Equiconsistencies at subcompact cardinals

We present equiconsistency results at the level of subcompact cardinals. Assuming SBH (delta) , a special case of the Strategic Branches Hypothesis, we prove that if delta is a Woodin cardinal and both a-(delta) and a- (delta) fail, then delta is subcompact in a class inner model. If in addition a-(delta (+)) fails, we prove that delta is subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary we also see that assuming the existence of a Woodin cardinal delta so that SBH (delta) holds, the Proper Forcing Axiom implies the existence of a class inner model with a subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to delta is delta (+(n)) supercompact for all n < omega. We state some results at this level, and indicate how they are proved.