ag.algebraic geometry - For which hypersurfaces in projective space does the complement admit an algebraic group structure?

Not an answer, but here's a Hodge-theoretic restriction on subvarieties $Z subset mathbf{P}^n$ such that $G = mathbf{P}^n - Z$ admits the structure of a linear algebraic group.

Under the above conditions, the natural mixed Hodge structure (MHS) on $RGamma(G,mathbf{Z})$ is of mixed Tate type (by the Bruhat decomposition, say); this condition on a Hodge structure means, roughly, that only (n,n) classes show up. As $G$ is smooth as a variety, it follows by duality that the same is true for compactly supported cohomology $RGamma_c(G,mathbf{Z})$. On the other hand, there is an exact triangle of MHSs

$RGamma_c(G,mathbf{Z}) to RGamma(mathbf{P}^n,mathbf{Z}) to RGamma(Z,mathbf{Z})$

This means that the MHS on $RGamma(Z,mathbf{Z})$ also has to be of mixed Tate type. This restriction rules out any Z with "interesting" cohomology.

Now if one further assumes that Z is smooth hypersurface, then the mixed Tate condition forces $h^{p,q}(Z) = 0$ unless p=q. Standard calculatons with Hodge numbers of hypersurfaces (see, eg, page 126 of "A Survey of the Hodge Conjecture" by Lewis) then show that Z has degree 1 or 2, at least when the ambient projective space is at least 6 dimensional.

(Edited to include degree restrictions in last paragraph.)