We introduce a new technological framework for high-contrast coronagraphy, namely digital adaptive coronagraphy (DAC) using spatial light modulators (SLMs), taking advantage of recent advances in this technology. We present proof-of-principle experimental results in the visible, using a transmissive twisted nematic liquid crystal SLM display to show that SLMs can be successfully implemented as focal-plane phase-mask coronagraphs (4QPM, 8OPM,...), and that the technology is essentially in place to address realistic instrumental configurations. We explore a specific application where SLM-based adaptive coronagraphy might be particularly competitive, which is direct imaging of multiple stars systems, by simultaneously nulling multiple point sources in the field. Using a simple approach to compute a brightness-weighted synthetized coronagraphic phase map, we show that in the case of binaries the contrast gain over using a regular phase map can exceed 4 stellar magnitudes for a 1:1 binaries down to separation as close as 1 λ/D. Thanks to video-rate update frequency of the SLM, the technique is in principle compatible with sky rotation in the case of altitude-azimuth telescope mounts, and can address multiple target configurations with no actual mechanical or hardware change.