To the geometry of spaces of plurisubharmonic functions on a K\"ahler manifold

László Lempert

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arXiv · 2201.12445

To the geometry of spaces of plurisubharmonic functions on a K\"ahler manifold

László Lempert

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Abstract

Consider a compact K\"ahler manifold $(X,\omega)$ and the space $\cal E(X,\omega)=\cal E$ of $\omega$--plurisubharmonic functions of full Monge--Amp\`ere mass on it. We introduce a quantity $\rho[u,v]$ to measure the distance between $u, v\in\cal E$; $\rho[u,v]$ is not a number but rather a decreasing function on a certain interval $(0,V)\subset\mathbb R$. We explore properties of $\rho[u,v]$, and using them we study Lagrangians and associated energy spaces of $\omega$--plurisubharmonic functions. Many results here generalize Darvas's findings about his metrics $d_\chi$.

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BibTeX RIS

László Lempert. 2022-01-28. To the geometry of spaces of plurisubharmonic functions on a K\"ahler manifold. https://arxiv.org/abs/2201.12445

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