Nash entropy, Calabi energy and geometric regularization of singular K\"ahler metrics

We prove uniform Sobolev bounds for solutions of the Laplace equation on a general family of K\"ahler manifolds with bounded Nash entropy and Calabi energy. These estimates establish a connection to the theory of RCD spaces and provide abundant examples of RCD spaces topologically and holomorphically equivalent to projective varieties. Suppose $X$ is a normal projective variety that admits a resolution of singularities with relative nef or relative effective anti-canonical bundle. Then every admissible singular K\"ahler metric on $X$ with Ricci curvature bounded below induces a non-collapsed RCD space homeomorphic to the projective variety $X$ itself.