocol. General bounds on ${\mathrm{\Pi }}_c\left(\rho \right)$ are derived by combining relative entropy bounds on coherent ergotropy with geometric bounds on the coherent extraction distance. We also formulate coherence-measure-induced bounds and protocol-corrected capacities involving the effective speed of a given Hamiltonian. Qubit and qutrit examples demonstrate that ${\mathrm{\Pi }}_c\left(\rho \right)$ captures a resource-geometric feature of coherent discharging beyond coherent ergotropy or coherence measures alone.

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• Quantum information processing
• Quantum information theory
• Quantum thermodynamics