Singular K\"ahler-Ricci Shrinkers are Complex Analytic

We prove that any singular K\"ahler--Ricci shrinker $X$ arising as a noncollapsed limit of K\"ahler--Ricci flows admits a natural structure of a locally algebraic complex-analytic variety with log terminal singularities. We then derive several geometric consequences: $X$ is simply connected, has a unique end, has unique tangent cones at every point, and is a smooth orbifold outside a subset of complex codimension at least three. As a further application, we prove a new long-time pseudolocality theorem for almost-selfsimilar K\"ahler--Ricci flows.