With the recent progress in generative artificial intelligence (Generative AI), particularly in the development of large language models, recommendation systems are evolving to become more versatile. Unlike traditional techniques, generative AI not only learns patterns and representations from complex data but also enables content generation, data synthesis, and personalized experiences. This generative capability plays a crucial role in the field of recommendation systems, helping to address the issue of data sparsity and improving the overall performance of recommendation systems. Numerous studies on generative AI have already emerged in the field of recommendation systems. Meanwhile, the current requirements for recommendation systems have surpassed the single utility of accuracy, leading to a proliferation of multi-objective research that considers various goals in recommendation systems. However, to the best of our knowledge, there remains a lack of comprehensive studies on multi-objective recommendation systems based on generative AI technologies, leaving a significant gap in the literature. Therefore, we investigate the existing research on multi-objective recommendation systems involving generative AI to bridge this gap. We compile current research on multi-objective recommendation systems based on generative techniques, categorizing them by objectives. Additionally, we summarize relevant evaluation metrics and commonly used datasets, concluding with an analysis of the challenges and future directions in this domain.