Although 3D Gaussian Splatting (3DGS) enables high-fidelity real-time rendering, its prohibitive storage overhead severely hinders practical deployment. Recent anchor-based 3DGS compression schemes reduce gaussian redundancy through some advanced context models. However, they overlook explicit geometric dependencies, leading to structural degradation and suboptimal ratedistortion performance. In this paper, we propose a Local Geometry-aware Hierarchical Context Compression framework for 3DGS(LG-HCC) that incorporates inter-anchor geometric correlations into anchor pruning and entropy coding for compact representation. Specifically, we introduce an Neighborhood-Aware Anchor Pruning (NAAP) strategy, which evaluates anchor importance via weighted neighborhood feature aggregation and then merges low-contribution anchors into salient neighbors, yielding a compact yet geometry-consistent anchor set. Moreover, we further develop a hierarchical entropy coding scheme, in which coarse-to-fine priors are exploited through a lightweight Geometry-Guided Convolution(GG-Conv) operator to enable spatially adaptive context modeling and rate-distortion optimization. Extensive experiments show that LG-HCC effectively alleviates structural preservation issues,achieving superior geometric integrity and rendering fidelity while reducing storage by up to 30.85x compared to the Scaffold-GS baseline on the Mip-NeRF360 dataset