Multiplier ideal sheaves and the Kähler-Ricci flow

Abstract

Multiplier ideal sheaves are constructed as obstructions to the convergence of the Kähler-Ricci flow on Fano manifolds, following earlier constructions of Kohn, Siu, and Nadel, and using the recent estimates of Kolodziej and Perelman.

On a Calabi-type estimate for pluriclosed flow

“…All of these results exploit some underlying structure of the given background to obtain a priori L ∞ estimates for the metric tensor along the flow. In the setting of Kähler-Ricci flow, reduced to a parabolic Monge-Ampere equation, this corresponds to having a C 1,1 estimate for the potential, at which point one applies either the Evans-Krylov method [5,7] to obtain a C 2,α estimate, or Calabi's C 3 estimate [2,18,13], after which Schauder estimates can be applied to obtain C ∞ estimates. As pluriclosed metrics cannot be described locally by a single function, the pluriclosed flow does not admit a scalar reduction, so the method of Evans-Krylov cannot be applied.…”

The Kähler–Ricci flow on Hirzebruch surfaces

“…By a conjecture of Yau [40], a necessary and su‰cient condition for M to admit a Kähler-Einstein metric is that M be 'stable in the sense of geometric invariant theory'. Indeed, the problem of using stabil-ity conditions to prove convergence properties of the Kähler-Ricci flow is an area of considerable current interest and we refer the reader to [21], [19], [22], [23], [25], [31], [24], [36] and [5] for some recent advances (however, this list of references is far from complete). One might expect that the su‰ciency part of the Yau-Tian-Donaldson conjecture can be proved via the flow (1.1).…”

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.…”