What obligations does the provider of an AI system have under Annex III of the AI Act?

You determine this yourself

• Applies where the intended purpose of the AI system falls within one of the eight areas listed in Annex III: biometrics, critical infrastructure, education and vocational training, employment and workers management, access to essential private and public services, law enforcement, migration and border control management, or administration of justice and democratic processes.
• Classification follows Article 6(2). For points 1, 6 and 7 the text adds the condition that the use is permitted under relevant Union or national law. We read that condition as meaning that a prohibition under Article 5 comes before the question whether the system is high-risk; that is our reading and not the literal text, which states the condition without naming the order.
• The duty rests on the provider, but the role can shift. Under Article 25(1) a distributor, importer, deployer or third party becomes a provider itself where it puts its name or trademark on the system, makes a substantial modification, or changes the intended purpose such that the system becomes high-risk under Article 6. From that moment it carries the mapping to a point of Annex III itself.
• Article 6(3) takes a system listed in Annex III back outside high-risk where it does not pose a significant risk of harm to health, safety or fundamental rights, including by not materially influencing the outcome of decision making, and one of four conditions is met: the system performs a narrow procedural task, it improves the result of a previously completed human activity, it detects decision making patterns or deviations from prior patterns without replacing or influencing the previously completed human assessment without proper human review, or it performs a preparatory task to an assessment relevant for the purposes of the use cases listed in Annex III. A system that performs profiling of natural persons is always high-risk.
• The derogation is not free. Article 6(4) requires a provider who considers that a system referred to in Annex III is not high-risk to document its assessment before the system is placed on the market or put into service, subjects that provider to the registration obligation in Article 49(2), and requires it to provide the documentation to national competent authorities on request.