Fano manifolds with weak almost K\"ahler-Ricci solitons

TL;DR

This paper proves that sequences of weak almost K"ahler-Ricci solitons on Fano manifolds converge to genuine K"ahler-Ricci solitons with controlled singularities, advancing understanding of geometric limits in complex differential geometry.

Contribution

It establishes convergence of weak almost K"ahler-Ricci solitons to true solitons with singularities of codimension at least 2, under suitable conditions, on Fano manifolds.

Findings

Sequences of weak almost K"ahler-Ricci solitons converge to K"ahler-Ricci solitons.

Convergence occurs in the Gromov-Hausdorff topology.

Limit solitons have singularities of complex codimension at least 2.

Abstract

In this paper, we prove that a sequence of weak almost K\"ahler-Ricci solitons under further suitable conditions converge to a K\"ahler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded below, there exists a sequence of weak almost K\"ahler-Ricci solitons which converge to a K\"ahler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology.