Geometric deep learning has shown great potential for cortical surface analysis, but its performance often depends on a large-scale training set of cortical surfaces, which are traditionally derived from MRI scans through complex and time-consuming preprocessing pipelines. Although deep learning-based surface reconstruction methods have streamlined this process, they still rely on MRI data, limiting the availability of training data. To address this, we propose CortexGen, a geometric generative framework that synthesizes highly realistic cortical surfaces without requiring MRI scans. CortexGen employs geometric variational encoders to map cortical surfaces into a latent space, where latent flow matching models efficiently learn the true data distribution. This enables a two-stage cortical surface synthesis process: first, deforming an icosahedron-discretized sphere into a coarse cortical surface, and second, refining it into a high-resolution surface. Experiments show that CortexGen generates diverse, realistic cortical surfaces with 163,842 vertices in just 1.4 seconds per surface. Using these synthetic surfaces as augmented training data significantly improved learning-based cortical surface parcellation in few-shot settings. Our code and pretrained models are available at https://github.com/ladderlab-xjtu/CortexGen .