Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting

Abstract

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Ampère mass $\mathcal{E}(X,\theta ,\varphi )$, for a model potential $\varphi$. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class ${\mathcal{E}}^{\frac{n}{n-1}}(X,\theta ,\varphi )$ (provided $n>1$), while they have the same singularities of $\varphi$ when $n=1$.

References in corpus (5)

• $L^1$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds
• Kähler-Einstein metrics with prescribed singularities on Fano manifolds
• Continuity method with movable singularities for classical Monge-Ampère equations
• A complete metric topology on relative low energy spaces
• Relative pluripotential theory on compact Kähler manifolds