Principal component analysis (PCA) is warmly welcomed in dimensionality reduction and its applications. Due to the high sensitivity of PCA to outliers, a series of PCA methods are proposed to enhance the robustness of PCA. Besides, the representation ability of the existing PCA methods has limitations as well. To enhance the robustness and representation ability of robust PCA, we elaborate a novel Graph Convolution Robust PCA method (GRPCA) to incorporate the manifold structure into PCA. It constructs a sparse graph based on the local connectivity structure of samples. Graph auto-encoder is utilized to solve the robust PCA problem under the low-rank and sparse constraints. With the dual-decoder, GRPCA learns the low-dimensional embeddings that reconstruct the manifold structure and low-rank approximation simultaneously. Furthermore, since the graph suffers from misconnection triggered by occlusions, the local connectivity structure of low-dimensional embeddings is utilized to modify the graph. Our proposed method excels in both the clustering of low-dimensional embeddings and the low-rank recovery. Lastly, extensive experiments conducted on six real-world datasets demonstrated the efficiency and superiority of the proposed GRPCA.