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arXiv · 2310.10152

Entropy for Monge-Amp\`ere Measures in the Prescribed Singularities Setting

Eleonora Di Nezza·Stefano Trapani·Antonio Trusiani
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Abstract

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Amp\`ere mass $\mathcal{E}(X, \theta, \phi)$, for a model potential $\phi$. 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 $\mathcal{E}^{\frac{n}{n-1}}(X, \theta, \phi)$ (provided $n>1$), while they have the same singularities of $\phi$ when $n=1$.

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Eleonora Di Nezza, Stefano Trapani, Antonio Trusiani. 2023-10-16. Entropy for Monge-Amp\`ere Measures in the Prescribed Singularities Setting. https://doi.org/10.3842/sigma.2024.039

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