Are there PMs in Scrum? No (but...)

Scrum has only 3 roles: Product Owner, ScrumMaster and Implementer.
Who is responsible for 'the project'?  Well, that would be the whole Team.
What do we do with George, who was a Project Manager (PM)?  Well, we have to talk to George, but maybe he would make a great ScrumMaster or a great Product Owner.  Or something else.
The ones who liked removing impediments tend to make good SMs. The ones who liked managing the requirements often make good POs.
Next, several comments.
1. Some of the best agile people I know were Project Managers once.  (I was one once.)
2. It is very common for the phrase 'project manager' to imply to some people that that person is 'responsible' for project success.  This is not the case with agile. The whole Team is responsible. Together.  Each person in a slightly different way. But, they win together or they lose together.
Now, often the PM knows well that only the Team has success. And sometimes the Team also understands this well. But as soon as we call someone 'PM', then a manager outside the Team starts talking as if this is George's project to win or lose. And it starts to hurt things, hurt the chances for success. So, I suggest avoiding the term completely.
It is a nice idea that one person only could be responsible. It might seem to make another manager's work easier.  And maybe in some areas, it is true. But in our knowledge work, the whole Team must contribute. So, they win or lose as a Team.
3. Some PMs have years of experience trying to drive a 'team'. They have their checklists. They have their command-and-control ways. I think this way of working is wrong, both as a way to treat people, and as a way to success. In any case, it is not what we do in Scrum.  And, many PMs cannot break these habits. So, watch out for that.
4. Scrum is a simple framework. Scrum does not even try to give the Team, by itself, everything that the Team will need to succeed.  They Team is expected to add things to Scrum, appropriate things for their work and their situation.
In this regards, many PMs have dealt with and studied many of these 'possible things to add'.  So, often George is good at suggesting these things. And other things can also be suggested by other 'disciplines too.
5. Some companies still have PMOs after getting well started in Scrum. (PMO means Project Management Office, although I find each PMO group...usually several PM types grouped together...each PMO group can be run quite differently.)  This may not be the best pattern, but it is not clear that this is a terrible pattern. Or at least, it might be a necessary pattern for a temporary period.  In any case, some PMs can still work in a PMO kind of office, and this does not seem to be totally contrary to Scrum, or to clearly hurt Scrum teams. (But there are occasionally issues.)
6. Most of the duties of the old PM are divided in Scrum amongst the PO, the SM, and the Team.
7. Should we put George (a PM) on a Team that already has a PO and a SM?  And give George mainly his old PM duties?  (We could make a long list of those....)  No, it seems very unlikely this will work well. George will be in too much unnecessary conflict with the PO and the SM. And, while there may be a few other duties, they are not enough to keep George busy full-time.  Maybe, in a system that still also demands Teams to comply with waterfall reporting, maybe then there is enough work for George (the former PM) as a 'pig'. Maybe. But we ought to get away from that as soon as possible.
8. Could Scrum Teams in large organizations need help doing 'project management reporting' and such?  Yes. And often George has a great role to assist several teams in handling this work.  In this case, we might say George is a 'chicken' to several teams.  Still, we would hope that most of this role would go away with time.