Which of the following is correct?
What he is looking for are books written by Jane Austin.
What he is looking for is books written by Jane Austin.
Is it are to agree with the object books or is to agree with the subject he?
I believe either one is actually correct, since the thing that determines the verb’s case is the noun that comes first in the predicate nominative expression (on the left side of the imaginary equals sign). In this sentence, that first noun is what, which is technically a pronoun, but stands in for the noun that comes later. But of course, at this point in the sentence, it has not yet been determined whether the predicate noun that what is referring to is singular or plural, so the verb is essentially given the benefit of the doubt and is allowed to take either case, regardless of what the predicate noun turns out to be. This flexibility really only arises out of the fact that what is naturally ambiguous in number. If the sentence had begun The things he is looking for…, the predicate would have had to have been are books. Similarly, if the sentence had begun The thing he is looking for…, the predicate would have been singular – is books.