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In band insulators, where the Fermi surface is absent, adiabatic transport isallowed only due to the geometry of the Hilbert space. By driving the system ata small but finite frequency $\omega$, transport is still expected to dependsensitively on the quantum geometry. Here we show that this expectation iscorrect and can be made precise by expressing the Kubo formula for conductivityas the variation of the \emph{time-dependent polarization} with respect to theapplied field. In particular, a little appreciated effect is that at linearorder in frequency, the longitudinal conductivity results from an intrinsiccapacitance, determined by the ratio of the quantum metric and the spectralgap. We demonstrate that this intrinsic capacitance has a measurable effect ina wide range of insulators with non-negligible metric, including the electrongas in a quantizing magnetic field, the gapped bands of hBN-aligned twistedbilayer graphene, and obstructed atomic insulators such as diamond whose largerefractive index has a topological origin. We also discuss the influence ofquantum geometry on the dielectric constant.

The quantum geometric origin of capacitance in insulators