Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties

In [26], henceforth referred to as Part I, we suggested an approach to the P vs. NP and related lower bound problems in complexity theory through geometric invariant theory. In particular, it reduces the arithmetic (characteristic zero) version of the NP 6⊆ P conjecture to the problem of showing that a variety associated with the complexity class NP cannot be embedded in a variety associated with the complexity class P . We shall call these class varieties associated with the complexity classes P and NP . This paper develops this approach further, reducing these lower bound problems– which are all nonexistence problems–to some existence problems: specifically to proving existence of obstructions to such embeddings among class varieties. It gives two results towards explicit construction of such obstructions. The first result is a generalization of the Borel-Weil theorem to a class of orbit