Finite time singularities of the Kähler-Ricci flow

• 成果类型: Article; Early Access
• 署名作者: Jian, Wangjian; Song, Jian; Tian, Gang
• 署名单位: Chinese Academy of Sciences; Nanjing Institute of Geology & Paleontology, CAS; Academy of Mathematics & System Sciences, CAS; Chinese Academy of Sciences; Nanjing Institute of Geology & Paleontology, CAS; Academy of Mathematics & System Sciences, CAS; Rutgers University System; Rutgers University New Brunswick; Peking University; Peking University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910; 1432-1297
• DOI: 10.1007/s00222-026-01449-x
• 发表日期: 2026-09-15
• 关键词: KAHLER-RICCI FLOW; CONTRACTING EXCEPTIONAL DIVISORS; BOUNDING SCALAR CURVATURE; GROMOV-HAUSDORFF LIMITS; einstein metrics; sobolev inequalities; logarithmic sobolev; MANIFOLDS; CONVERGENCE; conjecture

摘要

We establish the scalar curvature and distance bounds, extending Perelman's work on the Fano K & auml;hler-Ricci flow to general finite time solutions of the K & auml;hler-Ricci flow. These bounds are achieved by our Li-Yau type and Harnack estimates for weighted Ricci potential functions of the K & auml;hler-Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov-Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is obtained for a fibre over every relatively compact open subset of the regular base of the fibration. We also apply our estimates to show that every solution of the K & auml;hler-Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.