Restricted volumes on Kähler manifolds

Collins, Tristan C.; Tosatti, Valentino

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 3, pp. 907-947.

Nous étudions les volumes restreints numériques de classes $(1,1)$ sur des variétés kähleriennes compactes, introduits par Boucksom. Inspirés par les travaux de Ein–Lazarsfeld–Mustaţă–Nakamaye–Popa sur les volumes restreints de fibrés en droites sur des variétés projectives, nous proposons la conjecture naturelle que les composantes irréductibles du lieu non-kählerien d’une classe grosse ont un volume restreint numérique identiquement nul. Nous établissons cette conjecture sous l’hypothèse que la classe admet une décomposition de Zariski, puis donnons plusieurs applications.

We study numerical restricted volumes of $(1,1)$ classes on compact Kähler manifolds, as introduced by Boucksom. Inspired by work of Ein–Lazarsfeld–Mustaţă–Nakamaye–Popa on restricted volumes of line bundles on projective manifolds, we pose a natural conjecture to the effect that irreducible components of the non-Kähler locus of a big class should have vanishing numerical restricted volume. We prove this conjecture when the class has a Zariski decomposition, and give several applications.

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