No, according to the reference material, the Death Note is not a fake.
Initially, Light Yagami, the main character, thought the Death Note was a hoax. However, he discovered that its claims were real after testing its powers. This discovery proved that the notebook could indeed kill anyone whose name and face were known to its user.
Here's a breakdown:
- Initial Skepticism: Light Yagami initially dismisses the Death Note as a fake.
- Testing the Claims: He tests the notebook and finds that it works as claimed.
- Confirmed Reality: The notebook's powers are indeed real; it can kill people whose names are written within it while the writer has their face in mind.
Therefore, while the concept of a notebook that kills may be fictional, within the context of the story, the Death Note is a very real and potent object.