In a Network Function Virtualization (NFV) system, network functions (NFs) are implemented on general-purpose hardware, including CPU, GPU, and FPGA. Studies have shown that there is no one-size-fits-all processor, as each processor demonstrates performance advantages to implement certain types of NFs. With more general-purpose processors such as GPUs being deployed in data center servers, the best practice to build a high-performance NFV service chain should employ available heterogeneous processors. However, current NFV systems fail to utilize these processors for acceleration. This is because, due to separate memory spaces, data synchronization is demanded to guarantee correctness, which can incur non-trivial overhead and result in low performance. This paper proposes AdaptChain, a data management facility that enables adaptive data sharing and synchronization for hybrid NFV systems on heterogeneous architectures. AdaptChain shares the host and device memory among NFs in a service chain. With adaptive synchronization plan generation and NF code adaptation, AdaptChain exploits three classes of opportunities to reduce the amount of synchronized data while guaranteeing correctness. Experimental results show that AdaptChain improves the overall throughput by up to 3.2× and reduces the latency by up to 52%.