  We adapt tools from the algebraic approach to constraint satisfaction
problems to answer descriptive set theoretic questions about Borel CSPs. We
show that if a structure $\mathcal D$ does not have a Taylor polymorphism, then
the corresponding Borel CSP is $\mathbf{\Sigma}^1_2$-complete. In particular,
by the CSP Dichotomy Theorem, if $\operatorname{CSP}(\mathcal D)$ is
$\mathrm{NP}$-complete, then the Borel version, $\operatorname{csp}_B(\mathcal
D)$, is $\mathbf{\Sigma}^1_2$-complete (assuming $\mathrm{P}\not=\mathrm{NP}$).
We also have partial converses, such as a descriptive analogue of the
Hell--Ne\v set\v ril theorem characterizing $\mathbf{\Sigma}^1_2$-complete
graph homomorphism problems. We show that the structures where every solvable
Borel instance of their CSP has a Borel solution are exactly the width 1
structures. And, we prove a handful of results bounding the projective
complexity of certain bounded width structures.
