Connectivity of joins, cohomological quantifier elimination, and an algebraic Toda’s theorem

Abstract

In this article, we use cohomological techniques to obtain an algebraic version of Toda’s theorem in complexity theory valid over algebraically closed fields of arbitrary characteristic. This result follows from a general ‘connectivity’ result in cohomology. More precisely, given a closed subvariety $X \subset {\mathbb {P}}^{n}$ over an algebraically closed field k, and denoting by $\mathrm{J}^{[p]}(X) = \mathrm{J}(X,\mathrm{J}(X,\ldots ,\mathrm{J}(X,X)\cdots )$ the p-fold iterated join of X with itself, we prove that the restriction homomorphism on (singular or $\ell$-adic etale) cohomology $\mathrm{H}^{i}({\mathbb {P}}^{N}) \rightarrow \mathrm{H}^{i}(\mathrm{J}^{[p]}(X))$, with $N = (p+1)(n+1) - 1$, is an isomorphism for $0 \le i < p$, and injective for $i=p$. We also prove this result in the more general setting of relative joins for X over a base scheme S, where S is of finite type over k. We give several other applications of this connectivity result including a cohomological version of classical quantifier elimination in the first order theory of algebraically closed fields of arbitrary characteristic, and to obtain effective bounds on the Betti numbers of images of projective varieties under projection maps.