Projections of extreme precipitation based on modern climate models suffer from large uncertainties. Specifically, unresolved physics and natural variability limit the ability of climate models to provide actionable information on impacts and risks at the regional, watershed and city scales relevant for practical applications. Here we show that the interaction of precipitating systems with local features can constrain the statistical description of extreme precipitation. These observational constraints can be used to project local extremes of low yearly exceedance probability (e.g., 100-year events) using synoptic-scale information from climate models, which is generally represented more accurately than the local-scales, and without requiring climate models to explicitly resolve extremes. The novel approach offers a path for improving the predictability of local statistics of extremes in a changing climate, independent of pending improvements in climate models at regional and local scales.