Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 039, 19 pages      arXiv:2310.10152
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting

Eleonora Di Nezza a, Stefano Trapani b and Antonio Trusiani c
a) IMJ-PRG, Sorbonne Université & DMA, École Normale Supérieure, Université PSL, CNRS, 4 place Jussieu & 45 Rue d'Ulm, 75005 Paris, France
b) Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy
c) Chalmers University of Technology, Chalmers tvärgata 3, 41296 Göteborg, Sweden

Received October 16, 2023, in final form May 04, 2024; Published online May 08, 2024

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Ampère 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 $n=1$.

Key words: Kähler manifolds; Monge-Ampère energy; entropy; big classes.

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