Wasserstein Metric Normalization of Projective Cycle Spaces

Let $X$ be an irreducible reduced projective subvariety of a Chow variety. On a generic smooth parameter locus, normal motions of resolved components define a quadratic Wasserstein metric, even when the cycles are singular, reducible, or carry multiplicities. Its metric completion is canonically $X^ν$: resolution fibers collapse, whereas distinct normalization branches over the same Chow cycle remain separated. The completed metric admits an exact ambient action formula. For degree-$d$ hypersurfaces, the inner $W_q$ geometry is compact and induces the projective topology for every $1\le q\le2$. For $d\ge3$, the uniform Hölder exponent $1/d$ is attained and sharp for each such $q$; when $d=1$, the comparison is Lipschitz. Elliptic quartics furnish an explicit model of the boundary branching.