Do Sums of Squares Dream of Free Resolutions?

Abstract

We associate to a real projective variety $X$ two convex cones which are fundamental in real algebraic geometry: the cone $P_X$ of quadratic forms nonnegative on $X$, and the cone $Σ_X$ of sums of squares of linear forms. The dual cone $Σ_X^\ast$ is a spectrahedron and we show that its convexity properties are closely related to homological properties of $X$. For instance, we show that all extreme rays of $Σ_X^\ast$ have rank one if and only if X has Castelnuovo-Mumford regularity two. More generally, if $Σ_X^\ast$ has an extreme ray of rank $p > 1$, then $X$ does not satisfy the property $N_{2,p}$. We show that the converse also holds in a wide variety of situations: the smallest $p$ for which property $N_{2,p}$ does not hold is equal to the smallest rank of an extreme ray of $Σ_X^\ast$ greater than one. These results allow us to generalize the work of Blekherman-Smith-Velasco on equality of nonnegative polynomials and sums of squares from irreducible varieties to reduced schemes and to classify all spectrahedral cones with only rank one extreme rays. Our results have applications to the positive semidefinite matrix completion problem and to the truncated moment problem on projective varieties.