prophet's task does not start. ends the previous task, but the task does not start.game playing as a vampire I could not pass it a kind of duty. Waiting for your help. thank you

Did you leave the castle before Harkon finished his speech? Or does it not start after you finish The Bloodstone Chalice? My recommendation would be to load up a save from before turning in Bloodstone Chalice, go in and speak to Garan and then go and speak to Harkon. You may also try and get the Bloodstone Chalice buff (click on the Chalice in the little room off the dining hall) before speaking to Harkon. You need to make sure he goes through his entire speech and then speak to Serana and let her give you the 'ask at the College of Winterhold' option to the quest.

When asking for quest help it's a good idea to say whether you're on PC or not, in case only console commands will fix your problem.