# Questions about P vs NP and geometric complexity theory

Reading through various papers on geometric complexity theory (GCT), there is one thing, which pops up, while claimed in various places, that it is an approach to P vs NP, all the results seems to accumulate in conditional statements for VP vs VNP or NC vs #P. Are there any implications known, which connect the obstruction-based approach to VP vs VNP with P vs NP?

One more thing I keep wondering about is whether the attempts at separation through obstructions are strict, i.e. is there a class of obstructions, which exists if and only if $$VP \neq VNP$$?

Note: I use P vs NP here in the precise sense of Cook's hypothesis, not in the sense of P v NP in fields of certain characteristic as it sometimes appears in GCT papers.

• There's a related question on implications between VP vs VNP and P vs NP, but it doesn't address your more specific questions about GCT. – Robert Andrews Mar 31 at 15:16
• yes, that's precisely why i am asking this question – Nift Mar 31 at 20:58

The short answer is no these are not known, though they are certainly not out of the question. There are no direct implications known to P vs NP, and we do not even have a conjecture (let alone theorem) that such-and-such a representation is an obstruction iff $$VP \neq VNP$$.
Even in characteristic zero, we don't know the "general converse" about obstructions, namely, we don't know that $$VP \neq VNP$$ implies existence of multiplicity obstructions. But we do know that if $$VNP \not\subseteq \overline{VP}$$ then a separating module (equivalently, HWV obstruction, see Bürgisser-Ikenmeyer STOC '13, arXiv) must exist.