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Due to the instability and limitations of unimodal biometric systems, multimodal systems have attracted more and more attention from researchers. However, how to exploit the independent and complementary information between different modalities remains a key and challenging problem. In this paper, we propose a multimodal biometric fusion recognition algorithm based on fingerprints and finger veins (Fingerprint Finger Veins-Channel Spatial Attention Fusion Module, FPV-CSAFM). Specifically, for each pair of fingerprint and finger vein images, we first propose a simple and effective Convolutional Neural Network (CNN) to extract features. Then, we build a multimodal feature fusion module (Channel Spatial Attention Fusion Module, CSAFM) to fully fuse the complementary information between fingerprints and finger veins. Different from existing fusion strategies, our fusion method can dynamically adjust the fusion weights according to the importance of different modalities in channel and spatial dimensions, so as to better combine the information between different modalities and improve the overall recognition performance. To evaluate the performance of our method, we conduct a series of experiments on multiple public datasets. Experimental results show that the proposed FPV-CSAFM achieves excellent recognition performance on three multimodal datasets based on fingerprints and finger veins.