According to three grammar sources,*
the subject complement in this kind of sentence may be either the full
infinitive to study or the bare infinitive study. None
of the sources gives preference to one choice over the other.
It is interesting that when I studied English in secondary school many
decades ago, the infinitive with to in this kind of sentence
(pseudo-cleft) was judged absolutely wrong. In the intervening years,
usage seems to have made the full infinitive with to completely
acceptable, as well as the bare infinitive. The explanation may be that
speakers tend to use the full infinitive to study so as to be
parallel to the infinitive in the main clause:
What I want to do is to study English.
Two further points:
1) If the verb in the main clause is in the progressive, the verb
form in the complement is the progressive participle:
What he’s doing
is making a
fool of himself
2) By way of contrast, with the grammatical subject
ALL (plus an adjective clause), the bare infinitive is much more
common:
All you ever do is complain about the neighbors
Marilyn Martin
*Quirk et al., A Comprehensive Grammar of the English Language (Longman,
1985).
Biber et al., Longman
Grammar of Spoken and Written English (1999).
Collins COBUILD English
Grammar (1990).