The Ricci iteration and its applications

Abstract

Abstract.In this Note we introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics as discretizations of certain geometric flows. We pose a conjecture on their convergence towards canonical Kähler metrics and study the case where the first Chern class is negative, zero or positive. This construction has several applications in Kähler geometry, among them an answer to a question of Nadel and a construction of multiplier ideal sheaves. Résumé.Dans cette Note nous introduisons etétudions des systèmes dynamiques reliéesà l'opérateur de Ricci sur l'espace des métriques kählériennes comme discrétisations des certains flots géométriques. Nous posons une conjecture concernant leurs convergence vers des métriques kählériennes canoniques and nousétudions le cas où la première classe de Chern est négative, zéro ou positive. Cette construction a plusieurs applications en géométrie kählérienne, parmi elles une réponseà une question de Nadel et une construction des faisceaux d'idéaux multiplicateurs.

Convergence of the Kähler–Ricci iteration

“…One of the original motivations for introducing the Ricci iteration, going back to [26,27], is its relation to the Ricci flow. Hamilton's Ricci flow on a Kähler manifold of definite or zero first Chern class is defined as {ω(t)} t∈R + satisfying the evolution equation…”

Ricci iteration on homogeneous spaces

“…In essence, the Ricci iteration aims to reduce the Einstein equation to a sequence of prescribed Ricci curvature equations. Introduced by the second-named author [36,37] as a discretization of the Ricci flow, the Ricci iteration has been since studied by a number of authors, see the survey [39, §6.5] and references therein. In all previous works, the underlying manifold (M, g 1 ) is assumed to be Kähler; essentially nothing is known about the Ricci iteration in the general Riemannian setting.…”

Multiplier ideal sheaves and integral invariants on toric Fano manifolds

“…Multiplier ideal sheaves can also be constructed from the Kähler-Ricci flow ( [14], [17], [8]) and its discretization ( [16], [18]). As mentioned in Remark 1.4, we shall discuss the multiplier ideal sheaves constructed from the Kähler-Ricci flow on the toric del Pezzo surfaces.…”