"The book is serious."
Translation:Le livre est sérieux.
You wouldn't use "grave" with that structure, i.e. "le livre est grave".
You could say "C'est un livre grave", to express either "That is a major book, of great importance", or "that is a dangerous book" because it was allowed to be published whereas its content is racist, may cause damage, be gravely polemical, etc.
Saying "Le livre est grave" is too direct, and in French we'd use that gallicism "C'est..." to precisely refer to the context, or the consequences of that book.
While, with "le livre est sérieux", you are just qualifying the type of book it is: not a funny book, not a comedy, or also that it was not written "lightly".
Imagine dozens of books were written about, say, Facebook and you want to express that this very book, as opposed to others, is serious; the subject is researched, treated meticulously, full of information - yet that book could be funny in the same time, or be about a "light" subject.
If I an not mistaken, it is all according to the context. If the book is "serious" in terms of "not comedic", or serious as in "tackling some heavy issues", then I don't think that "sombre", which means "serious" in terms of "dark" (i.e. Gothic, tragic) is the best adjective to use. You will find, if you haven't already, that just as in English, there are various words that can all correlate to an umbrella idea, but have their own details (ex. "happy" can be further explained as "giddy", "euphoric", or "jubilant"). It's all about the context of the sentence, if Duolingo considers such a thing.
P.S. I'm sorry for being long-winded. I'm learning French, too, and wanted to be sure that I was clear.