Being a good specialist in one area our having an overview of many different subjects which is good for a project manager?

Not sure if you are asking about cross industry, or within a specific industry ....

You can certainly be a specialist in a single area, but should have knowledge in multiple areas ...

Think of a General Contractor (GC) - they may be a specialist when it comes to framing or electrical, but it is necessary for them to be competent in all areas pertaining to a GC.

First, congratulations on your choice to pursue Project Management. Now, you are asking a very loaded question, and the answer has a lot to do with the organization you are working in, the projects you are managing, and your own experience level.

At the beginning, most project managers are very specialized in a particular area and are "promoted" to PM because of that expertise. It is often assumed that a person with specific technical knowledge can run a team whose focus is in that realm. This approach, however, doesn't take into account the variety of PM skills missing from a technical expertise (perhaps a completely different conversation). For most people entering PM world, it is because they are a specialist.

As a career progresses, the PM may find himself (herself) engaging in larger and more complex projects. Eventually, the PM will not be able to be a specialist in every aspect of the project and instead have to rely on their PM skills rather than the technical skills. As a PM whose been called in to take over FEED and DED projects for entire drilling platforms there was no way I could be an expert on the mechanical, electrical, piping, control, etc. systems of the rig. I had to rely on the expertise within the project team. I can also tell you that many times the previous PM was too wrapped up in the technical skills with which they were comfortable rather than focusing on the PM work.

Since you are starting out, I would recommend finding an industry niche that you can learn a lot about and prove your knowledge in. It is more likely that people will want a new PM with comprehension of the topic. Once you get started, work to grow your PM skills instead of relying solely on the technical, looking to get to the larger, more complex projects.

Best of luck!