Autonomous driving perceives surroundings with line-of-sight sensors that are compromised under environmental uncertainties. To achieve real time global information in high definition map, we investigate to share perception information among connected and automated vehicles. However, it is challenging to achieve real time perception sharing under varying network dynamics in automotive edge computing. In this paper, we propose a novel real time dynamic map, named LiveMap to detect, match, and track objects on the road. We design the data plane of LiveMap to efficiently process individual vehicle data with multiple sequential computation components, including detection, projection, extraction, matching and combination. We design the control plane of LiveMap to achieve adaptive vehicular offloading with two new algorithms (central and distributed) to balance the latency and coverage performance based on deep reinforcement learning techniques. We conduct extensive evaluation through both realistic experiments on a small-scale physical testbed and network simulations on an edge network simulator. The results suggest that LiveMap significantly outperforms existing solutions in terms of latency, coverage, and accuracy.