Diameter estimates for long-time solutions of the Kähler–Ricci flow

Diameter estimates in Kähler geometry

“…Furthermore, for any 𝑝 > 𝑛, there exist 𝐴, 𝐵, 𝐾 > 0 such that for all 𝑡 ≥ 0, 𝜔(𝑡) ∈ (𝑋, 𝜔 0 , 𝑛, 𝐴, 𝑝, 𝐾; 𝐵 [29], the corollary can be proved by applying the same argument of Ref. [20].…”

“…We let 𝑌 • be the set of regular points of Φ, then 𝑌 • is an open dense Zariski subset of 𝑌. We also let 𝑋 • = Φ −1 (𝑌 [20] for the long-time solutions of the Kähler-Ricci flow (see also [35,39]).…”

The entropy of Ricci flows with Type-I scalar curvature bounds

“…It was shown that such limits are infinitesimally self-similar, and have singularities of codimension four. This structure theory was combined with geometric estimates for projective Kähler-Ricci flows in [JST23] to prove a partial C 0 estimate near certain points called Ricci vertices. Using this, it was shown in [JST23,HJST23] that Gromov-Hausdorff limits of Kähler-Ricci flow based at Ricci vertices are infinitesimally conical, continuous in time, and that their time slices are normal analytic varieties.…”

“…This structure theory was combined with geometric estimates for projective Kähler-Ricci flows in [JST23] to prove a partial C 0 estimate near certain points called Ricci vertices. Using this, it was shown in [JST23,HJST23] that Gromov-Hausdorff limits of Kähler-Ricci flow based at Ricci vertices are infinitesimally conical, continuous in time, and that their time slices are normal analytic varieties. Away from Ricci vertices (whose locations are so far difficult to determine in general), or without the assumption of projectivity, there is little known for general limits of Kähler-Ricci flow that is not present in the Riemannian setting.…”

“…In [HJ22], the Kähler structure was used to improve the description of the singular sets and tangent flows of singularity models, analogous to results for Ricci limit spaces proved in [CCT02], [Liu18]. However, the conjectural pictureat least in the projective case (see [JST23]) -is much stronger: any Kähler-Ricci tangent flow should be a normal analytic variety.…”