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Autonomous driving needs various line-of-sight sensors to perceive surroundings that could be impaired under diverse environment uncertainties such as visual occlusion and extreme weather. To improve driving safety, we explore to wirelessly share perception information among connected vehicles within automotive edge computing networks. Sharing massive perception data in real time, however, is challenging under dynamic networking conditions and varying computation workloads. In this paper, we propose LiveMap, a real-time dynamic map, that detects, matches, and tracks objects on the road with crowdsourcing data from connected vehicles in sub-second. We develop the data plane of LiveMap that efficiently processes individual vehicle data with object detection, projection, feature extraction, object matching, and effectively integrates objects from multiple vehicles with object combination. We design the control plane of LiveMap that allows adaptive offloading of vehicle computations, and develop an intelligent vehicle scheduling and offloading algorithm to reduce the offloading latency of vehicles based on deep reinforcement learning (DRL) techniques. We implement LiveMap on a small-scale testbed and develop a large-scale network simulator. We evaluate the performance of LiveMap with both experiments and simulations, and the results show LiveMap reduces 34.1% average latency than the baseline solution.