"It is imperative that he writes a letter to his sister as soon as possible."
In this case, is the correct form write? If so, why?
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If it's necessary to use that construction, I would write
It is imperative that he write a letter ...
and then I would stray past the notice which reads "Here be dragons" and say that that use of write is not the infinitive mood of the verb but what may be conveniently called the subjunctive. Some decry calling it subjunctive because it's not exactly the same as the Latin subjunctive; but we're not discussing Latin.
Perhaps it's safest to say that write is uninflected.
Barrie England's answer to a previous question is rather good and explains the subjunctive.
Let’s take as an example the sentence We demand that he resign. It is used here as part of a mandative construction, that is, one that, in Huddleston and Pullum’s words ‘includes a component of meaning comparable to that expressed by the modal verb must’. Subjunctive resign is identical to the form of the verb used as the infinitive and as the present tense in all persons except the third person singular. However, in such sentences the indicative is also available and many native speakers, particularly native speakers of British English, will choose it and say, in all but the most formal contexts, We demand that he resigns. In truth, the subjunctive is rare in contemporary English, and may well disappear altogether over the next 50 to 100 years.
[I would probably say writes in that sentence, or resigns in Barrie's, because speech tends to be less careful. Or less pedantic.]